\(\int \frac {(e x)^{-1+3 n}}{a+b \sec (c+d x^n)} \, dx\) [80]

Optimal result

Integrand size = 24, antiderivative size = 485 \[ \int \frac {(e x)^{-1+3 n}}{a+b \sec \left (c+d x^n\right )} \, dx=\frac {(e x)^{3 n}}{3 a e n}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {2 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}+\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3 e n}-\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3 e n} \]

[Out]

1/3*(e*x)^(3*n)/a/e/n+I*b*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a/d/e/n/(x^n)/(-a^2+b^2)^(
1/2)-I*b*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a/d/e/n/(x^n)/(-a^2+b^2)^(1/2)+2*b*(e*x)^(3
*n)*polylog(2,-a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)-2*b*(e*x)^(3*n)*p
olylog(2,-a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)+2*I*b*(e*x)^(3*n)*poly
log(3,-a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)-2*I*b*(e*x)^(3*n)*polylog
(3,-a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4293, 4289, 4276, 3402, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {(e x)^{-1+3 n}}{a+b \sec \left (c+d x^n\right )} \, dx=\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}-\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^3 e n \sqrt {b^2-a^2}}+\frac {2 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d e n \sqrt {b^2-a^2}}-\frac {i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{3 n}}{3 a e n} \]

[In]

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

[Out]

(e*x)^(3*n)/(3*a*e*n) + (I*b*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 +
b^2]*d*e*n*x^n) - (I*b*(e*x)^(3*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]
*d*e*n*x^n) + (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2
]*d^2*e*n*x^(2*n)) - (2*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^
2 + b^2]*d^2*e*n*x^(2*n)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/
(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2
+ b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3*e*n*x^(3*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 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 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 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 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}}{a+b \sec \left (c+d x^n\right )} \, 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)} \, 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}-\frac {b x^2}{a (b+a \cos (c+d x))}\right ) \, dx,x,x^n\right )}{e n} \\ & = \frac {(e x)^{3 n}}{3 a e n}-\frac {\left (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 e n} \\ & = \frac {(e x)^{3 n}}{3 a e n}-\frac {\left (2 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 e n} \\ & = \frac {(e x)^{3 n}}{3 a e n}-\frac {\left (2 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 )}{\sqrt {-a^2+b^2} e n}+\frac {\left (2 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 )}{\sqrt {-a^2+b^2} e n} \\ & = \frac {(e x)^{3 n}}{3 a e n}+\frac {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 \sqrt {-a^2+b^2} d e n}-\frac {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 \sqrt {-a^2+b^2} d e n}-\frac {\left (2 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 \sqrt {-a^2+b^2} d e n}+\frac {\left (2 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 \sqrt {-a^2+b^2} d e n} \\ & = \frac {(e x)^{3 n}}{3 a e n}+\frac {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 \sqrt {-a^2+b^2} d e n}-\frac {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 \sqrt {-a^2+b^2} d e n}+\frac {2 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 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 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 \sqrt {-a^2+b^2} d^2 e n}-\frac {\left (2 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 \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 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 \sqrt {-a^2+b^2} d^2 e n} \\ & = \frac {(e x)^{3 n}}{3 a e n}+\frac {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 \sqrt {-a^2+b^2} d e n}-\frac {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 \sqrt {-a^2+b^2} d e n}+\frac {2 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 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 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 \sqrt {-a^2+b^2} d^2 e n}+\frac {\left (2 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 \sqrt {-a^2+b^2} d^3 e n}-\frac {\left (2 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 \sqrt {-a^2+b^2} d^3 e n} \\ & = \frac {(e x)^{3 n}}{3 a e n}+\frac {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 \sqrt {-a^2+b^2} d e n}-\frac {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 \sqrt {-a^2+b^2} d e n}+\frac {2 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 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 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 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 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 \sqrt {-a^2+b^2} d^3 e n}-\frac {2 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 \sqrt {-a^2+b^2} d^3 e n} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

int((e*x)^(3*n-1)/(a+b*sec(c+d*x^n)),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. 1711 vs. \(2 (445) = 890\).

Time = 0.51 (sec) , antiderivative size = 1711, normalized size of antiderivative = 3.53 \[ \int \frac {(e x)^{-1+3 n}}{a+b \sec \left (c+d x^n\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*(6*a*b*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2)*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*a*b*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2)*
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*a*b*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2)*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*a*b*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a
^2)*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*a*b*c^2*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2)*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*a*b*c^2*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2)*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*a*b*c^2*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2)*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*a*b*c^2*e^(3*n - 1)*sqrt(-(a
^2 - b^2)/a^2)*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) - 2*(a^2 - b
^2)*d^3*e^(3*n - 1)*x^(3*n) + 6*I*a*b*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*I*a*b*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*I*a*b*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*I*a*b*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) - 3*(I*a*b*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) - I*a*b*c^2*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^
2))*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*a*b*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) + I*a*b*c^2*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2)
)*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*a*b*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) - I*a*b*c^2*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2))*
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*a*b*d^2*e^(3*n - 1)*x^(2*n)*sqrt(-(a^2 - b^2)/a^2) + I*a*b*c^2*e^(3*n - 1)*sqrt(-(a^2 - b^2)/a^2))*l
og(-((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
))/((a^3 - a*b^2)*d^3*n)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/3*(6*a*b*e^(3*n + 1)*n*integrate((a*x^(3*n)*cos(2*d*x^n + 2*c)*cos(d*x^n + c) + 2*b*x^(3*n)*cos(d*x^n + c)^
2 + a*x^(3*n)*sin(2*d*x^n + 2*c)*sin(d*x^n + c) + 2*b*x^(3*n)*sin(d*x^n + c)^2 + a*x^(3*n)*cos(d*x^n + c))/(a^
3*e*x*cos(2*d*x^n + 2*c)^2 + 4*a*b^2*e*x*cos(d*x^n + c)^2 + a^3*e*x*sin(2*d*x^n + 2*c)^2 + 4*a^2*b*e*x*sin(2*d
*x^n + 2*c)*sin(d*x^n + c) + 4*a*b^2*e*x*sin(d*x^n + c)^2 + 4*a^2*b*e*x*cos(d*x^n + c) + a^3*e*x + 2*(2*a^2*b*
e*x*cos(d*x^n + c) + a^3*e*x)*cos(2*d*x^n + 2*c)), x) - e^(3*n)*x^(3*n))/(a*e*n)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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