\(\int \frac {(e x)^{-1+3 n}}{(a+b \sec (c+d x^n))^2} \, dx\) [83]

Optimal result

Integrand size = 24, antiderivative size = 1384 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 i b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}+\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {2 i b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )} \]

[Out]

1/3*(e*x)^(3*n)/a^2/e/n-2*I*b^2*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d*x^n))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)
/d^3/e/n/(x^(3*n))+2*b^2*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/(x^(
2*n))+2*b^2*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/(x^(2*n))+2*I*b^3
*(e*x)^(3*n)*polylog(3,-a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))+I*b^3*
(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)-2*I*b^2*(e*x)^(3*n)
*polylog(2,-a*exp(I*(c+d*x^n))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))+2*I*b*(e*x)^(3*n)*ln(1+a
*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)-2*b^3*(e*x)^(3*n)*polylog(2,-a*exp(I*
(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))+2*b^3*(e*x)^(3*n)*polylog(2,-a*exp(I*(
c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))-I*b^2*(e*x)^(3*n)/a^2/(a^2-b^2)/d/e/n/(
x^n)-4*I*b*(e*x)^(3*n)*polylog(3,-a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1
/2)+b^2*(e*x)^(3*n)*sin(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(b+a*cos(c+d*x^n))-2*I*b^3*(e*x)^(3*n)*polylog(3,-a*e
xp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))-I*b^3*(e*x)^(3*n)*ln(1+a*exp(I*(c
+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)+4*b*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d*x^n))/
(b-(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)-4*b*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d*x^n))/(b+
(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)+4*I*b*(e*x)^(3*n)*polylog(3,-a*exp(I*(c+d*x^n))/(b-(
-a^2+b^2)^(1/2)))/a^2/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)-2*I*b*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b+(-a^2+b^
2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 2.87 (sec) , antiderivative size = 1384, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {4293, 4289, 4276, 3405, 3402, 2296, 2221, 2611, 2320, 6724, 4618, 2317, 2438} \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=-\frac {2 i b^2 (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-i \sqrt {a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+i \sqrt {a^2-b^2}}\right ) x^{-3 n}}{a^2 \left (a^2-b^2\right ) d^3 e n}+\frac {4 i b (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}-\frac {2 i b^3 (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}-\frac {4 i b (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}+\frac {2 i b^3 (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b-i \sqrt {a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b+i \sqrt {a^2-b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {4 b (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}-\frac {2 b^3 (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac {4 b (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}+\frac {2 b^3 (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}-\frac {i b^2 (e x)^{3 n} x^{-n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 i b (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}-\frac {i b^3 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}-\frac {2 i b (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}+\frac {i b^3 (e x)^{3 n} \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}+\frac {b^2 (e x)^{3 n} \sin \left (d x^n+c\right ) x^{-n}}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (d x^n+c\right )\right )}+\frac {(e x)^{3 n}}{3 a^2 e n} \]

[In]

Int[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2,x]

[Out]

(e*x)^(3*n)/(3*a^2*e*n) - (I*b^2*(e*x)^(3*n))/(a^2*(a^2 - b^2)*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(I
*(c + d*x^n)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(
I*(c + d*x^n)))/(b + I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) - (I*b^3*(e*x)^(3*n)*Log[1 + (a*E^
(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) + ((2*I)*b*(e*x)^(3*n)*Log[1 + (a
*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) + (I*b^3*(e*x)^(3*n)*Log[1 + (a*
E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d*e*n*x^n) - ((2*I)*b*(e*x)^(3*n)*Log[1 +
(a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d*e*n*x^n) - ((2*I)*b^2*(e*x)^(3*n)*PolyL
og[2, -((a*E^(I*(c + d*x^n)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3*e*n*x^(3*n)) - ((2*I)*b^2*(e*x)^
(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3*e*n*x^(3*n)) - (2*b^3
*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2*e*n*x^(2
*n)) + (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2
*e*n*x^(2*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b
^2)^(3/2)*d^2*e*n*x^(2*n)) - (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a^
2*Sqrt[-a^2 + b^2]*d^2*e*n*x^(2*n)) - ((2*I)*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2
+ b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) + ((4*I)*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))
/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) + ((2*I)*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(
I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) - ((4*I)*b*(e*x)^(3*n)*Poly
Log[3, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) + (b^2*(e*x)^(
3*n)*Sin[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cos[c + d*x^n]))

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3402

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c
+ d*x)^m*E^(I*Pi*(k - 1/2))*(E^(I*(e + f*x))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(2
*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 4289

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 4293

Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[e^IntPart[m]*((e*x
)^FracPart[m]/x^FracPart[m]), Int[x^m*(a + b*Sec[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]

Rule 4618

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :>
Simp[I*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x)))), x
]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac {x^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx}{e} \\ & = \frac {\left (x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {x^2}{(a+b \sec (c+d x))^2} \, dx,x,x^n\right )}{e n} \\ & = \frac {\left (x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \left (\frac {x^2}{a^2}+\frac {b^2 x^2}{a^2 (b+a \cos (c+d x))^2}-\frac {2 b x^2}{a^2 (b+a \cos (c+d x))}\right ) \, dx,x,x^n\right )}{e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}-\frac {\left (2 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {x^2}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {x^2}{(b+a \cos (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a^2 e n}-\frac {\left (b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {x^2}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {x \sin (c+d x)}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}-\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} e n}+\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \sqrt {-a^2+b^2} e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i b-\sqrt {a^2-b^2}+i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{i b+\sqrt {a^2-b^2}+i a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) d e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}-\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}+\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x^2}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) \sqrt {-a^2+b^2} e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {i a e^{i (c+d x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \log \left (1+\frac {i a e^{i (c+d x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}+\frac {\left (2 i b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{i b-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}+\frac {\left (2 i b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i a x}{i b+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (4 b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n}+\frac {\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}+\frac {\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (4 i b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^2 e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}-\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )}+\frac {\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (2 i b^3 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{a^2 \left (a^2-b^2\right ) \sqrt {-a^2+b^2} d^3 e n} \\ & = \frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 i b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}+\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {2 i b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )} \\ \end{align*}

Mathematica [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx \]

[In]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2,x]

[Out]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2, x]

Maple [F]

\[\int \frac {\left (e x \right )^{3 n -1}}{{\left (a +b \sec \left (c +d \,x^{n}\right )\right )}^{2}}d x\]

[In]

int((e*x)^(3*n-1)/(a+b*sec(c+d*x^n))^2,x)

[Out]

int((e*x)^(3*n-1)/(a+b*sec(c+d*x^n))^2,x)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3831 vs. \(2 (1282) = 2564\).

Time = 0.68 (sec) , antiderivative size = 3831, normalized size of antiderivative = 2.77 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x)^(-1+3*n)/(a+b*sec(c+d*x^n))^2,x, algorithm="fricas")

[Out]

1/6*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d^3*e^(3*n - 1)*x^(3*n)*cos(d*x^n + c) + 2*(a^4*b - 2*a^2*b^3 + b^5)*d^3*e^(3
*n - 1)*x^(3*n) + 6*(a^3*b^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*sin(d*x^n + c) - 6*((2*a^3*b^2 - a*b^4)*d*e^(3*n
- 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + (-I*a^2*b^3 + I*b^5)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d*e^(3*n - 1)*x^n*s
qrt(-(a^2 - b^2)/a^2) + (-I*a^3*b^2 + I*a*b^4)*e^(3*n - 1))*cos(d*x^n + c))*dilog(-((a*sqrt(-(a^2 - b^2)/a^2)
+ b)*cos(d*x^n + c) - (I*a*sqrt(-(a^2 - b^2)/a^2) + I*b)*sin(d*x^n + c) + a)/a + 1) - 6*((2*a^3*b^2 - a*b^4)*d
*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + (I*a^2*b^3 - I*b^5)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d*e^(3*n - 1)
*x^n*sqrt(-(a^2 - b^2)/a^2) + (I*a^3*b^2 - I*a*b^4)*e^(3*n - 1))*cos(d*x^n + c))*dilog(-((a*sqrt(-(a^2 - b^2)/
a^2) + b)*cos(d*x^n + c) - (-I*a*sqrt(-(a^2 - b^2)/a^2) - I*b)*sin(d*x^n + c) + a)/a + 1) + 6*((2*a^3*b^2 - a*
b^4)*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - (I*a^2*b^3 - I*b^5)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d*e^(3*
n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - (I*a^3*b^2 - I*a*b^4)*e^(3*n - 1))*cos(d*x^n + c))*dilog(((a*sqrt(-(a^2 -
b^2)/a^2) - b)*cos(d*x^n + c) + (I*a*sqrt(-(a^2 - b^2)/a^2) - I*b)*sin(d*x^n + c) - a)/a + 1) + 6*((2*a^3*b^2
- a*b^4)*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - (-I*a^2*b^3 + I*b^5)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d*
e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - (-I*a^3*b^2 + I*a*b^4)*e^(3*n - 1))*cos(d*x^n + c))*dilog(((a*sqrt(-(
a^2 - b^2)/a^2) - b)*cos(d*x^n + c) + (-I*a*sqrt(-(a^2 - b^2)/a^2) + I*b)*sin(d*x^n + c) - a)/a + 1) - 3*((-I*
(2*a^4*b - a^2*b^3)*c^2*sqrt(-(a^2 - b^2)/a^2) + 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1)*cos(d*x^n + c) + (-I*(2*a^
3*b^2 - a*b^4)*c^2*sqrt(-(a^2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1))*log(2*a*cos(d*x^n + c) + 2*I*a*s
in(d*x^n + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 3*((I*(2*a^4*b - a^2*b^3)*c^2*sqrt(-(a^2 - b^2)/a^2) + 2*(
a^3*b^2 - a*b^4)*c)*e^(3*n - 1)*cos(d*x^n + c) + (I*(2*a^3*b^2 - a*b^4)*c^2*sqrt(-(a^2 - b^2)/a^2) + 2*(a^2*b^
3 - b^5)*c)*e^(3*n - 1))*log(2*a*cos(d*x^n + c) - 2*I*a*sin(d*x^n + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) + 2*b) - 3
*((-I*(2*a^4*b - a^2*b^3)*c^2*sqrt(-(a^2 - b^2)/a^2) + 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1)*cos(d*x^n + c) + (-I
*(2*a^3*b^2 - a*b^4)*c^2*sqrt(-(a^2 - b^2)/a^2) + 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1))*log(-2*a*cos(d*x^n + c) +
2*I*a*sin(d*x^n + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) - 2*b) - 3*((I*(2*a^4*b - a^2*b^3)*c^2*sqrt(-(a^2 - b^2)/a^2
) + 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1)*cos(d*x^n + c) + (I*(2*a^3*b^2 - a*b^4)*c^2*sqrt(-(a^2 - b^2)/a^2) + 2*
(a^2*b^3 - b^5)*c)*e^(3*n - 1))*log(-2*a*cos(d*x^n + c) - 2*I*a*sin(d*x^n + c) + 2*a*sqrt(-(a^2 - b^2)/a^2) -
2*b) - 3*(-I*(2*a^3*b^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*d*e^(3*n -
1)*x^n + (I*(2*a^3*b^2 - a*b^4)*c^2*sqrt(-(a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1) + (-I*(2*a^4*b
- a^2*b^3)*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n + (I*(2*a^4*
b - a^2*b^3)*c^2*sqrt(-(a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1))*cos(d*x^n + c))*log(((a*sqrt(-(a
^2 - b^2)/a^2) + b)*cos(d*x^n + c) - (I*a*sqrt(-(a^2 - b^2)/a^2) + I*b)*sin(d*x^n + c) + a)/a) - 3*(I*(2*a^3*b
^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*d*e^(3*n - 1)*x^n + (-I*(2*a^3*
b^2 - a*b^4)*c^2*sqrt(-(a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1) + (I*(2*a^4*b - a^2*b^3)*d^2*e^(3*n
- 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n + (-I*(2*a^4*b - a^2*b^3)*c^2*sqr
t(-(a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1))*cos(d*x^n + c))*log(((a*sqrt(-(a^2 - b^2)/a^2) + b)*
cos(d*x^n + c) - (-I*a*sqrt(-(a^2 - b^2)/a^2) - I*b)*sin(d*x^n + c) + a)/a) - 3*(-I*(2*a^3*b^2 - a*b^4)*d^2*e^
(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*d*e^(3*n - 1)*x^n + (I*(2*a^3*b^2 - a*b^4)*c^2*sq
rt(-(a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1) + (-I*(2*a^4*b - a^2*b^3)*d^2*e^(3*n - 1)*x^(2*n)*sqrt
(-(a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n + (I*(2*a^4*b - a^2*b^3)*c^2*sqrt(-(a^2 - b^2)/a^2)
- 2*(a^3*b^2 - a*b^4)*c)*e^(3*n - 1))*cos(d*x^n + c))*log(-((a*sqrt(-(a^2 - b^2)/a^2) - b)*cos(d*x^n + c) + (
I*a*sqrt(-(a^2 - b^2)/a^2) - I*b)*sin(d*x^n + c) - a)/a) - 3*(I*(2*a^3*b^2 - a*b^4)*d^2*e^(3*n - 1)*x^(2*n)*sq
rt(-(a^2 - b^2)/a^2) - 2*(a^2*b^3 - b^5)*d*e^(3*n - 1)*x^n + (-I*(2*a^3*b^2 - a*b^4)*c^2*sqrt(-(a^2 - b^2)/a^2
) - 2*(a^2*b^3 - b^5)*c)*e^(3*n - 1) + (I*(2*a^4*b - a^2*b^3)*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) -
2*(a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n + (-I*(2*a^4*b - a^2*b^3)*c^2*sqrt(-(a^2 - b^2)/a^2) - 2*(a^3*b^2 - a*b
^4)*c)*e^(3*n - 1))*cos(d*x^n + c))*log(-((a*sqrt(-(a^2 - b^2)/a^2) - b)*cos(d*x^n + c) + (-I*a*sqrt(-(a^2 - b
^2)/a^2) + I*b)*sin(d*x^n + c) - a)/a) - 6*((2*I*a^4*b - I*a^2*b^3)*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x
^n + c) + (2*I*a^3*b^2 - I*a*b^4)*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2))*polylog(3, -((a*sqrt(-(a^2 - b^2)/a^2) +
b)*cos(d*x^n + c) + (I*a*sqrt(-(a^2 - b^2)/a^2) + I*b)*sin(d*x^n + c))/a) - 6*((-2*I*a^4*b + I*a^2*b^3)*e^(3*
n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x^n + c) + (-2*I*a^3*b^2 + I*a*b^4)*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2))*po
lylog(3, -((a*sqrt(-(a^2 - b^2)/a^2) + b)*cos(d*x^n + c) + (-I*a*sqrt(-(a^2 - b^2)/a^2) - I*b)*sin(d*x^n + c))
/a) - 6*((2*I*a^4*b - I*a^2*b^3)*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x^n + c) + (2*I*a^3*b^2 - I*a*b^4)*e
^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2))*polylog(3, ((a*sqrt(-(a^2 - b^2)/a^2) - b)*cos(d*x^n + c) - (I*a*sqrt(-(a^2
- b^2)/a^2) - I*b)*sin(d*x^n + c))/a) - 6*((-2*I*a^4*b + I*a^2*b^3)*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*
x^n + c) + (-2*I*a^3*b^2 + I*a*b^4)*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2))*polylog(3, ((a*sqrt(-(a^2 - b^2)/a^2)
- b)*cos(d*x^n + c) - (-I*a*sqrt(-(a^2 - b^2)/a^2) + I*b)*sin(d*x^n + c))/a))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^3
*n*cos(d*x^n + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^3*n)

Sympy [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{3 n - 1}}{\left (a + b \sec {\left (c + d x^{n} \right )}\right )^{2}}\, dx \]

[In]

integrate((e*x)**(-1+3*n)/(a+b*sec(c+d*x**n))**2,x)

[Out]

Integral((e*x)**(3*n - 1)/(a + b*sec(c + d*x**n))**2, x)

Maxima [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((e*x)^(-1+3*n)/(a+b*sec(c+d*x^n))^2,x, algorithm="maxima")

[Out]

1/3*(6*a*b^3*e^(3*n)*x^(2*n)*sin(d*x^n + c) + (a^4 - a^2*b^2)*d*e^(3*n)*x^(3*n)*cos(2*d*x^n + 2*c)^2 + 4*(a^2*
b^2 - b^4)*d*e^(3*n)*x^(3*n)*cos(d*x^n + c)^2 + (a^4 - a^2*b^2)*d*e^(3*n)*x^(3*n)*sin(2*d*x^n + 2*c)^2 + 4*(a^
2*b^2 - b^4)*d*e^(3*n)*x^(3*n)*sin(d*x^n + c)^2 + 4*(a^3*b - a*b^3)*d*e^(3*n)*x^(3*n)*cos(d*x^n + c) + (a^4 -
a^2*b^2)*d*e^(3*n)*x^(3*n) - 2*(3*a*b^3*e^(3*n)*x^(2*n)*sin(d*x^n + c) - 2*(a^3*b - a*b^3)*d*e^(3*n)*x^(3*n)*c
os(d*x^n + c) - (a^4 - a^2*b^2)*d*e^(3*n)*x^(3*n))*cos(2*d*x^n + 2*c) + 3*((a^6 - a^4*b^2)*d*e*n*cos(2*d*x^n +
2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*e*n*cos(d*x^n + c)^2 + (a^6 - a^4*b^2)*d*e*n*sin(2*d*x^n + 2*c)^2 + 4*(a^5*b
- a^3*b^3)*d*e*n*sin(2*d*x^n + 2*c)*sin(d*x^n + c) + 4*(a^4*b^2 - a^2*b^4)*d*e*n*sin(d*x^n + c)^2 + 4*(a^5*b
- a^3*b^3)*d*e*n*cos(d*x^n + c) + (a^6 - a^4*b^2)*d*e*n + 2*(2*(a^5*b - a^3*b^3)*d*e*n*cos(d*x^n + c) + (a^6 -
a^4*b^2)*d*e*n)*cos(2*d*x^n + 2*c))*integrate(2*(2*a^2*b^4*e^(3*n)*x^(2*n)*cos(2*c)*sin(2*d*x^n) + 2*a^2*b^4*
e^(3*n)*x^(2*n)*cos(2*d*x^n)*sin(2*c) - 4*(a^3*b^3 - a*b^5)*e^(3*n)*x^(2*n)*cos(c)*sin(d*x^n) - 4*(a^3*b^3 - a
*b^5)*e^(3*n)*x^(2*n)*cos(d*x^n)*sin(c) + (2*a^3*b^3*e^(3*n)*x^(2*n)*sin(d*x^n + c) - (2*a^5*b - a^3*b^3)*d*e^
(3*n)*x^(3*n)*cos(d*x^n + c))*cos(2*d*x^n + 2*c) - ((2*a^5*b - 3*a^3*b^3 + a*b^5)*d*e^(3*n)*x^(3*n) - (2*a*b^5
*e^(3*n)*x^(2*n)*sin(2*c) + (2*a^3*b^3 - a*b^5)*d*e^(3*n)*x^(3*n)*cos(2*c))*cos(2*d*x^n) + 2*((2*a^4*b^2 - 3*a
^2*b^4 + b^6)*d*e^(3*n)*x^(3*n)*cos(c) + 2*(a^2*b^4 - b^6)*e^(3*n)*x^(2*n)*sin(c))*cos(d*x^n) - (2*a*b^5*e^(3*
n)*x^(2*n)*cos(2*c) - (2*a^3*b^3 - a*b^5)*d*e^(3*n)*x^(3*n)*sin(2*c))*sin(2*d*x^n) - 2*((2*a^4*b^2 - 3*a^2*b^4
+ b^6)*d*e^(3*n)*x^(3*n)*sin(c) - 2*(a^2*b^4 - b^6)*e^(3*n)*x^(2*n)*cos(c))*sin(d*x^n))*cos(d*x^n + c) - (2*a
^3*b^3*e^(3*n)*x^(2*n)*cos(d*x^n + c) + 2*a^4*b^2*e^(3*n)*x^(2*n) + (2*a^5*b - a^3*b^3)*d*e^(3*n)*x^(3*n)*sin(
d*x^n + c))*sin(2*d*x^n + 2*c) + (2*(a^3*b^3 - a*b^5)*e^(3*n)*x^(2*n) - (2*a*b^5*e^(3*n)*x^(2*n)*cos(2*c) - (2
*a^3*b^3 - a*b^5)*d*e^(3*n)*x^(3*n)*sin(2*c))*cos(2*d*x^n) - 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(3*n)*x^(3*n
)*sin(c) - 2*(a^2*b^4 - b^6)*e^(3*n)*x^(2*n)*cos(c))*cos(d*x^n) + (2*a*b^5*e^(3*n)*x^(2*n)*sin(2*c) + (2*a^3*b
^3 - a*b^5)*d*e^(3*n)*x^(3*n)*cos(2*c))*sin(2*d*x^n) - 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(3*n)*x^(3*n)*cos(
c) + 2*(a^2*b^4 - b^6)*e^(3*n)*x^(2*n)*sin(c))*sin(d*x^n))*sin(d*x^n + c))/(a^8*d*e*x*cos(2*d*x^n + 2*c)^2 + a
^8*d*e*x*sin(2*d*x^n + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*e*x*cos(2*d*x^n)^2 + 4*((a^6*b^2 -
2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*e*x*cos(d*x^n)^2 + 4*(a^7*b - 2*a
^5*b^3 + a^3*b^5)*d*e*x*cos(d*x^n)*cos(c) + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*d*e*x*sin(2*d*x^n)^2 + 4
*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*d*e*x*sin(d*x^n)^2 - 4*
(a^7*b - 2*a^5*b^3 + a^3*b^5)*d*e*x*sin(d*x^n)*sin(c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*d*e*x - 2*(2*((a^5*b^3 - a
^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*e*x*cos(d*x^n) + (a^6*b^2 - a^4*b^4)*d*e*x*co
s(2*c) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*e*x*sin(d*x^n))*cos(2
*d*x^n) - 2*(a^6*b^2*d*e*x*cos(2*d*x^n)*cos(2*c) - a^6*b^2*d*e*x*sin(2*d*x^n)*sin(2*c) - 2*(a^7*b - a^5*b^3)*d
*e*x*cos(d*x^n)*cos(c) + 2*(a^7*b - a^5*b^3)*d*e*x*sin(d*x^n)*sin(c) - (a^8 - a^6*b^2)*d*e*x)*cos(2*d*x^n + 2*
c) + 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*d*e*x*cos(d*x^n) - 2*((a
^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*d*e*x*sin(d*x^n) + (a^6*b^2 - a^4*b^4
)*d*e*x*sin(2*c))*sin(2*d*x^n) - 2*(a^6*b^2*d*e*x*cos(2*c)*sin(2*d*x^n) + a^6*b^2*d*e*x*cos(2*d*x^n)*sin(2*c)
- 2*(a^7*b - a^5*b^3)*d*e*x*cos(c)*sin(d*x^n) - 2*(a^7*b - a^5*b^3)*d*e*x*cos(d*x^n)*sin(c))*sin(2*d*x^n + 2*c
)), x) + 2*(3*a*b^3*e^(3*n)*x^(2*n)*cos(d*x^n + c) + 3*a^2*b^2*e^(3*n)*x^(2*n) + 2*(a^3*b - a*b^3)*d*e^(3*n)*x
^(3*n)*sin(d*x^n + c))*sin(2*d*x^n + 2*c))/((a^6 - a^4*b^2)*d*e*n*cos(2*d*x^n + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)
*d*e*n*cos(d*x^n + c)^2 + (a^6 - a^4*b^2)*d*e*n*sin(2*d*x^n + 2*c)^2 + 4*(a^5*b - a^3*b^3)*d*e*n*sin(2*d*x^n +
2*c)*sin(d*x^n + c) + 4*(a^4*b^2 - a^2*b^4)*d*e*n*sin(d*x^n + c)^2 + 4*(a^5*b - a^3*b^3)*d*e*n*cos(d*x^n + c)
+ (a^6 - a^4*b^2)*d*e*n + 2*(2*(a^5*b - a^3*b^3)*d*e*n*cos(d*x^n + c) + (a^6 - a^4*b^2)*d*e*n)*cos(2*d*x^n +
2*c))

Giac [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((e*x)^(-1+3*n)/(a+b*sec(c+d*x^n))^2,x, algorithm="giac")

[Out]

integrate((e*x)^(3*n - 1)/(b*sec(d*x^n + c) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{3\,n-1}}{{\left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )}^2} \,d x \]

[In]

int((e*x)^(3*n - 1)/(a + b/cos(c + d*x^n))^2,x)

[Out]

int((e*x)^(3*n - 1)/(a + b/cos(c + d*x^n))^2, x)