\(\int \frac {(e x)^{-1+2 n}}{(a+b \sec (c+d x^n))^2} \, dx\) [82]

Optimal result

Integrand size = 24, antiderivative size = 757 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \cos \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 1.68 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {4293, 4289, 4276, 3405, 3402, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \sqrt {b^2-a^2}}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (a \cos \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2-b^2\right )}+\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d e n \sqrt {b^2-a^2}}-\frac {2 i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d e n \sqrt {b^2-a^2}}+\frac {b^2 x^{-n} (e x)^{2 n} \sin \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a \cos \left (c+d x^n\right )+b\right )}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 d^2 e n \left (b^2-a^2\right )^{3/2}}-\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a^2 d e n \left (b^2-a^2\right )^{3/2}}+\frac {(e x)^{2 n}}{2 a^2 e n} \]

[In]

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

[Out]

(e*x)^(2*n)/(2*a^2*e*n) - (I*b^3*(e*x)^(2*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)^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sq
rt[-a^2 + b^2]*d*e*n*x^n) + (I*b^3*(e*x)^(2*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)^(2*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) + (b^2*(e*x)^(2*n)*Log[b + a*Cos[c + d*x^n]])/(a^2*(a^2 - b^2)*d^2*e*n*x^(2*n)) -
(b^3*(e*x)^(2*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)) + (2*b*(e*x)^(2*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)) + (b^3*(e*x)^(2*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)) - (2*b*(e*x)^(2*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)) + (b^2*(e*x)^(2*n)*Sin[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cos[c
+ d*x^n]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac {x^{-1+2 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx}{e} \\ & = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(a+b \sec (c+d x))^2} \, dx,x,x^n\right )}{e n} \\ & = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \left (\frac {x}{a^2}+\frac {b^2 x}{a^2 (b+a \cos (c+d x))^2}-\frac {2 b x}{a^2 (b+a \cos (c+d x))}\right ) \, dx,x,x^n\right )}{e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{(b+a \cos (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {x}{b+a \cos (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n}-\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\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)^{2 n}}{2 a^2 e n}+\frac {b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{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 (b^2 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \cos \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}+\frac {2 i b x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \cos \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{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^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {e^{i (c+d x)} x}{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 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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 (2 i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 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)^{2 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)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \cos \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 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 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \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^2 e n}+\frac {\left (2 b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \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^2 e n}-\frac {\left (i b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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 (i b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \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)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 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)^{2 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)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \cos \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 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 x^{-2 n} (e x)^{2 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)^{2 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 (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \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^2 e n}+\frac {\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \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^2 e n} \\ & = \frac {(e x)^{2 n}}{2 a^2 e n}-\frac {i b^3 x^{-n} (e x)^{2 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)^{2 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)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \cos \left (c+d x^n\right )\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 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 {2 b x^{-2 n} (e x)^{2 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^3 x^{-2 n} (e x)^{2 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 {2 b x^{-2 n} (e x)^{2 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)^{2 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 [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2450\) vs. \(2(757)=1514\).

Time = 10.69 (sec) , antiderivative size = 2450, normalized size of antiderivative = 3.24 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(-2*b*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*(2*(c + d*x^n)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2]
)/Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(
b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sq
rt[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Cos[c + d*x^n]])] + (Ar
cCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)
/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]
])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2
- b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + (-Ar
cCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(
a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + I*(PolyLog[2,
((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d
*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sq
rt[a^2 - b^2]*Tan[(c + d*x^n)/2]))]))*Sec[c + d*x^n]^2)/((a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2) + (
b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*(2*(c + d*x^n)*ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/
Sqrt[a^2 - b^2]] - 2*(c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/
a)] - (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt
[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Cos[c + d*x^n]])] + (ArcC
os[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(c + d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(c + d*x^n)/2
])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]])
] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2 -
b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + (-ArcC
os[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a
+ b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] + I*(PolyLog[2, (
(b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x
^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt
[a^2 - b^2]*Tan[(c + d*x^n)/2]))]))*Sec[c + d*x^n]^2)/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2) +
(x^(1 - n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*Sec[c + d*x^n]^2*(a^2*d*x^n*Cos[c] - b^2*d*x^n*Cos[c] +
2*b^2*Sin[c]))/(2*a^2*(a - b)*(a + b)*d*n*(a + b*Sec[c + d*x^n])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])
) + (b^2*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^2*Sec[c]*Sec[c + d*x^n]^2*(a*Cos[c]*Log[b + a*Cos
[c]*Cos[d*x^n] - a*Sin[c]*Sin[d*x^n]] + a*d*x^n*Sin[c] - ((2*I)*a*b*ArcTan[((-I)*a*Sin[c] - I*(-b + a*Cos[c])*
Tan[(d*x^n)/2])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2]]*Sin[c])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2]))/(
a*(a^2 - b^2)*d^2*n*(a + b*Sec[c + d*x^n])^2*(a^2*Cos[c]^2 + a^2*Sin[c]^2)) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*
(b + a*Cos[c + d*x^n])*Sec[c + d*x^n]^2*(b*Sin[c] - a*Sin[d*x^n]))/(a^2*(-a + b)*(a + b)*d*n*(a + b*Sec[c + d*
x^n])^2*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*(b + a*Cos[c + d*x^n])^
2*Sec[c + d*x^n]^2*Tan[c])/(a^2*(-a^2 + b^2)*d*n*(a + b*Sec[c + d*x^n])^2) - ((2*I)*b^3*x^(1 - 2*n)*(e*x)^(-1
+ 2*n)*ArcTan[(b + a*Cos[c + d*x^n] + I*a*Sin[c + d*x^n])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x^n])^2*Sec[c + d*
x^n]^2*Tan[c])/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Sec[c + d*x^n])^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.74 (sec) , antiderivative size = 3010, normalized size of antiderivative = 3.98

method	result	size
risch	\(\text {Expression too large to display}\)	\(3010\)

[In]

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

[Out]

1/2/a^2/n*x*exp(1/2*(2*n-1)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*Pi*csgn(I*e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)
*csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2*ln(x)+2*ln(e)))+2*I*b^2/a^2/(a^2-b^2)/d/n*x^n/(a*exp(2*I*(c+d*x^n))+2*b*ex
p(I*(c+d*x^n))+a)*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(e^n)^2*(b*exp(1/2*I*(-2*Pi*n*csgn(I*e)*csgn(I*x)
*csgn(I*e*x)+2*Pi*n*csgn(I*e)*csgn(I*e*x)^2+2*Pi*n*csgn(I*x)*csgn(I*e*x)^2-2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*c
sgn(I*e*x)^2-Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e*x)^3+2*d*x^n+2*c))+exp(1/2*I*Pi*csgn(I*e*x)*(-2*csgn(I*e)*
csgn(I*x)*n+2*csgn(I*e)*csgn(I*e*x)*n+2*csgn(I*x)*csgn(I*e*x)*n-2*csgn(I*e*x)^2*n-csgn(I*e)*csgn(I*e*x)-csgn(I
*x)*csgn(I*e*x)+csgn(I*e*x)^2))*a)/e-2*I*b/(a^2-b^2)^2/d*(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)/n/e*(e^n)^2*exp
(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2+
2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*csgn(I*e*x)^2-
Pi*csgn(I*e*x)^3+2*c))*x^n*ln((-a*exp(I*(d*x^n+2*c))-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))/(-exp(I
*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))+I*b^3/a^2/(a^2-b^2)^2/d*(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)/n/
e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*c
sgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*c
sgn(I*e*x)^2-Pi*csgn(I*e*x)^3+2*c))*x^n*ln((-a*exp(I*(d*x^n+2*c))-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(
1/2))/(-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))+2*I*b/(a^2-b^2)^2/d*(exp(2*I*c)*b^2-a^2*exp(2*I*c))
^(1/2)/n/e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*cs
gn(I*x)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*cs
gn(I*x)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3+2*c))*x^n*ln((a*exp(I*(d*x^n+2*c))+exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2
*I*c))^(1/2))/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))-I*b^3/a^2/(a^2-b^2)^2/d*(exp(2*I*c)*b^2-a^2*
exp(2*I*c))^(1/2)/n/e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*e)*csgn(I*e*x)^
2-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*e)*csgn(I*e
*x)^2+Pi*csgn(I*x)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3+2*c))*x^n*ln((a*exp(I*(d*x^n+2*c))+exp(I*c)*b+(exp(2*I*c)*b^
2-a^2*exp(2*I*c))^(1/2))/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)))-2*b/(a^2-b^2)^2/d^2*(exp(2*I*c)*b
^2-a^2*exp(2*I*c))^(1/2)/n/e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*e)*csgn(
I*e*x)^2-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csgn(I*e)*c
sgn(I*e*x)^2+Pi*csgn(I*x)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3+2*c))*dilog(-a/(-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2
*I*c))^(1/2))*exp(I*(d*x^n+2*c))-1/(-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*exp(I*c)*b+1/(-exp(I*c)
*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))+b^3/a^2/(a^2-b^2)^2/d^2*(exp(
2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)/n/e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csgn(I*
e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi*csg
n(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3+2*c))*dilog(-a/(-exp(I*c)*b+(exp(2*I*c)*b^2-a
^2*exp(2*I*c))^(1/2))*exp(I*(d*x^n+2*c))-1/(-exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*exp(I*c)*b+1/(-
exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))+2*b/(a^2-b^2)^2/d^2*(
exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)/n/e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n*csg
n(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+Pi
*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3+2*c))*dilog(a/(exp(I*c)*b+(exp(2*I*c)*b^2
-a^2*exp(2*I*c))^(1/2))*exp(I*(d*x^n+2*c))+1/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*exp(I*c)*b+1/(
exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))-b^3/a^2/(a^2-b^2)^2/d
^2*(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2)/n/e*(e^n)^2*exp(-1/2*I*(2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-2*Pi*n
*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x
)+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*csgn(I*e*x)^2-Pi*csgn(I*e*x)^3+2*c))*dilog(a/(exp(I*c)*b+(exp(2*I*c)
*b^2-a^2*exp(2*I*c))^(1/2))*exp(I*(d*x^n+2*c))+1/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*exp(I*c)*b
+1/(exp(I*c)*b+(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))*(exp(2*I*c)*b^2-a^2*exp(2*I*c))^(1/2))-2*b^2/a^2/(a^2-b^
2)/d^2*ln(exp(I*x^n*d))/e*(e^n)^2/n*exp(1/2*I*csgn(I*e*x)*Pi*(2*n-1)*(csgn(I*e*x)-csgn(I*x))*(-csgn(I*e*x)+csg
n(I*e)))+b^2/a^2/(a^2-b^2)/d^2*ln(a*exp(2*I*(c+d*x^n))+2*b*exp(I*(c+d*x^n))+a)/e*(e^n)^2/n*exp(1/2*I*csgn(I*e*
x)*Pi*(2*n-1)*(csgn(I*e*x)-csgn(I*x))*(-csgn(I*e*x)+csgn(I*e)))

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. 2503 vs. \(2 (705) = 1410\).

Time = 0.57 (sec) , antiderivative size = 2503, normalized size of antiderivative = 3.31 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*((a^5 - 2*a^3*b^2 + a*b^4)*d^2*e^(2*n - 1)*x^(2*n)*cos(d*x^n + c) + (a^4*b - 2*a^2*b^3 + b^5)*d^2*e^(2*n -
1)*x^(2*n) + 2*(a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sin(d*x^n + c) - ((2*a^4*b - a^2*b^3)*e^(2*n - 1)*sqrt(-(a
^2 - b^2)/a^2)*cos(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(2*n - 1)*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) - ((2*a^4*b -
a^2*b^3)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x^n + c) + (2*a^3*b^2 - a*b^4)*e^(2*n - 1)*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) + ((2*a^4*b - a^2*b^3)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x^n + c) + (2*a^3*b^2 - a*b^4)
*e^(2*n - 1)*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) + ((2*a^4*b - a^2*b^3)*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2)*cos(d*x^
n + c) + (2*a^3*b^2 - a*b^4)*e^(2*n - 1)*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) + ((a^3*b^2 - a*b^4 - I*(2*a^4*b - a^
2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1)*cos(d*x^n + c) + (a^2*b^3 - b^5 - I*(2*a^3*b^2 - a*b^4)*c*sqrt(-(
a^2 - b^2)/a^2))*e^(2*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
) + ((a^3*b^2 - a*b^4 + I*(2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1)*cos(d*x^n + c) + (a^2*b^3
- b^5 + I*(2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*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) + ((a^3*b^2 - a*b^4 - I*(2*a^4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))
*e^(2*n - 1)*cos(d*x^n + c) + (a^2*b^3 - b^5 - I*(2*a^3*b^2 - a*b^4)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1))*lo
g(-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) + ((a^3*b^2 - a*b^4 + I*(2*a^
4*b - a^2*b^3)*c*sqrt(-(a^2 - b^2)/a^2))*e^(2*n - 1)*cos(d*x^n + c) + (a^2*b^3 - b^5 + I*(2*a^3*b^2 - a*b^4)*c
*sqrt(-(a^2 - b^2)/a^2))*e^(2*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) - (-I*(2*a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - I*(2*a^3*b^2 - a*b^4)*c*e^(2*n
- 1)*sqrt(-(a^2 - b^2)/a^2) + (-I*(2*a^4*b - a^2*b^3)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - I*(2*a^4*b -
a^2*b^3)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*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) - (I*(2*a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sq
rt(-(a^2 - b^2)/a^2) + I*(2*a^3*b^2 - a*b^4)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2) + (I*(2*a^4*b - a^2*b^3)*d*e
^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + I*(2*a^4*b - a^2*b^3)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*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) - (-I*(2*a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - I*(2*a^3*b^2 - a*b^4)*c*e^(2*n -
1)*sqrt(-(a^2 - b^2)/a^2) + (-I*(2*a^4*b - a^2*b^3)*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - I*(2*a^4*b - a^
2*b^3)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*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) - (I*(2*a^3*b^2 - a*b^4)*d*e^(2*n - 1)*x^n*sqrt
(-(a^2 - b^2)/a^2) + I*(2*a^3*b^2 - a*b^4)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2) + (I*(2*a^4*b - a^2*b^3)*d*e^(
2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + I*(2*a^4*b - a^2*b^3)*c*e^(2*n - 1)*sqrt(-(a^2 - b^2)/a^2))*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))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d^2*n*cos(d*x^n + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d^2*n)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*(4*a*b^3*e^(2*n)*x^n*sin(d*x^n + c) + (a^4 - a^2*b^2)*d*e^(2*n)*x^(2*n)*cos(2*d*x^n + 2*c)^2 + 4*(a^2*b^2
- b^4)*d*e^(2*n)*x^(2*n)*cos(d*x^n + c)^2 + (a^4 - a^2*b^2)*d*e^(2*n)*x^(2*n)*sin(2*d*x^n + 2*c)^2 + 4*(a^2*b^
2 - b^4)*d*e^(2*n)*x^(2*n)*sin(d*x^n + c)^2 + 4*(a^3*b - a*b^3)*d*e^(2*n)*x^(2*n)*cos(d*x^n + c) + (a^4 - a^2*
b^2)*d*e^(2*n)*x^(2*n) - 2*(2*a*b^3*e^(2*n)*x^n*sin(d*x^n + c) - 2*(a^3*b - a*b^3)*d*e^(2*n)*x^(2*n)*cos(d*x^n
+ c) - (a^4 - a^2*b^2)*d*e^(2*n)*x^(2*n))*cos(2*d*x^n + 2*c) + 2*((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*(a^2*b^4*e^(2*n)*x^n*cos(2*c)*sin(2*d*x^n) + a^2*b^4*e^(2*n)*x^n*cos(
2*d*x^n)*sin(2*c) - 2*(a^3*b^3 - a*b^5)*e^(2*n)*x^n*cos(c)*sin(d*x^n) - 2*(a^3*b^3 - a*b^5)*e^(2*n)*x^n*cos(d*
x^n)*sin(c) + (a^3*b^3*e^(2*n)*x^n*sin(d*x^n + c) - (2*a^5*b - a^3*b^3)*d*e^(2*n)*x^(2*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^(2*n)*x^(2*n) - (a*b^5*e^(2*n)*x^n*sin(2*c) + (2*a^3*b^3 -
a*b^5)*d*e^(2*n)*x^(2*n)*cos(2*c))*cos(2*d*x^n) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(2*n)*x^(2*n)*cos(c) +
(a^2*b^4 - b^6)*e^(2*n)*x^n*sin(c))*cos(d*x^n) - (a*b^5*e^(2*n)*x^n*cos(2*c) - (2*a^3*b^3 - a*b^5)*d*e^(2*n)*
x^(2*n)*sin(2*c))*sin(2*d*x^n) - 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(2*n)*x^(2*n)*sin(c) - (a^2*b^4 - b^6)*e
^(2*n)*x^n*cos(c))*sin(d*x^n))*cos(d*x^n + c) - (a^3*b^3*e^(2*n)*x^n*cos(d*x^n + c) + a^4*b^2*e^(2*n)*x^n + (2
*a^5*b - a^3*b^3)*d*e^(2*n)*x^(2*n)*sin(d*x^n + c))*sin(2*d*x^n + 2*c) + ((a^3*b^3 - a*b^5)*e^(2*n)*x^n - (a*b
^5*e^(2*n)*x^n*cos(2*c) - (2*a^3*b^3 - a*b^5)*d*e^(2*n)*x^(2*n)*sin(2*c))*cos(2*d*x^n) - 2*((2*a^4*b^2 - 3*a^2
*b^4 + b^6)*d*e^(2*n)*x^(2*n)*sin(c) - (a^2*b^4 - b^6)*e^(2*n)*x^n*cos(c))*cos(d*x^n) + (a*b^5*e^(2*n)*x^n*sin
(2*c) + (2*a^3*b^3 - a*b^5)*d*e^(2*n)*x^(2*n)*cos(2*c))*sin(2*d*x^n) - 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(2
*n)*x^(2*n)*cos(c) + (a^2*b^4 - b^6)*e^(2*n)*x^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*cos(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) + 4*(a*b^3*e^(2*n)*x^n*cos(d*x^n + c) + a^2*b^2*e^(2*n)*x^n + (a^3*b - a*b^3)*d*e^(2*n)*x^(2*
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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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