\(\int \frac {(e x)^{-1+2 n}}{a+b \sec (c+d x^n)} \, dx\) [79]

Optimal result

Integrand size = 24, antiderivative size = 328 \[ \int \frac {(e x)^{-1+2 n}}{a+b \sec \left (c+d x^n\right )} \, dx=\frac {(e x)^{2 n}}{2 a e n}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d e n}+\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n}-\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2 e n} \]

[Out]

1/2*(e*x)^(2*n)/a/e/n+I*b*(e*x)^(2*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)^(2*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)+b*(e*x)^(2*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)-b*(e*x)^(2*n)*polyl
og(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)

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4293, 4289, 4276, 3402, 2296, 2221, 2317, 2438} \[ \int \frac {(e x)^{-1+2 n}}{a+b \sec \left (c+d x^n\right )} \, dx=\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}-\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a d^2 e n \sqrt {b^2-a^2}}+\frac {i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {b^2-a^2}}\right )}{a d e n \sqrt {b^2-a^2}}-\frac {i b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{\sqrt {b^2-a^2}+b}\right )}{a d e n \sqrt {b^2-a^2}}+\frac {(e x)^{2 n}}{2 a e n} \]

[In]

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

[Out]

(e*x)^(2*n)/(2*a*e*n) + (I*b*(e*x)^(2*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)^(2*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) + (b*(e*x)^(2*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)) - (b*(e*x)^(2*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))

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 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]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(861\) vs. \(2(328)=656\).

Time = 2.36 (sec) , antiderivative size = 861, normalized size of antiderivative = 2.62 \[ \int \frac {(e x)^{-1+2 n}}{a+b \sec \left (c+d x^n\right )} \, dx=\frac {(e x)^{2 n} \left (b+a \cos \left (c+d x^n\right )\right ) \left (1-\frac {2 b x^{-2 n} \left (2 \left (c+d x^n\right ) \text {arctanh}\left (\frac {(a+b) \cot \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )-2 \left (c+\arccos \left (-\frac {b}{a}\right )\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (\arccos \left (-\frac {b}{a}\right )-2 i \text {arctanh}\left (\frac {(a+b) \cot \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+2 i \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {\sqrt {a^2-b^2} e^{-\frac {1}{2} i \left (c+d x^n\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \cos \left (c+d x^n\right )}}\right )+\left (\arccos \left (-\frac {b}{a}\right )+2 i \left (\text {arctanh}\left (\frac {(a+b) \cot \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )-\text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt {a^2-b^2} e^{\frac {1}{2} i \left (c+d x^n\right )}}{\sqrt {2} \sqrt {a} \sqrt {b+a \cos \left (c+d x^n\right )}}\right )-\left (\arccos \left (-\frac {b}{a}\right )-2 i \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (a-b-i \sqrt {a^2-b^2}\right ) \left (1+i \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}\right )-\left (\arccos \left (-\frac {b}{a}\right )+2 i \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \log \left (\frac {(a+b) \left (-i a+i b+\sqrt {a^2-b^2}\right ) \left (i+\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (b-i \sqrt {a^2-b^2}\right ) \left (a+b-\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (b+i \sqrt {a^2-b^2}\right ) \left (a+b-\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a \left (a+b+\sqrt {a^2-b^2} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}\right )\right )\right )}{\sqrt {a^2-b^2} d^2}\right ) \sec \left (c+d x^n\right )}{2 a e n \left (a+b \sec \left (c+d x^n\right )\right )} \]

[In]

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

[Out]

((e*x)^(2*n)*(b + a*Cos[c + d*x^n])*(1 - (2*b*(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]] + (2*I)*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]])] + (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]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Cos[c + d*x^n]])] - (Arc
Cos[-(b/a)] - (2*I)*ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[((a + b)*(a - b - I*Sqrt[a^2 -
b^2])*(1 + I*Tan[(c + d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(c + d*x^n)/2]))] - (ArcCos[-(b/a)] + (2*I)*
ArcTanh[((a - b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[((a + b)*((-I)*a + I*b + Sqrt[a^2 - b^2])*(I + 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]))])))/(Sqrt[a^2 - b^2]*d^2*x^(2*n)))*Sec[c + d*x^n])/(2*a*e*n*(a + b*Sec[c + d*x^n]))

Maple [C] (warning: unable to verify)

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

Time = 0.66 (sec) , antiderivative size = 752, normalized size of antiderivative = 2.29

method	result	size
risch	\(\frac {x \,{\mathrm e}^{\frac {\left (2 n -1\right ) \left (-i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-i \pi \operatorname {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{2 a n}+\frac {\left (i x^{n} d \ln \left (\frac {a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}+{\mathrm e}^{i c} b +\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}{{\mathrm e}^{i c} b +\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}\right )-i x^{n} d \ln \left (\frac {a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}+{\mathrm e}^{i c} b -\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}{{\mathrm e}^{i c} b -\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}\right )-\operatorname {dilog}\left (\frac {a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}}{{\mathrm e}^{i c} b -\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}+\frac {{\mathrm e}^{i c} b}{{\mathrm e}^{i c} b -\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}-\frac {\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}{{\mathrm e}^{i c} b -\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}\right )+\operatorname {dilog}\left (\frac {a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}}{{\mathrm e}^{i c} b +\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}+\frac {{\mathrm e}^{i c} b}{{\mathrm e}^{i c} b +\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}+\frac {\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}{{\mathrm e}^{i c} b +\sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}}\right )\right ) \sqrt {{\mathrm e}^{2 i c} b^{2}-a^{2} {\mathrm e}^{2 i c}}\, e^{2 n} b \,{\mathrm e}^{-\frac {i \left (2 \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )-2 \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}-2 \pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}+2 \pi n \operatorname {csgn}\left (i e x \right )^{3}-\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi \operatorname {csgn}\left (i e x \right )^{3}+2 c \right )}{2}}}{\left (a^{2}-b^{2}\right ) d^{2} n e a}\)	\(752\)

[In]

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

[Out]

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

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. 1268 vs. \(2 (300) = 600\).

Time = 0.53 (sec) , antiderivative size = 1268, normalized size of antiderivative = 3.87 \[ \int \frac {(e x)^{-1+2 n}}{a+b \sec \left (c+d x^n\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(-I*a*b*c*e^(2*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) + I*a*b*c*e^(2*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) - I*a*b*c*e^(2*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) + I*a*b*c*e^(2*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) + (a^2 - b^2)*d^2*e^(2*n - 1)*x^(2*n
) - a*b*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*b*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*b*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*b*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) + (I*a*b*d*e^(2*n - 1)
*x^n*sqrt(-(a^2 - b^2)/a^2) + I*a*b*c*e^(2*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) + (-I*a*b*d*e^(2*n - 1)*x^n*sqrt(-(
a^2 - b^2)/a^2) - I*a*b*c*e^(2*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) + (I*a*b*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a
^2) + I*a*b*c*e^(2*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*s
qrt(-(a^2 - b^2)/a^2) - I*b)*sin(d*x^n + c) - a)/a) + (-I*a*b*d*e^(2*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - I*a*b
*c*e^(2*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))/((a^3 - a*b^2)*d^2*n)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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