3.5.95 \(\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx\) [495]

Optimal. Leaf size=85 \[ -\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \text {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \text {PolyLog}\left (3,e^{4 (a+b x)}\right )}{8 b^4} \]

[Out]

-2*x^3/b-2*x^3*coth(2*b*x+2*a)/b+3*x^2*ln(1-exp(4*b*x+4*a))/b^2+3/2*x*polylog(2,exp(4*b*x+4*a))/b^3-3/8*polylo
g(3,exp(4*b*x+4*a))/b^4

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Rubi [A]
time = 0.16, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5569, 4269, 3797, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {3 \text {Li}_3\left (e^{4 (a+b x)}\right )}{8 b^4}+\frac {3 x \text {Li}_2\left (e^{4 (a+b x)}\right )}{2 b^3}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}-\frac {2 x^3 \coth (2 a+2 b x)}{b}-\frac {2 x^3}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Csch[a + b*x]^2*Sech[a + b*x]^2,x]

[Out]

(-2*x^3)/b - (2*x^3*Coth[2*a + 2*b*x])/b + (3*x^2*Log[1 - E^(4*(a + b*x))])/b^2 + (3*x*PolyLog[2, E^(4*(a + b*
x))])/(2*b^3) - (3*PolyLog[3, E^(4*(a + b*x))])/(8*b^4)

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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4269

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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

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*} \int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx &=4 \int x^3 \text {csch}^2(2 a+2 b x) \, dx\\ &=-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {6 \int x^2 \coth (2 a+2 b x) \, dx}{b}\\ &=-\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}-\frac {12 \int \frac {e^{2 (2 a+2 b x)} x^2}{1-e^{2 (2 a+2 b x)}} \, dx}{b}\\ &=-\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}-\frac {6 \int x \log \left (1-e^{2 (2 a+2 b x)}\right ) \, dx}{b^2}\\ &=-\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \text {Li}_2\left (e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \int \text {Li}_2\left (e^{2 (2 a+2 b x)}\right ) \, dx}{2 b^3}\\ &=-\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \text {Li}_2\left (e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (2 a+2 b x)}\right )}{8 b^4}\\ &=-\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \text {Li}_2\left (e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \text {Li}_3\left (e^{4 (a+b x)}\right )}{8 b^4}\\ \end {align*}

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Mathematica [A]
time = 3.69, size = 100, normalized size = 1.18 \begin {gather*} \frac {12 b x \text {PolyLog}\left (2,e^{4 (a+b x)}\right )-3 \text {PolyLog}\left (3,e^{4 (a+b x)}\right )+8 b^2 x^2 \left (-\frac {4 b e^{4 a} x}{-1+e^{4 a}}+3 \log \left (1-e^{4 (a+b x)}\right )+2 b x \text {csch}(2 a) \text {csch}(2 (a+b x)) \sinh (2 b x)\right )}{8 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Csch[a + b*x]^2*Sech[a + b*x]^2,x]

[Out]

(12*b*x*PolyLog[2, E^(4*(a + b*x))] - 3*PolyLog[3, E^(4*(a + b*x))] + 8*b^2*x^2*((-4*b*E^(4*a)*x)/(-1 + E^(4*a
)) + 3*Log[1 - E^(4*(a + b*x))] + 2*b*x*Csch[2*a]*Csch[2*(a + b*x)]*Sinh[2*b*x]))/(8*b^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(262\) vs. \(2(81)=162\).
time = 1.89, size = 263, normalized size = 3.09

method	result	size
risch	\(-\frac {4 x^{3}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right ) \left ({\mathrm e}^{2 b x +2 a}+1\right )}-\frac {4 x^{3}}{b}+\frac {12 x \,a^{2}}{b^{3}}+\frac {3 x^{2} \ln \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b^{2}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}+\frac {6 x \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}-\frac {6 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \polylog \left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}-\frac {6 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {8 a^{3}}{b^{4}}+\frac {3 x \polylog \left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {6 x \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {12 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}-\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}\)	\(263\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-4*x^3/b/(exp(2*b*x+2*a)-1)/(exp(2*b*x+2*a)+1)-4/b*x^3+12/b^3*x*a^2+3*x^2*ln(exp(2*b*x+2*a)+1)/b^2+3/b^2*ln(1-
exp(b*x+a))*x^2+6*x*polylog(2,exp(b*x+a))/b^3+3/b^2*ln(exp(b*x+a)+1)*x^2-6*polylog(3,exp(b*x+a))/b^4-3/2*polyl
og(3,-exp(2*b*x+2*a))/b^4-6*polylog(3,-exp(b*x+a))/b^4+8/b^4*a^3+3*x*polylog(2,-exp(2*b*x+2*a))/b^3+6*x*polylo
g(2,-exp(b*x+a))/b^3-12/b^4*a^2*ln(exp(b*x+a))+3/b^4*a^2*ln(exp(b*x+a)-1)-3/b^4*ln(1-exp(b*x+a))*a^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (80) = 160\).
time = 0.28, size = 180, normalized size = 2.12 \begin {gather*} -\frac {4 \, x^{3}}{b e^{\left (4 \, b x + 4 \, a\right )} - b} - \frac {4 \, x^{3}}{b} + \frac {3 \, {\left (2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})\right )}}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="maxima")

[Out]

-4*x^3/(b*e^(4*b*x + 4*a) - b) - 4*x^3/b + 3/2*(2*b^2*x^2*log(e^(2*b*x + 2*a) + 1) + 2*b*x*dilog(-e^(2*b*x + 2
*a)) - polylog(3, -e^(2*b*x + 2*a)))/b^4 + 3*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*dilog(-e^(b*x + a)) - 2*pol
ylog(3, -e^(b*x + a)))/b^4 + 3*(b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*dilog(e^(b*x + a)) - 2*polylog(3, e^(b*x
+ a)))/b^4

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Fricas [C] Result contains complex when optimal does not.
time = 0.39, size = 1924, normalized size = 22.64 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="fricas")

[Out]

-(4*(b^3*x^3 + a^3)*cosh(b*x + a)^4 + 16*(b^3*x^3 + a^3)*cosh(b*x + a)^3*sinh(b*x + a) + 24*(b^3*x^3 + a^3)*co
sh(b*x + a)^2*sinh(b*x + a)^2 + 16*(b^3*x^3 + a^3)*cosh(b*x + a)*sinh(b*x + a)^3 + 4*(b^3*x^3 + a^3)*sinh(b*x
+ a)^4 - 4*a^3 - 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x
+ a)^2 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(cosh(b*x + a) + sinh(b*x + a)
) - 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b
*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 6*(b*
x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b*x*cosh(b
*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - 6*(b*x*cosh(b
*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b*x*cosh(b*x + a)*
sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 3*(b^2*x^2*cosh(b*x + a)^
4 + 4*b^2*x^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*b^2*x^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b^2*x^2*cosh(b*x +
a)*sinh(b*x + a)^3 + b^2*x^2*sinh(b*x + a)^4 - b^2*x^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 3*(a^2*cosh(
b*x + a)^4 + 4*a^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*a^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*a^2*cosh(b*x + a)
*sinh(b*x + a)^3 + a^2*sinh(b*x + a)^4 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) + I) - 3*(a^2*cosh(b*x + a)^4
+ 4*a^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*a^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*a^2*cosh(b*x + a)*sinh(b*x +
a)^3 + a^2*sinh(b*x + a)^4 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) - I) - 3*(a^2*cosh(b*x + a)^4 + 4*a^2*cos
h(b*x + a)^3*sinh(b*x + a) + 6*a^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*a^2*cosh(b*x + a)*sinh(b*x + a)^3 + a^2
*sinh(b*x + a)^4 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 3*((b^2*x^2 - a^2)*cosh(b*x + a)^4 + 4*(b^2*x
^2 - a^2)*cosh(b*x + a)^3*sinh(b*x + a) + 6*(b^2*x^2 - a^2)*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*(b^2*x^2 - a^2
)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - a^2)*sinh(b*x + a)^4 - b^2*x^2 + a^2)*log(I*cosh(b*x + a) + I*sin
h(b*x + a) + 1) - 3*((b^2*x^2 - a^2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - a^2)*cosh(b*x + a)^3*sinh(b*x + a) + 6*(b^
2*x^2 - a^2)*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - a^
2)*sinh(b*x + a)^4 - b^2*x^2 + a^2)*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 3*((b^2*x^2 - a^2)*cosh(b*x
+ a)^4 + 4*(b^2*x^2 - a^2)*cosh(b*x + a)^3*sinh(b*x + a) + 6*(b^2*x^2 - a^2)*cosh(b*x + a)^2*sinh(b*x + a)^2 +
4*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - a^2)*sinh(b*x + a)^4 - b^2*x^2 + a^2)*log(-cosh(
b*x + a) - sinh(b*x + a) + 1) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(
b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*polylog(3, cosh(b*x + a) + sinh(b*x + a))
+ 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*s
inh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*polylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) + 6*(cosh(b*x + a)^4 + 4*c
osh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x
+ a)^4 - 1)*polylog(3, -I*cosh(b*x + a) - I*sinh(b*x + a)) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x +
a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*polylog(3, -c
osh(b*x + a) - sinh(b*x + a)))/(b^4*cosh(b*x + a)^4 + 4*b^4*cosh(b*x + a)^3*sinh(b*x + a) + 6*b^4*cosh(b*x + a
)^2*sinh(b*x + a)^2 + 4*b^4*cosh(b*x + a)*sinh(b*x + a)^3 + b^4*sinh(b*x + a)^4 - b^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*csch(b*x+a)**2*sech(b*x+a)**2,x)

[Out]

Integral(x**3*csch(a + b*x)**2*sech(a + b*x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x^3*csch(b*x + a)^2*sech(b*x + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(cosh(a + b*x)^2*sinh(a + b*x)^2),x)

[Out]

int(x^3/(cosh(a + b*x)^2*sinh(a + b*x)^2), x)

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