Integrand size = 18, antiderivative size = 83 \[ \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {3 x^2}{2 b^2}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {3 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4} \]
-3/2*x^2/b^2-3/2*x^2*coth(b*x+a)/b^2-1/2*x^3*csch(b*x+a)^2/b+3*x*ln(1-exp( 2*b*x+2*a))/b^3+3/2*polylog(2,exp(2*b*x+2*a))/b^4
Time = 1.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.42 \[ \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\frac {-6 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )-6 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+b x \left (-b^2 x^2 \text {csch}^2(a+b x)+6 \left (-\frac {b x}{-1+e^{2 a}}+\log \left (1-e^{-a-b x}\right )+\log \left (1+e^{-a-b x}\right )\right )+3 b x \text {csch}(a) \text {csch}(a+b x) \sinh (b x)\right )}{2 b^4} \]
(-6*PolyLog[2, -E^(-a - b*x)] - 6*PolyLog[2, E^(-a - b*x)] + b*x*(-(b^2*x^ 2*Csch[a + b*x]^2) + 6*(-((b*x)/(-1 + E^(2*a))) + Log[1 - E^(-a - b*x)] + Log[1 + E^(-a - b*x)]) + 3*b*x*Csch[a]*Csch[a + b*x]*Sinh[b*x]))/(2*b^4)
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.37, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5942, 3042, 25, 4672, 26, 3042, 26, 4201, 2620, 2715, 2838}
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.
\(\displaystyle \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx\) |
\(\Big \downarrow \) 5942 |
\(\displaystyle \frac {3 \int x^2 \text {csch}^2(a+b x)dx}{2 b}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}+\frac {3 \int -x^2 \csc (i a+i b x)^2dx}{2 b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \int x^2 \csc (i a+i b x)^2dx}{2 b}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}-\frac {2 i \int -i x \coth (a+b x)dx}{b}\right )}{2 b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}-\frac {2 \int x \coth (a+b x)dx}{b}\right )}{2 b}-\frac {x^3 \text {csch}^2(a+b x)}{2 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}-\frac {2 \int -i x \tan \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}\right )}{2 b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}+\frac {2 i \int x \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx}{b}\right )}{2 b}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}+\frac {2 i \left (2 i \int \frac {e^{2 a+2 b x-i \pi } x}{1+e^{2 a+2 b x-i \pi }}dx-\frac {i x^2}{2}\right )}{b}\right )}{2 b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}+\frac {2 i \left (2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{2 b}\right )-\frac {i x^2}{2}\right )}{b}\right )}{2 b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}+\frac {2 i \left (2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int e^{-2 a-2 b x+i \pi } \log \left (1+e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}\right )-\frac {i x^2}{2}\right )}{b}\right )}{2 b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {x^3 \text {csch}^2(a+b x)}{2 b}-\frac {3 \left (\frac {x^2 \coth (a+b x)}{b}+\frac {2 i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )}{b}\right )}{2 b}\) |
-1/2*(x^3*Csch[a + b*x]^2)/b - (3*((x^2*Coth[a + b*x])/b + ((2*I)*((-1/2*I )*x^2 + (2*I)*((x*Log[1 + E^(2*a - I*Pi + 2*b*x)])/(2*b) + PolyLog[2, -E^( 2*a - I*Pi + 2*b*x)]/(4*b^2))))/b))/(2*b)
3.5.46.3.1 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[(((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] - Si mp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
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] + Simp[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]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(75)=150\).
Time = 0.54 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.13
method | result | size |
risch | \(-\frac {x^{2} \left (2 \,{\mathrm e}^{2 b x +2 a} b x +3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 x^{2}}{b^{2}}-\frac {6 a x}{b^{3}}-\frac {3 a^{2}}{b^{4}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}+\frac {3 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {6 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}\) | \(177\) |
-x^2*(2*exp(2*b*x+2*a)*b*x+3*exp(2*b*x+2*a)-3)/b^2/(exp(2*b*x+2*a)-1)^2-3/ b^2*x^2-6/b^3*a*x-3/b^4*a^2+3/b^3*ln(exp(b*x+a)+1)*x+3*polylog(2,-exp(b*x+ a))/b^4+3/b^3*ln(1-exp(b*x+a))*x+3/b^4*ln(1-exp(b*x+a))*a+3*polylog(2,exp( b*x+a))/b^4-3/b^4*a*ln(exp(b*x+a)-1)+6/b^4*a*ln(exp(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 979 vs. \(2 (74) = 148\).
Time = 0.27 (sec) , antiderivative size = 979, normalized size of antiderivative = 11.80 \[ \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\text {Too large to display} \]
-(3*(b^2*x^2 - a^2)*cosh(b*x + a)^4 + 12*(b^2*x^2 - a^2)*cosh(b*x + a)*sin h(b*x + a)^3 + 3*(b^2*x^2 - a^2)*sinh(b*x + a)^4 + (2*b^3*x^3 - 3*b^2*x^2 + 6*a^2)*cosh(b*x + a)^2 + (2*b^3*x^3 - 3*b^2*x^2 + 18*(b^2*x^2 - a^2)*cos h(b*x + a)^2 + 6*a^2)*sinh(b*x + a)^2 - 3*a^2 - 3*(cosh(b*x + a)^4 + 4*cos h(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*s inh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b*x + a))*s inh(b*x + a) + 1)*dilog(cosh(b*x + a) + sinh(b*x + a)) - 3*(cosh(b*x + a)^ 4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a) ^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b* x + a))*sinh(b*x + a) + 1)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 3*(b*x* cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^ 4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^ 2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*log(c osh(b*x + a) + sinh(b*x + a) + 1) + 3*(a*cosh(b*x + a)^4 + 4*a*cosh(b*x + a)*sinh(b*x + a)^3 + a*sinh(b*x + a)^4 - 2*a*cosh(b*x + a)^2 + 2*(3*a*cosh (b*x + a)^2 - a)*sinh(b*x + a)^2 + 4*(a*cosh(b*x + a)^3 - a*cosh(b*x + a)) *sinh(b*x + a) + a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 3*((b*x + a)* cosh(b*x + a)^4 + 4*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^3 + (b*x + a)*si nh(b*x + a)^4 - 2*(b*x + a)*cosh(b*x + a)^2 + 2*(3*(b*x + a)*cosh(b*x + a) ^2 - b*x - a)*sinh(b*x + a)^2 + b*x + 4*((b*x + a)*cosh(b*x + a)^3 - (b...
Timed out. \[ \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\text {Timed out} \]
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.57 \[ \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\frac {3 \, x^{2} - {\left (2 \, b x^{3} e^{\left (2 \, a\right )} + 3 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {3 \, x^{2}}{b^{2}} + \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \]
(3*x^2 - (2*b*x^3*e^(2*a) + 3*x^2*e^(2*a))*e^(2*b*x))/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^(2*b*x + 2*a) + b^2) - 3*x^2/b^2 + 3*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^4 + 3*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^4
\[ \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\int { x^{3} \cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int x^3 \coth (a+b x) \text {csch}^2(a+b x) \, dx=\int \frac {x^3\,\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]