Integrand size = 16, antiderivative size = 88 \[ \int x \cosh ^2(a+b x) \coth (a+b x) \, dx=\frac {x}{4 b}-\frac {x^2}{2}+\frac {x \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {\operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {\cosh (a+b x) \sinh (a+b x)}{4 b^2}+\frac {x \sinh ^2(a+b x)}{2 b} \]
1/4*x/b-1/2*x^2+x*ln(1-exp(2*b*x+2*a))/b+1/2*polylog(2,exp(2*b*x+2*a))/b^2 -1/4*cosh(b*x+a)*sinh(b*x+a)/b^2+1/2*x*sinh(b*x+a)^2/b
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.82 \[ \int x \cosh ^2(a+b x) \coth (a+b x) \, dx=-\frac {4 a^2-4 b^2 x^2-2 b x \cosh (2 (a+b x))-8 b x \log \left (1-e^{-2 (a+b x)}\right )+4 \operatorname {PolyLog}\left (2,e^{-2 (a+b x)}\right )+\sinh (2 (a+b x))}{8 b^2} \]
-1/8*(4*a^2 - 4*b^2*x^2 - 2*b*x*Cosh[2*(a + b*x)] - 8*b*x*Log[1 - E^(-2*(a + b*x))] + 4*PolyLog[2, E^(-2*(a + b*x))] + Sinh[2*(a + b*x)])/b^2
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5973, 3042, 26, 4201, 2620, 2715, 2838, 5895, 3042, 25, 3115, 24}
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.
| \(\displaystyle \int x \cosh ^2(a+b x) \coth (a+b x) \, dx\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \int x \coth (a+b x)dx+\int x \cosh (a+b x) \sinh (a+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x \cosh (a+b x) \sinh (a+b x)dx+\int -i x \tan \left (i a+i b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x \cosh (a+b x) \sinh (a+b x)dx-i \int x \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \int x \cosh (a+b x) \sinh (a+b x)dx-i \left (2 i \int \frac {e^{2 a+2 b x-i \pi } x}{1+e^{2 a+2 b x-i \pi }}dx-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \int x \cosh (a+b x) \sinh (a+b x)dx-i \left (2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{2 b}\right )-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \int x \cosh (a+b x) \sinh (a+b x)dx-i \left (2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int e^{-2 a-2 b x+i \pi } \log \left (1+e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}\right )-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \int x \cosh (a+b x) \sinh (a+b x)dx-i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )\) |
| \(\Big \downarrow \) 5895 |
| \(\displaystyle -\frac {\int \sinh ^2(a+b x)dx}{2 b}-i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )+\frac {x \sinh ^2(a+b x)}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int -\sin (i a+i b x)^2dx}{2 b}-i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )+\frac {x \sinh ^2(a+b x)}{2 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \sin (i a+i b x)^2dx}{2 b}-i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )+\frac {x \sinh ^2(a+b x)}{2 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {\int 1dx}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b}-i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )+\frac {x \sinh ^2(a+b x)}{2 b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )+\frac {x \sinh ^2(a+b x)}{2 b}+\frac {\frac {x}{2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b}\) |
(-I)*((-1/2*I)*x^2 + (2*I)*((x*Log[1 + E^(2*a - I*Pi + 2*b*x)])/(2*b) + Po lyLog[2, -E^(2*a - I*Pi + 2*b*x)]/(4*b^2))) + (x*Sinh[a + b*x]^2)/(2*b) + (x/2 - (Cosh[a + b*x]*Sinh[a + b*x])/(2*b))/(2*b)
3.5.14.3.1 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_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) ]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*Sinh[a + b*x^n]^ (p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(161\) vs. \(2(78)=156\).
Time = 1.02 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(-\frac {x^{2}}{2}+\frac {\left (2 b x -1\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{2}}+\frac {\left (2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{2}}-\frac {2 a x}{b}-\frac {a^{2}}{b^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}+\frac {\operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}+\frac {\operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\) | \(162\) |
-1/2*x^2+1/16*(2*b*x-1)/b^2*exp(2*b*x+2*a)+1/16*(2*b*x+1)/b^2*exp(-2*b*x-2 *a)-2/b*a*x-a^2/b^2+1/b*ln(exp(b*x+a)+1)*x+polylog(2,-exp(b*x+a))/b^2+1/b* ln(1-exp(b*x+a))*x+1/b^2*ln(1-exp(b*x+a))*a+polylog(2,exp(b*x+a))/b^2-1/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. 488 vs. \(2 (77) = 154\).
Time = 0.27 (sec) , antiderivative size = 488, normalized size of antiderivative = 5.55 \[ \int x \cosh ^2(a+b x) \coth (a+b x) \, dx=\frac {{\left (2 \, b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (2 \, b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (2 \, b x - 1\right )} \sinh \left (b x + a\right )^{4} - 8 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (4 \, b^{2} x^{2} - 3 \, {\left (2 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} - 8 \, a^{2}\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x + 16 \, {\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}\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 16 \, {\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}\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 16 \, {\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}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 16 \, {\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 16 \, {\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b x + a\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 4 \, {\left ({\left (2 \, b x - 1\right )} \cosh \left (b x + a\right )^{3} - 4 \, {\left (b^{2} x^{2} - 2 \, a^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{16 \, {\left (b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2}\right )}} \]
1/16*((2*b*x - 1)*cosh(b*x + a)^4 + 4*(2*b*x - 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (2*b*x - 1)*sinh(b*x + a)^4 - 8*(b^2*x^2 - 2*a^2)*cosh(b*x + a)^2 - 2*(4*b^2*x^2 - 3*(2*b*x - 1)*cosh(b*x + a)^2 - 8*a^2)*sinh(b*x + a)^2 + 2*b*x + 16*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a )^2)*dilog(cosh(b*x + a) + sinh(b*x + a)) + 16*(cosh(b*x + a)^2 + 2*cosh(b *x + a)*sinh(b*x + a) + sinh(b*x + a)^2)*dilog(-cosh(b*x + a) - sinh(b*x + a)) + 16*(b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*s inh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 16*(a*cosh(b*x + a)^2 + 2*a*cosh(b*x + a)*sinh(b*x + a) + a*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 16*((b*x + a)*cosh(b*x + a)^2 + 2*(b*x + a)*cos h(b*x + a)*sinh(b*x + a) + (b*x + a)*sinh(b*x + a)^2)*log(-cosh(b*x + a) - sinh(b*x + a) + 1) + 4*((2*b*x - 1)*cosh(b*x + a)^3 - 4*(b^2*x^2 - 2*a^2) *cosh(b*x + a))*sinh(b*x + a) + 1)/(b^2*cosh(b*x + a)^2 + 2*b^2*cosh(b*x + a)*sinh(b*x + a) + b^2*sinh(b*x + a)^2)
\[ \int x \cosh ^2(a+b x) \coth (a+b x) \, dx=\int x \cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28 \[ \int x \cosh ^2(a+b x) \coth (a+b x) \, dx=-x^{2} + \frac {{\left (8 \, b^{2} x^{2} e^{\left (2 \, a\right )} + {\left (2 \, b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} + {\left (2 \, b x + 1\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{16 \, b^{2}} + \frac {b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac {b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \]
-x^2 + 1/16*(8*b^2*x^2*e^(2*a) + (2*b*x*e^(4*a) - e^(4*a))*e^(2*b*x) + (2* b*x + 1)*e^(-2*b*x))*e^(-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
\[ \int x \cosh ^2(a+b x) \coth (a+b x) \, dx=\int { x \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x \cosh ^2(a+b x) \coth (a+b x) \, dx=\int \frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]