Integrand size = 16, antiderivative size = 93 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {6 x^2 \text {arctanh}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {6 x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac {6 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^4} \]
-6*x^2*arctanh(exp(b*x+a))/b^2-x^3*csch(b*x+a)/b-6*x*polylog(2,-exp(b*x+a) )/b^3+6*x*polylog(2,exp(b*x+a))/b^3+6*polylog(3,-exp(b*x+a))/b^4-6*polylog (3,exp(b*x+a))/b^4
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.17 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\frac {-b^3 x^3 \text {csch}(a+b x)+3 b^2 x^2 \log \left (1-e^{a+b x}\right )-3 b^2 x^2 \log \left (1+e^{a+b x}\right )-6 b x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+6 b x \operatorname {PolyLog}\left (2,e^{a+b x}\right )+6 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-6 \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^4} \]
(-(b^3*x^3*Csch[a + b*x]) + 3*b^2*x^2*Log[1 - E^(a + b*x)] - 3*b^2*x^2*Log [1 + E^(a + b*x)] - 6*b*x*PolyLog[2, -E^(a + b*x)] + 6*b*x*PolyLog[2, E^(a + b*x)] + 6*PolyLog[3, -E^(a + b*x)] - 6*PolyLog[3, E^(a + b*x)])/b^4
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5942, 3042, 26, 4670, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx\) |
| \(\Big \downarrow \) 5942 |
| \(\displaystyle \frac {3 \int x^2 \text {csch}(a+b x)dx}{b}-\frac {x^3 \text {csch}(a+b x)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 \int i x^2 \csc (i a+i b x)dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 i \int x^2 \csc (i a+i b x)dx}{b}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 i \left (\frac {2 i \int x \log \left (1-e^{a+b x}\right )dx}{b}-\frac {2 i \int x \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 i \left (-\frac {2 i \left (\frac {\int \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i \left (\frac {\int \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 i \left (-\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,-e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i \left (\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (2,e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 i \left (\frac {2 i x^2 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {2 i \left (\frac {\operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i \left (\frac {\operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^2}-\frac {x \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}\right )}{b}\) |
-((x^3*Csch[a + b*x])/b) + ((3*I)*(((2*I)*x^2*ArcTanh[E^(a + b*x)])/b - (( 2*I)*(-((x*PolyLog[2, -E^(a + b*x)])/b) + PolyLog[3, -E^(a + b*x)]/b^2))/b + ((2*I)*(-((x*PolyLog[2, E^(a + b*x)])/b) + PolyLog[3, E^(a + b*x)]/b^2) )/b))/b
3.5.25.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[Coth[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csch[(a_.) + (b_.)*(x_)^(n_.)]^(p_ .)*(x_)^(m_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Csch[a + b*x^n]^p/(b*n*p )), x] + Simp[(m - n + 1)/(b*n*p) Int[x^(m - n)*Csch[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Time = 0.44 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.87
| method | result | size |
| risch | \(-\frac {2 x^{3} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {6 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}+\frac {6 x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{b^{4}}-\frac {6 x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}\) | \(174\) |
-2/b*x^3*exp(b*x+a)/(exp(2*b*x+2*a)-1)-6/b^4*a^2*arctanh(exp(b*x+a))+3/b^2 *ln(1-exp(b*x+a))*x^2-3/b^4*ln(1-exp(b*x+a))*a^2+6*x*polylog(2,exp(b*x+a)) /b^3-6*polylog(3,exp(b*x+a))/b^4-3/b^2*ln(exp(b*x+a)+1)*x^2+3/b^4*ln(exp(b *x+a)+1)*a^2-6*x*polylog(2,-exp(b*x+a))/b^3+6*polylog(3,-exp(b*x+a))/b^4
Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (86) = 172\).
Time = 0.27 (sec) , antiderivative size = 551, normalized size of antiderivative = 5.92 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, b^{3} x^{3} \cosh \left (b x + a\right ) + 2 \, b^{3} x^{3} \sinh \left (b x + a\right ) - 6 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 3 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2} - b^{2} x^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 3 \, {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 3 \, {\left (b^{2} x^{2} - {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{4} \cosh \left (b x + a\right )^{2} + 2 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )^{2} - b^{4}} \]
-(2*b^3*x^3*cosh(b*x + a) + 2*b^3*x^3*sinh(b*x + a) - 6*(b*x*cosh(b*x + a) ^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*sinh(b*x + a)^2 - b*x)*dilog( cosh(b*x + a) + sinh(b*x + a)) + 6*(b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*sinh(b*x + a)^2 - b*x)*dilog(-cosh(b*x + a) - sinh (b*x + a)) + 3*(b^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2*cosh(b*x + a)*sinh(b*x + a) + b^2*x^2*sinh(b*x + a)^2 - b^2*x^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 3*(a^2*cosh(b*x + a)^2 + 2*a^2*cosh(b*x + a)*sinh(b*x + a) + a^2 *sinh(b*x + a)^2 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 3*(b^2*x^ 2 - (b^2*x^2 - a^2)*cosh(b*x + a)^2 - 2*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh (b*x + a) - (b^2*x^2 - a^2)*sinh(b*x + a)^2 - a^2)*log(-cosh(b*x + a) - si nh(b*x + a) + 1) + 6*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + si nh(b*x + a)^2 - 1)*polylog(3, cosh(b*x + a) + sinh(b*x + a)) - 6*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*polylog(3, -cosh(b*x + a) - sinh(b*x + a)))/(b^4*cosh(b*x + a)^2 + 2*b^4*cosh(b*x + a )*sinh(b*x + a) + b^4*sinh(b*x + a)^2 - b^4)
\[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\int x^{3} \cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30 \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, x^{3} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{4}} \]
-2*x^3*e^(b*x + a)/(b*e^(2*b*x + 2*a) - b) - 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 + 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
\[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\int { x^{3} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2} \,d x } \]
Timed out. \[ \int x^3 \coth (a+b x) \text {csch}(a+b x) \, dx=\int \frac {x^3\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]