Integrand size = 13, antiderivative size = 36 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=-\frac {3}{8} \text {arctanh}(\cosh (x))+\frac {1}{5} i \coth ^5(x)-\frac {3}{8} \coth (x) \text {csch}(x)-\frac {1}{4} \coth ^3(x) \text {csch}(x) \] Output:
-3/8*arctanh(cosh(x))+1/5*I*coth(x)^5-3/8*coth(x)*csch(x)-1/4*coth(x)^3*cs ch(x)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(36)=72\).
Time = 0.07 (sec) , antiderivative size = 175, normalized size of antiderivative = 4.86 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {1}{10} i \coth \left (\frac {x}{2}\right )-\frac {5}{32} \text {csch}^2\left (\frac {x}{2}\right )+\frac {7}{160} i \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{64} \text {csch}^4\left (\frac {x}{2}\right )+\frac {1}{160} i \coth \left (\frac {x}{2}\right ) \text {csch}^4\left (\frac {x}{2}\right )-\frac {3}{8} \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {3}{8} \log \left (\sinh \left (\frac {x}{2}\right )\right )-\frac {5}{32} \text {sech}^2\left (\frac {x}{2}\right )+\frac {1}{64} \text {sech}^4\left (\frac {x}{2}\right )+\frac {1}{10} i \tanh \left (\frac {x}{2}\right )-\frac {7}{160} i \text {sech}^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right )+\frac {1}{160} i \text {sech}^4\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \] Input:
Integrate[Coth[x]^6/(I + Sinh[x]),x]
Output:
(I/10)*Coth[x/2] - (5*Csch[x/2]^2)/32 + ((7*I)/160)*Coth[x/2]*Csch[x/2]^2 - Csch[x/2]^4/64 + (I/160)*Coth[x/2]*Csch[x/2]^4 - (3*Log[Cosh[x/2]])/8 + (3*Log[Sinh[x/2]])/8 - (5*Sech[x/2]^2)/32 + Sech[x/2]^4/64 + (I/10)*Tanh[x /2] - ((7*I)/160)*Sech[x/2]^2*Tanh[x/2] + (I/160)*Sech[x/2]^4*Tanh[x/2]
Time = 0.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.538, Rules used = {3042, 25, 26, 3185, 25, 26, 3042, 25, 26, 3087, 15, 3091, 26, 3042, 26, 3091, 26, 3042, 26, 4257}
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.
| \(\displaystyle \int \frac {\coth ^6(x)}{\sinh (x)+i} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{(i-i \sin (i x)) \tan (i x)^6}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {i}{(1-\sin (i x)) \tan (i x)^6}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{(1-\sin (i x)) \tan (i x)^6}dx\) |
| \(\Big \downarrow \) 3185 |
| \(\displaystyle i \left (\int -\coth ^4(x) \text {csch}^2(x)dx+\int -i \coth ^4(x) \text {csch}(x)dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\int \coth ^4(x) \text {csch}^2(x)dx+\int -i \coth ^4(x) \text {csch}(x)dx\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\int \coth ^4(x) \text {csch}^2(x)dx-i \int \coth ^4(x) \text {csch}(x)dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-i \int i \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx-\int -\sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx-i \int i \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx+\int \sec \left (i x-\frac {\pi }{2}\right )^2 \tan \left (i x-\frac {\pi }{2}\right )^4dx\right )\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle i \left (\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx-i \int \coth ^4(x)d(i \coth (x))\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {\coth ^5(x)}{5}+\int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^4dx\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle i \left (-\frac {3}{4} \int i \coth ^2(x) \text {csch}(x)dx+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {3}{4} i \int \coth ^2(x) \text {csch}(x)dx+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {3}{4} i \int -i \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {3}{4} \int \sec \left (i x-\frac {\pi }{2}\right ) \tan \left (i x-\frac {\pi }{2}\right )^2dx+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle i \left (-\frac {3}{4} \left (-\frac {1}{2} \int -i \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {3}{4} \left (\frac {1}{2} i \int \text {csch}(x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {3}{4} \left (\frac {1}{2} i \int i \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (i x)dx-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle i \left (-\frac {3}{4} \left (-\frac {1}{2} i \text {arctanh}(\cosh (x))-\frac {1}{2} i \coth (x) \text {csch}(x)\right )+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth ^3(x) \text {csch}(x)\right )\) |
Input:
Int[Coth[x]^6/(I + Sinh[x]),x]
Output:
I*(Coth[x]^5/5 + (I/4)*Coth[x]^3*Csch[x] - (3*((-1/2*I)*ArcTanh[Cosh[x]] - (I/2)*Coth[x]*Csch[x]))/4)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)]), x_Symbol] :> Simp[1/a Int[Sec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x ] - Simp[1/(b*g) Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /; Fre eQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (27 ) = 54\).
Time = 12.51 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.81
| method | result | size |
| risch | \(-\frac {-40 i {\mathrm e}^{8 x}+25 \,{\mathrm e}^{9 x}-10 \,{\mathrm e}^{7 x}-80 i {\mathrm e}^{4 x}+10 \,{\mathrm e}^{3 x}-8 i-25 \,{\mathrm e}^{x}}{20 \left ({\mathrm e}^{2 x}-1\right )^{5}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8}\) | \(65\) |
| default | \(\frac {i \tanh \left (\frac {x}{2}\right )}{16}+\frac {i \tanh \left (\frac {x}{2}\right )^{5}}{160}+\frac {\tanh \left (\frac {x}{2}\right )^{4}}{64}+\frac {i \tanh \left (\frac {x}{2}\right )^{3}}{32}+\frac {\tanh \left (\frac {x}{2}\right )^{2}}{8}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {i}{32 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {i}{160 \tanh \left (\frac {x}{2}\right )^{5}}-\frac {1}{64 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {i}{16 \tanh \left (\frac {x}{2}\right )}\) | \(93\) |
Input:
int(coth(x)^6/(I+sinh(x)),x,method=_RETURNVERBOSE)
Output:
-1/20*(-40*I*exp(x)^8+25*exp(x)^9-10*exp(x)^7-80*I*exp(x)^4+10*exp(x)^3-8* I-25*exp(x))/(exp(x)^2-1)^5-3/8*ln(exp(x)+1)+3/8*ln(exp(x)-1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (26) = 52\).
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.00 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=-\frac {15 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} + 1\right ) - 15 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )} \log \left (e^{x} - 1\right ) + 50 \, e^{\left (9 \, x\right )} - 80 i \, e^{\left (8 \, x\right )} - 20 \, e^{\left (7 \, x\right )} - 160 i \, e^{\left (4 \, x\right )} + 20 \, e^{\left (3 \, x\right )} - 50 \, e^{x} - 16 i}{40 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}} \] Input:
integrate(coth(x)^6/(I+sinh(x)),x, algorithm="fricas")
Output:
-1/40*(15*(e^(10*x) - 5*e^(8*x) + 10*e^(6*x) - 10*e^(4*x) + 5*e^(2*x) - 1) *log(e^x + 1) - 15*(e^(10*x) - 5*e^(8*x) + 10*e^(6*x) - 10*e^(4*x) + 5*e^( 2*x) - 1)*log(e^x - 1) + 50*e^(9*x) - 80*I*e^(8*x) - 20*e^(7*x) - 160*I*e^ (4*x) + 20*e^(3*x) - 50*e^x - 16*I)/(e^(10*x) - 5*e^(8*x) + 10*e^(6*x) - 1 0*e^(4*x) + 5*e^(2*x) - 1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (36) = 72\).
Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.78 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {3 \log {\left (e^{x} - 1 \right )}}{8} - \frac {3 \log {\left (e^{x} + 1 \right )}}{8} + \frac {- 25 e^{9 x} + 40 i e^{8 x} + 10 e^{7 x} + 80 i e^{4 x} - 10 e^{3 x} + 25 e^{x} + 8 i}{20 e^{10 x} - 100 e^{8 x} + 200 e^{6 x} - 200 e^{4 x} + 100 e^{2 x} - 20} \] Input:
integrate(coth(x)**6/(I+sinh(x)),x)
Output:
3*log(exp(x) - 1)/8 - 3*log(exp(x) + 1)/8 + (-25*exp(9*x) + 40*I*exp(8*x) + 10*exp(7*x) + 80*I*exp(4*x) - 10*exp(3*x) + 25*exp(x) + 8*I)/(20*exp(10* x) - 100*exp(8*x) + 200*exp(6*x) - 200*exp(4*x) + 100*exp(2*x) - 20)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (26) = 52\).
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.53 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {25 \, e^{\left (-x\right )} - 10 \, e^{\left (-3 \, x\right )} - 80 i \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-7 \, x\right )} - 40 i \, e^{\left (-8 \, x\right )} - 25 \, e^{\left (-9 \, x\right )} - 8 i}{20 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} - \frac {3}{8} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{8} \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate(coth(x)^6/(I+sinh(x)),x, algorithm="maxima")
Output:
1/20*(25*e^(-x) - 10*e^(-3*x) - 80*I*e^(-4*x) + 10*e^(-7*x) - 40*I*e^(-8*x ) - 25*e^(-9*x) - 8*I)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x ) + e^(-10*x) - 1) - 3/8*log(e^(-x) + 1) + 3/8*log(e^(-x) - 1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.72 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=-\frac {25 \, e^{\left (9 \, x\right )} - 40 i \, e^{\left (8 \, x\right )} - 10 \, e^{\left (7 \, x\right )} - 80 i \, e^{\left (4 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 25 \, e^{x} - 8 i}{20 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} - \frac {3}{8} \, \log \left (e^{x} + 1\right ) + \frac {3}{8} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate(coth(x)^6/(I+sinh(x)),x, algorithm="giac")
Output:
-1/20*(25*e^(9*x) - 40*I*e^(8*x) - 10*e^(7*x) - 80*I*e^(4*x) + 10*e^(3*x) - 25*e^x - 8*I)/(e^(2*x) - 1)^5 - 3/8*log(e^x + 1) + 3/8*log(abs(e^x - 1))
Time = 1.80 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.44 \[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\frac {3\,\ln \left (\frac {3}{4}-\frac {3\,{\mathrm {e}}^x}{4}\right )}{8}-\frac {3\,\ln \left (\frac {3\,{\mathrm {e}}^x}{4}+\frac {3}{4}\right )}{8}-\frac {5\,{\mathrm {e}}^x}{4\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {9\,{\mathrm {e}}^x}{2\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {6\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^3}-\frac {4\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^4}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {8{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {16{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^3}+\frac {16{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^4}+\frac {32{}\mathrm {i}}{5\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5} \] Input:
int(coth(x)^6/(sinh(x) + 1i),x)
Output:
(3*log(3/4 - (3*exp(x))/4))/8 - (3*log((3*exp(x))/4 + 3/4))/8 - (5*exp(x)) /(4*(exp(2*x) - 1)) - (9*exp(x))/(2*(exp(2*x) - 1)^2) - (6*exp(x))/(exp(2* x) - 1)^3 - (4*exp(x))/(exp(2*x) - 1)^4 + 2i/(exp(2*x) - 1) + 8i/(exp(2*x) - 1)^2 + 16i/(exp(2*x) - 1)^3 + 16i/(exp(2*x) - 1)^4 + 32i/(5*(exp(2*x) - 1)^5)
\[ \int \frac {\coth ^6(x)}{i+\sinh (x)} \, dx=\int \frac {\coth \left (x \right )^{6}}{\sinh \left (x \right )+i}d x \] Input:
int(coth(x)^6/(I+sinh(x)),x)
Output:
int(coth(x)**6/(sinh(x) + i),x)