Integrand size = 13, antiderivative size = 58 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {15 i x}{8}-4 \cosh (x)+\frac {4 \cosh ^3(x)}{3}+\frac {15}{8} i \cosh (x) \sinh (x)-\frac {5}{4} i \cosh (x) \sinh ^3(x)-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)} \] Output:
-15/8*I*x-4*cosh(x)+4/3*cosh(x)^3+15/8*I*cosh(x)*sinh(x)-5/4*I*cosh(x)*sin h(x)^3-cosh(x)*sinh(x)^3/(I+csch(x))
Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {1}{96} \left (-180 i x-168 \cosh (x)+8 \cosh (3 x)+\frac {192 \sinh \left (\frac {x}{2}\right )}{-i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}+48 i \sinh (2 x)-3 i \sinh (4 x)\right ) \] Input:
Integrate[Sinh[x]^4/(I + Csch[x]),x]
Output:
((-180*I)*x - 168*Cosh[x] + 8*Cosh[3*x] + (192*Sinh[x/2])/((-I)*Cosh[x/2] + Sinh[x/2]) + (48*I)*Sinh[2*x] - (3*I)*Sinh[4*x])/96
Time = 0.47 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.231, Rules used = {3042, 4306, 25, 3042, 4274, 26, 3042, 26, 3113, 2009, 3115, 25, 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.
| \(\displaystyle \int \frac {\sinh ^4(x)}{\text {csch}(x)+i} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(i \csc (i x)+i) \csc (i x)^4}dx\) |
| \(\Big \downarrow \) 4306 |
| \(\displaystyle \int -\left ((5 i-4 \text {csch}(x)) \sinh ^4(x)\right )dx-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int (5 i-4 \text {csch}(x)) \sinh ^4(x)dx-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {5 i-4 i \csc (i x)}{\csc (i x)^4}dx-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle -5 i \int \sinh ^4(x)dx+4 i \int -i \sinh ^3(x)dx-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -5 i \int \sinh ^4(x)dx+4 \int \sinh ^3(x)dx-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 \int i \sin (i x)^3dx-5 i \int \sin (i x)^4dx-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle 4 i \int \sin (i x)^3dx-5 i \int \sin (i x)^4dx-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -5 i \int \sin (i x)^4dx-4 \int \left (1-\cosh ^2(x)\right )d\cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 i \int \sin (i x)^4dx-4 \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -5 i \left (\frac {3}{4} \int -\sinh ^2(x)dx+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-4 \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 i \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int \sinh ^2(x)dx\right )-4 \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -5 i \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int -\sin (i x)^2dx\right )-4 \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -5 i \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \int \sin (i x)^2dx\right )-4 \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -5 i \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-4 \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -4 \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )-5 i \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )\right )-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i}\) |
Input:
Int[Sinh[x]^4/(I + Csch[x]),x]
Output:
-4*(Cosh[x] - Cosh[x]^3/3) - (Cosh[x]*Sinh[x]^3)/(I + Csch[x]) - (5*I)*((C osh[x]*Sinh[x]^3)/4 + (3*(x/2 - (Cosh[x]*Sinh[x])/2))/4)
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[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(a + b*Csc[e + f*x]))), x] - Simp[1/a^2 Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0 ]
Time = 1.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(-\frac {15 i x}{8}-\frac {i {\mathrm e}^{4 x}}{64}+\frac {{\mathrm e}^{3 x}}{24}+\frac {i {\mathrm e}^{2 x}}{4}-\frac {7 \,{\mathrm e}^{x}}{8}-\frac {7 \,{\mathrm e}^{-x}}{8}-\frac {i {\mathrm e}^{-2 x}}{4}+\frac {{\mathrm e}^{-3 x}}{24}+\frac {i {\mathrm e}^{-4 x}}{64}-\frac {2}{{\mathrm e}^{x}-i}\) | \(65\) |
| default | \(-\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {15 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {\frac {3}{2}+\frac {7 i}{8}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {-\frac {1}{2}+\frac {5 i}{8}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {-\frac {1}{3}-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )-i}-\frac {15 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {\frac {1}{3}-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {-\frac {1}{2}-\frac {5 i}{8}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {-\frac {3}{2}+\frac {7 i}{8}}{\tanh \left (\frac {x}{2}\right )+1}\) | \(128\) |
| parallelrisch | \(\frac {\left (-360 i \sinh \left (\frac {x}{2}\right )-360 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )+\left (360 i \sinh \left (\frac {x}{2}\right )+360 \cosh \left (\frac {x}{2}\right )\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-432 i \cosh \left (\frac {x}{2}\right )-120 i \cosh \left (\frac {3 x}{2}\right )-40 i \cosh \left (\frac {5 x}{2}\right )+5 i \cosh \left (\frac {7 x}{2}\right )+3 i \cosh \left (\frac {9 x}{2}\right )-5 \sinh \left (\frac {7 x}{2}\right )+3 \sinh \left (\frac {9 x}{2}\right )-288 \sinh \left (\frac {x}{2}\right )+120 \sinh \left (\frac {3 x}{2}\right )-40 \sinh \left (\frac {5 x}{2}\right )}{192 i \cosh \left (\frac {x}{2}\right )-192 \sinh \left (\frac {x}{2}\right )}\) | \(128\) |
Input:
int(sinh(x)^4/(I+csch(x)),x,method=_RETURNVERBOSE)
Output:
-15/8*I*x-1/64*I*exp(x)^4+1/24*exp(x)^3+1/4*I*exp(x)^2-7/8*exp(x)-7/8/exp( x)-1/4*I/exp(x)^2+1/24/exp(x)^3+1/64*I/exp(x)^4-2/(exp(x)-I)
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.36 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {24 \, {\left (15 i \, x - 7 i\right )} e^{\left (5 \, x\right )} + 24 \, {\left (15 \, x + 23\right )} e^{\left (4 \, x\right )} + 3 i \, e^{\left (9 \, x\right )} - 5 \, e^{\left (8 \, x\right )} - 40 i \, e^{\left (7 \, x\right )} + 120 \, e^{\left (6 \, x\right )} - 120 i \, e^{\left (3 \, x\right )} + 40 \, e^{\left (2 \, x\right )} + 5 i \, e^{x} - 3}{192 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )}\right )}} \] Input:
integrate(sinh(x)^4/(I+csch(x)),x, algorithm="fricas")
Output:
-1/192*(24*(15*I*x - 7*I)*e^(5*x) + 24*(15*x + 23)*e^(4*x) + 3*I*e^(9*x) - 5*e^(8*x) - 40*I*e^(7*x) + 120*e^(6*x) - 120*I*e^(3*x) + 40*e^(2*x) + 5*I *e^x - 3)/(e^(5*x) - I*e^(4*x))
\[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh ^{4}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \] Input:
integrate(sinh(x)**4/(I+csch(x)),x)
Output:
Integral(sinh(x)**4/(csch(x) + I), x)
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {15}{8} i \, x - \frac {-5 i \, e^{\left (-x\right )} + 40 \, e^{\left (-2 \, x\right )} + 120 i \, e^{\left (-3 \, x\right )} + 552 \, e^{\left (-4 \, x\right )} - 3}{16 \, {\left (12 i \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )}\right )}} - \frac {7}{8} \, e^{\left (-x\right )} - \frac {1}{4} i \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{64} i \, e^{\left (-4 \, x\right )} \] Input:
integrate(sinh(x)^4/(I+csch(x)),x, algorithm="maxima")
Output:
-15/8*I*x - 1/16*(-5*I*e^(-x) + 40*e^(-2*x) + 120*I*e^(-3*x) + 552*e^(-4*x ) - 3)/(12*I*e^(-4*x) + 12*e^(-5*x)) - 7/8*e^(-x) - 1/4*I*e^(-2*x) + 1/24* e^(-3*x) + 1/64*I*e^(-4*x)
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {15}{8} i \, x - \frac {{\left (552 \, e^{\left (4 \, x\right )} - 120 i \, e^{\left (3 \, x\right )} + 40 \, e^{\left (2 \, x\right )} + 5 i \, e^{x} - 3\right )} e^{\left (-4 \, x\right )}}{192 \, {\left (e^{x} - i\right )}} - \frac {1}{64} i \, e^{\left (4 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{4} i \, e^{\left (2 \, x\right )} - \frac {7}{8} \, e^{x} \] Input:
integrate(sinh(x)^4/(I+csch(x)),x, algorithm="giac")
Output:
-15/8*I*x - 1/192*(552*e^(4*x) - 120*I*e^(3*x) + 40*e^(2*x) + 5*I*e^x - 3) *e^(-4*x)/(e^x - I) - 1/64*I*e^(4*x) + 1/24*e^(3*x) + 1/4*I*e^(2*x) - 7/8* e^x
Time = 2.61 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {7\,{\mathrm {e}}^{-x}}{8}-\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{4}-\frac {x\,15{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {{\mathrm {e}}^{-4\,x}\,1{}\mathrm {i}}{64}-\frac {{\mathrm {e}}^{4\,x}\,1{}\mathrm {i}}{64}-\frac {7\,{\mathrm {e}}^x}{8}-\frac {2}{{\mathrm {e}}^x-\mathrm {i}} \] Input:
int(sinh(x)^4/(1/sinh(x) + 1i),x)
Output:
(exp(2*x)*1i)/4 - (7*exp(-x))/8 - (exp(-2*x)*1i)/4 - (x*15i)/8 + exp(-3*x) /24 + exp(3*x)/24 + (exp(-4*x)*1i)/64 - (exp(4*x)*1i)/64 - (7*exp(x))/8 - 2/(exp(x) - 1i)
\[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh \left (x \right )^{4}}{\mathrm {csch}\left (x \right )+i}d x \] Input:
int(sinh(x)^4/(I+csch(x)),x)
Output:
int(sinh(x)**4/(csch(x) + i),x)