Integrand size = 10, antiderivative size = 82 \[ \int x \coth ^3(a+b x) \, dx=\frac {x}{2 b}-\frac {x^2}{2}-\frac {\coth (a+b x)}{2 b^2}-\frac {x \coth ^2(a+b x)}{2 b}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2} \]
1/2*x/b-1/2*x^2-1/2*coth(b*x+a)/b^2-1/2*x*coth(b*x+a)^2/b+x*ln(1-exp(2*b*x +2*a))/b+1/2*polylog(2,exp(2*b*x+2*a))/b^2
Time = 1.53 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.60 \[ \int x \coth ^3(a+b x) \, dx=\frac {1}{2} \left (-\frac {2 x^2}{-1+e^{2 a}}+x^2 \coth (a)-\frac {x \text {csch}^2(a+b x)}{b}+\frac {2 x \log \left (1-e^{-a-b x}\right )}{b}+\frac {2 x \log \left (1+e^{-a-b x}\right )}{b}-\frac {2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )}{b^2}-\frac {2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )}{b^2}+\frac {\text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^2}\right ) \]
((-2*x^2)/(-1 + E^(2*a)) + x^2*Coth[a] - (x*Csch[a + b*x]^2)/b + (2*x*Log[ 1 - E^(-a - b*x)])/b + (2*x*Log[1 + E^(-a - b*x)])/b - (2*PolyLog[2, -E^(- a - b*x)])/b^2 - (2*PolyLog[2, E^(-a - b*x)])/b^2 + (Csch[a]*Csch[a + b*x] *Sinh[b*x])/b^2)/2
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.37, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3042, 26, 4203, 25, 26, 3042, 25, 26, 3954, 24, 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.
\(\displaystyle \int x \coth ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i x \tan \left (i a+i b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int x \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^3dx\) |
\(\Big \downarrow \) 4203 |
\(\displaystyle i \left (\frac {i \int -\coth ^2(a+b x)dx}{2 b}-\int i x \coth (a+b x)dx+\frac {i x \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (-\frac {i \int \coth ^2(a+b x)dx}{2 b}-\int i x \coth (a+b x)dx+\frac {i x \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (-\frac {i \int \coth ^2(a+b x)dx}{2 b}-i \int x \coth (a+b x)dx+\frac {i x \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle i \left (-i \int -i x \tan \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {i \int -\tan \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{2 b}+\frac {i x \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle i \left (-i \int -i x \tan \left (i a+i b x+\frac {\pi }{2}\right )dx+\frac {i \int \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx}{2 b}+\frac {i x \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \left (-\int x \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx+\frac {i \int \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx}{2 b}+\frac {i x \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 3954 |
\(\displaystyle i \left (-\int x \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx+\frac {i \left (\frac {\coth (a+b x)}{b}-\int 1dx\right )}{2 b}+\frac {i x \coth ^2(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle i \left (-\int x \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx+\frac {i x \coth ^2(a+b x)}{2 b}+\frac {i \left (\frac {\coth (a+b x)}{b}-x\right )}{2 b}\right )\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle i \left (-2 i \int \frac {e^{2 a+2 b x-i \pi } x}{1+e^{2 a+2 b x-i \pi }}dx+\frac {i x \coth ^2(a+b x)}{2 b}+\frac {i \left (\frac {\coth (a+b x)}{b}-x\right )}{2 b}+\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle i \left (-2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{2 b}\right )+\frac {i x \coth ^2(a+b x)}{2 b}+\frac {i \left (\frac {\coth (a+b x)}{b}-x\right )}{2 b}+\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle i \left (-2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int e^{-2 a-2 b x+i \pi } \log \left (1+e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}\right )+\frac {i x \coth ^2(a+b x)}{2 b}+\frac {i \left (\frac {\coth (a+b x)}{b}-x\right )}{2 b}+\frac {i x^2}{2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle i \left (-2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )+\frac {i x \coth ^2(a+b x)}{2 b}+\frac {i \left (\frac {\coth (a+b x)}{b}-x\right )}{2 b}+\frac {i x^2}{2}\right )\) |
I*((I/2)*x^2 + ((I/2)*x*Coth[a + b*x]^2)/b + ((I/2)*(-x + Coth[a + b*x]/b) )/b - (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)))
3.1.13.3.1 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[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 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[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(163\) vs. \(2(72)=144\).
Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.00
method | result | size |
risch | \(-\frac {x^{2}}{2}-\frac {2 \,{\mathrm e}^{2 b x +2 a} b x +{\mathrm e}^{2 b x +2 a}-1}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(164\) |
-1/2*x^2-(2*exp(2*b*x+2*a)*b*x+exp(2*b*x+2*a)-1)/b^2/(exp(2*b*x+2*a)-1)^2- 2/b*a*x-a^2/b^2+1/b*ln(exp(b*x+a)+1)*x+1/b^2*polylog(2,-exp(b*x+a))+1/b*ln (1-exp(b*x+a))*x+1/b^2*ln(1-exp(b*x+a))*a+1/b^2*polylog(2,exp(b*x+a))-1/b^ 2*a*ln(exp(b*x+a)-1)+2/b^2*a*ln(exp(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (71) = 142\).
Time = 0.32 (sec) , antiderivative size = 975, normalized size of antiderivative = 11.89 \[ \int x \coth ^3(a+b x) \, dx=\text {Too large to display} \]
-1/2*((b^2*x^2 - 2*a^2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 2*a^2)*cosh(b*x + a )*sinh(b*x + a)^3 + (b^2*x^2 - 2*a^2)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x ^2 - 2*a^2 - 2*b*x - 1)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 - 2*a^2) *cosh(b*x + a)^2 - 2*a^2 - 2*b*x - 1)*sinh(b*x + a)^2 - 2*a^2 - 2*(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)) - 2 *(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)) - 2*(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(cosh(b*x + a) + sinh(b*x + a) + 1) + 2*(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) - 2*((b*x + a)*cosh(b*x + a)^4 + 4*(b*x + a)*cosh(b*x + a)*sinh(b*x + a)^ 3 + (b*x + a)*sinh(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...
\[ \int x \coth ^3(a+b x) \, dx=\int x \coth ^{3}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (71) = 142\).
Time = 0.24 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.82 \[ \int x \coth ^3(a+b x) \, dx=-x^{2} + \frac {b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{2} - 2 \, {\left (b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 2}{2 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} + \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
-x^2 + 1/2*(b^2*x^2*e^(4*b*x + 4*a) + b^2*x^2 - 2*(b^2*x^2*e^(2*a) + 2*b*x *e^(2*a) + e^(2*a))*e^(2*b*x) + 2)/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^(2*b*x + 2*a) + b^2) + (b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^2 + (b*x *log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^2
\[ \int x \coth ^3(a+b x) \, dx=\int { x \coth \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int x \coth ^3(a+b x) \, dx=\int x\,{\mathrm {coth}\left (a+b\,x\right )}^3 \,d x \]