3.78 \(\int (e x)^{-1+3 n} (a+b \text{sech}(c+d x^n))^2 \, dx\)

Optimal. Leaf size=363 \[ \frac{4 i a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-i e^{c+d x^n}\right )}{d^3 e n}-\frac{4 i a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,i e^{c+d x^n}\right )}{d^3 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}-\frac{b^2 x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (2,-e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{4 a b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (e^{2 \left (c+d x^n\right )}+1\right )}{d^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac{b^2 x^{-n} (e x)^{3 n}}{d e n} \]

[Out]

(a^2*(e*x)^(3*n))/(3*e*n) + (b^2*(e*x)^(3*n))/(d*e*n*x^n) + (4*a*b*(e*x)^(3*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x
^n) - (2*b^2*(e*x)^(3*n)*Log[1 + E^(2*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, (-I
)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) -
(b^2*(e*x)^(3*n)*PolyLog[2, -E^(2*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^
(c + d*x^n)])/(d^3*e*n*x^(3*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[3, I*E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) + (b^2
*(e*x)^(3*n)*Tanh[c + d*x^n])/(d*e*n*x^n)

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Rubi [A]  time = 0.404006, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5440, 5436, 4190, 4180, 2531, 2282, 6589, 4184, 3718, 2190, 2279, 2391} \[ \frac{4 i a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-i e^{c+d x^n}\right )}{d^3 e n}-\frac{4 i a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,i e^{c+d x^n}\right )}{d^3 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}-\frac{b^2 x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (2,-e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{4 a b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (e^{2 \left (c+d x^n\right )}+1\right )}{d^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac{b^2 x^{-n} (e x)^{3 n}}{d e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(e*x)^(3*n))/(3*e*n) + (b^2*(e*x)^(3*n))/(d*e*n*x^n) + (4*a*b*(e*x)^(3*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x
^n) - (2*b^2*(e*x)^(3*n)*Log[1 + E^(2*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, (-I
)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) -
(b^2*(e*x)^(3*n)*PolyLog[2, -E^(2*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^
(c + d*x^n)])/(d^3*e*n*x^(3*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[3, I*E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) + (b^2
*(e*x)^(3*n)*Tanh[c + d*x^n])/(d*e*n*x^n)

Rule 5440

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

Rule 5436

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

Rule 4190

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

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2531

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 2282

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 6589

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]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int (e x)^{-1+3 n} \left (a+b \text{sech}\left (c+d x^n\right )\right )^2 \, dx &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \int x^{-1+3 n} \left (a+b \text{sech}\left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 (a+b \text{sech}(c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text{sech}(c+d x)+b^2 x^2 \text{sech}^2(c+d x)\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{\left (2 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 \text{sech}(c+d x) \, dx,x,x^n\right )}{e n}+\frac{\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 \text{sech}^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{4 a b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}-\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \tanh (c+d x) \, dx,x,x^n\right )}{d e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac{4 a b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d^2 e n}-\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{c+d x}\right ) \, dx,x,x^n\right )}{d^2 e n}-\frac{\left (4 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{2 (c+d x)} x}{1+e^{2 (c+d x)}} \, dx,x,x^n\right )}{d e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac{4 a b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}-\frac{\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}+\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac{4 a b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}+\frac{4 i a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-i e^{c+d x^n}\right )}{d^3 e n}-\frac{4 i a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (i e^{c+d x^n}\right )}{d^3 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac{\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac{4 a b x^{-n} (e x)^{3 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}-\frac{b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 i a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-i e^{c+d x^n}\right )}{d^3 e n}-\frac{4 i a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (i e^{c+d x^n}\right )}{d^3 e n}+\frac{b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}\\ \end{align*}

Mathematica [F]  time = 68.2869, size = 0, normalized size = 0. \[ \int (e x)^{-1+3 n} \left (a+b \text{sech}\left (c+d x^n\right )\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.306, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+3\,n} \left ( a+b{\rm sech} \left (c+d{x}^{n}\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.71074, size = 17207, normalized size = 47.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 - 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3*n -
1)*log(e)) + a^2*d^3*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^3 + (a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x)
)^3 + 6*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3*
n - 1)*log(e)) + a^2*d^3*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^3 + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*l
og(x)) + 2*b^2*d^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh(n*log(x)) + 2*b^2*d^2)*sinh((3*n - 1)*log(e)))*sinh(
n*log(x))^2 + (a^2*d^3*cosh(n*log(x))^3 + 6*b^2*d^2*cosh(n*log(x))^2 - 6*b^2*c^2)*sinh((3*n - 1)*log(e)) + 3*(
a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 + 4*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a^2*d^3*c
osh(n*log(x))^2 + 4*b^2*d^2*cosh(n*log(x)))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*
sinh(n*log(x)) + c)^2 + 2*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 + 6*b^2*d^2*cosh((3*n - 1)*log(e))*
cosh(n*log(x))^2 - 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3*n - 1)*log(e)) + a^2*d^3*sinh((3*n - 1)
*log(e)))*sinh(n*log(x))^3 + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + 2*b^2*d^2*cosh((3*n - 1)*log(e
)) + (a^2*d^3*cosh(n*log(x)) + 2*b^2*d^2)*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^3*cosh(n*log(x))^3
+ 6*b^2*d^2*cosh(n*log(x))^2 - 6*b^2*c^2)*sinh((3*n - 1)*log(e)) + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*l
og(x))^2 + 4*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a^2*d^3*cosh(n*log(x))^2 + 4*b^2*d^2*cosh(n*log(
x)))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x
)) + d*sinh(n*log(x)) + c) + (a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 + 6*b^2*d^2*cosh((3*n - 1)*log(e
))*cosh(n*log(x))^2 - 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3*n - 1)*log(e)) + a^2*d^3*sinh((3*n -
1)*log(e)))*sinh(n*log(x))^3 + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + 2*b^2*d^2*cosh((3*n - 1)*lo
g(e)) + (a^2*d^3*cosh(n*log(x)) + 2*b^2*d^2)*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^3*cosh(n*log(x)
)^3 + 6*b^2*d^2*cosh(n*log(x))^2 - 6*b^2*c^2)*sinh((3*n - 1)*log(e)) + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(
n*log(x))^2 + 4*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a^2*d^3*cosh(n*log(x))^2 + 4*b^2*d^2*cosh(n*l
og(x)))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 3*(a^2*d^3*c
osh((3*n - 1)*log(e))*cosh(n*log(x)) + a^2*d^3*cosh(n*log(x))*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (12*I
*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (12*I*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*cosh(
(3*n - 1)*log(e)) - 6*(-2*I*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (12*I*a*b*d*cosh((3*n - 1)*lo
g(e)) + 12*I*a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - 6
*b^2*cosh((3*n - 1)*log(e)) + (24*I*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 12*b^2*cosh((3*n - 1)*log(e)
) - 12*(-2*I*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (24*I*a*b*d*cosh((3*n - 1)*log(e)) + 24*I*a*
b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)
) + d*sinh(n*log(x)) + c) + (12*I*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*cosh((3*n - 1)*log(e)) -
6*(-2*I*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (12*I*a*b*d*cosh((3*n - 1)*log(e)) + 12*I*a*b*d*
sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - 6*(-2*I*a*b*d*cosh(n
*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (12*I*a*b*d*cosh((3*n - 1)*log(e)) + 12*I*a*b*d*sinh((3*n - 1)*log(e)
))*sinh(n*log(x)))*dilog(I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + I*sinh(d*cosh(n*log(x)) + d*sinh(n*
log(x)) + c)) + (-12*I*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (-12*I*a*b*d*cosh((3*n - 1)*log(e))*cosh(
n*log(x)) - 6*b^2*cosh((3*n - 1)*log(e)) - 6*(2*I*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (-12*I*
a*b*d*cosh((3*n - 1)*log(e)) - 12*I*a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c)^2 - 6*b^2*cosh((3*n - 1)*log(e)) + (-24*I*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 12*b
^2*cosh((3*n - 1)*log(e)) - 12*(2*I*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (-24*I*a*b*d*cosh((3*
n - 1)*log(e)) - 24*I*a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (-12*I*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2
*cosh((3*n - 1)*log(e)) - 6*(2*I*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (-12*I*a*b*d*cosh((3*n -
1)*log(e)) - 12*I*a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)
^2 - 6*(2*I*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + (-12*I*a*b*d*cosh((3*n - 1)*log(e)) - 12*I*a*
b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*dilog(-I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - I*sinh(d*
cosh(n*log(x)) + d*sinh(n*log(x)) + c)) + (((6*I*a*b*c^2 + 6*b^2*c)*cosh((3*n - 1)*log(e)) + (6*I*a*b*c^2 + 6*
b^2*c)*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((12*I*a*b*c^2 + 12*b^2*c)*co
sh((3*n - 1)*log(e)) + (12*I*a*b*c^2 + 12*b^2*c)*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(
x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((6*I*a*b*c^2 + 6*b^2*c)*cosh((3*n - 1)*log(e)) + (6*
I*a*b*c^2 + 6*b^2*c)*sinh((3*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (6*I*a*b*c^2 +
6*b^2*c)*cosh((3*n - 1)*log(e)) + (6*I*a*b*c^2 + 6*b^2*c)*sinh((3*n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + I) + (((-6*I*a*b*c^2 + 6*b^2*c)*cosh((
3*n - 1)*log(e)) + (-6*I*a*b*c^2 + 6*b^2*c)*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c)^2 + ((-12*I*a*b*c^2 + 12*b^2*c)*cosh((3*n - 1)*log(e)) + (-12*I*a*b*c^2 + 12*b^2*c)*sinh((3*n - 1)*log(e))
)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((-6*I*a*b*c^2
+ 6*b^2*c)*cosh((3*n - 1)*log(e)) + (-6*I*a*b*c^2 + 6*b^2*c)*sinh((3*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c)^2 + (-6*I*a*b*c^2 + 6*b^2*c)*cosh((3*n - 1)*log(e)) + (-6*I*a*b*c^2 + 6*b^2*c)*sinh((3*n
- 1)*log(e)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c) - I) + (-6*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 6*b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)
) + (-6*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 6*b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (-
6*I*a*b*d^2*cosh((3*n - 1)*log(e)) - 6*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (6*I*a*b*c^2 - 6*b
^2*c)*cosh((3*n - 1)*log(e)) + (-6*I*a*b*d^2*cosh(n*log(x))^2 + 6*I*a*b*c^2 - 6*b^2*d*cosh(n*log(x)) - 6*b^2*c
)*sinh((3*n - 1)*log(e)) + (-12*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*d*cosh((3*n - 1)*log(e
)) + (-12*I*a*b*d^2*cosh(n*log(x)) - 6*b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c)^2 + (-12*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 12*b^2*d*cosh((3*n - 1)*log
(e))*cosh(n*log(x)) + (-12*I*a*b*d^2*cosh((3*n - 1)*log(e)) - 12*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(
x))^2 + (12*I*a*b*c^2 - 12*b^2*c)*cosh((3*n - 1)*log(e)) + (-12*I*a*b*d^2*cosh(n*log(x))^2 + 12*I*a*b*c^2 - 12
*b^2*d*cosh(n*log(x)) - 12*b^2*c)*sinh((3*n - 1)*log(e)) + (-24*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)
) - 12*b^2*d*cosh((3*n - 1)*log(e)) + (-24*I*a*b*d^2*cosh(n*log(x)) - 12*b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n
*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (-6*I*
a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 6*b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (-6*I*a*b*d^
2*cosh((3*n - 1)*log(e)) - 6*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (6*I*a*b*c^2 - 6*b^2*c)*cosh
((3*n - 1)*log(e)) + (-6*I*a*b*d^2*cosh(n*log(x))^2 + 6*I*a*b*c^2 - 6*b^2*d*cosh(n*log(x)) - 6*b^2*c)*sinh((3*
n - 1)*log(e)) + (-12*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*d*cosh((3*n - 1)*log(e)) + (-12*
I*a*b*d^2*cosh(n*log(x)) - 6*b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*l
og(x)) + c)^2 + (-6*I*a*b*d^2*cosh((3*n - 1)*log(e)) - 6*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 +
(6*I*a*b*c^2 - 6*b^2*c)*cosh((3*n - 1)*log(e)) + (-6*I*a*b*d^2*cosh(n*log(x))^2 + 6*I*a*b*c^2 - 6*b^2*d*cosh(n
*log(x)) - 6*b^2*c)*sinh((3*n - 1)*log(e)) + (-12*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*d*co
sh((3*n - 1)*log(e)) + (-12*I*a*b*d^2*cosh(n*log(x)) - 6*b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*log(I*
cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + I*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) + (6*I*a*
b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 6*b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (6*I*a*b*d^2*c
osh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 6*b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (6*I*a*b*d^2*cosh((3*
n - 1)*log(e)) + 6*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (-6*I*a*b*c^2 - 6*b^2*c)*cosh((3*n - 1
)*log(e)) + (6*I*a*b*d^2*cosh(n*log(x))^2 - 6*I*a*b*c^2 - 6*b^2*d*cosh(n*log(x)) - 6*b^2*c)*sinh((3*n - 1)*log
(e)) + (12*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*d*cosh((3*n - 1)*log(e)) + (12*I*a*b*d^2*co
sh(n*log(x)) - 6*b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^
2 + (12*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 12*b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (
12*I*a*b*d^2*cosh((3*n - 1)*log(e)) + 12*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (-12*I*a*b*c^2 -
12*b^2*c)*cosh((3*n - 1)*log(e)) + (12*I*a*b*d^2*cosh(n*log(x))^2 - 12*I*a*b*c^2 - 12*b^2*d*cosh(n*log(x)) -
12*b^2*c)*sinh((3*n - 1)*log(e)) + (24*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 12*b^2*d*cosh((3*n -
1)*log(e)) + (24*I*a*b*d^2*cosh(n*log(x)) - 12*b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (6*I*a*b*d^2*cosh((3*n - 1)*log(
e))*cosh(n*log(x))^2 - 6*b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (6*I*a*b*d^2*cosh((3*n - 1)*log(e)) + 6
*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (-6*I*a*b*c^2 - 6*b^2*c)*cosh((3*n - 1)*log(e)) + (6*I*a
*b*d^2*cosh(n*log(x))^2 - 6*I*a*b*c^2 - 6*b^2*d*cosh(n*log(x)) - 6*b^2*c)*sinh((3*n - 1)*log(e)) + (12*I*a*b*d
^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*d*cosh((3*n - 1)*log(e)) + (12*I*a*b*d^2*cosh(n*log(x)) - 6*b
^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (6*I*a*b*d^2*c
osh((3*n - 1)*log(e)) + 6*I*a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (-6*I*a*b*c^2 - 6*b^2*c)*cosh((
3*n - 1)*log(e)) + (6*I*a*b*d^2*cosh(n*log(x))^2 - 6*I*a*b*c^2 - 6*b^2*d*cosh(n*log(x)) - 6*b^2*c)*sinh((3*n -
1)*log(e)) + (12*I*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 6*b^2*d*cosh((3*n - 1)*log(e)) + (12*I*a*b
*d^2*cosh(n*log(x)) - 6*b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*log(-I*cosh(d*cosh(n*log(x)) + d*sinh(n
*log(x)) + c) - I*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) + ((-12*I*a*b*cosh((3*n - 1)*log(e)) - 12
*I*a*b*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - 12*I*a*b*cosh((3*n - 1)*log(e
)) + (-24*I*a*b*cosh((3*n - 1)*log(e)) - 24*I*a*b*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log
(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (-12*I*a*b*cosh((3*n - 1)*log(e)) - 12*I*a*b*sinh((3
*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - 12*I*a*b*sinh((3*n - 1)*log(e)))*polylog(3,
I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + I*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) + ((12*I*a
*b*cosh((3*n - 1)*log(e)) + 12*I*a*b*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 +
12*I*a*b*cosh((3*n - 1)*log(e)) + (24*I*a*b*cosh((3*n - 1)*log(e)) + 24*I*a*b*sinh((3*n - 1)*log(e)))*cosh(d*
cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (12*I*a*b*cosh((3*n - 1
)*log(e)) + 12*I*a*b*sinh((3*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 12*I*a*b*sinh((
3*n - 1)*log(e)))*polylog(3, -I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - I*sinh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c)) + (a^2*d^3*cosh(n*log(x))^3 - 6*b^2*c^2)*sinh((3*n - 1)*log(e)) + 3*(a^2*d^3*cosh((3*n - 1)
*log(e))*cosh(n*log(x))^2 + a^2*d^3*cosh(n*log(x))^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))/(d^3*n*cosh(d*cos
h(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*d^3*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c) + d^3*n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + d^3*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{sech}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{3 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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