\(\int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx\) [62]

Optimal result

Integrand size = 13, antiderivative size = 58 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {15 i x}{8}-4 \cosh (x)+\frac {4 \cosh ^3(x)}{3}+\frac {15}{8} i \cosh (x) \sinh (x)-\frac {5}{4} i \cosh (x) \sinh ^3(x)-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)} \]

[Out]

-15/8*I*x-4*cosh(x)+4/3*cosh(x)^3+15/8*I*cosh(x)*sinh(x)-5/4*I*cosh(x)*sinh(x)^3-cosh(x)*sinh(x)^3/(I+csch(x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2715, 8, 2713} \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {15 i x}{8}+\frac {4 \cosh ^3(x)}{3}-4 \cosh (x)-\frac {5}{4} i \sinh ^3(x) \cosh (x)+\frac {15}{8} i \sinh (x) \cosh (x)-\frac {\sinh ^3(x) \cosh (x)}{\text {csch}(x)+i} \]

[In]

Int[Sinh[x]^4/(I + Csch[x]),x]

[Out]

((-15*I)/8)*x - 4*Cosh[x] + (4*Cosh[x]^3)/3 + ((15*I)/8)*Cosh[x]*Sinh[x] - ((5*I)/4)*Cosh[x]*Sinh[x]^3 - (Cosh
[x]*Sinh[x]^3)/(I + Csch[x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3872

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3904

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}+\int (-5 i+4 \text {csch}(x)) \sinh ^4(x) \, dx \\ & = -\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}-5 i \int \sinh ^4(x) \, dx+4 \int \sinh ^3(x) \, dx \\ & = -\frac {5}{4} i \cosh (x) \sinh ^3(x)-\frac {\cosh (x) \sinh ^3(x)}{i+\text {csch}(x)}+\frac {15}{4} i \int \sinh ^2(x) \, dx-4 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right ) \\ & = -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)}-\frac {15}{8} i \int 1 \, 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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {1}{96} \left (-180 i x-168 \cosh (x)+8 \cosh (3 x)+\frac {192 \sinh \left (\frac {x}{2}\right )}{-i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )}+48 i \sinh (2 x)-3 i \sinh (4 x)\right ) \]

[In]

Integrate[Sinh[x]^4/(I + Csch[x]),x]

[Out]

((-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

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.12

method	result	size
risch	\(-\frac {15 i x}{8}-\frac {i {\mathrm e}^{4 x}}{64}+\frac {{\mathrm e}^{3 x}}{24}+\frac {i {\mathrm e}^{2 x}}{4}-\frac {7 \,{\mathrm e}^{x}}{8}-\frac {7 \,{\mathrm e}^{-x}}{8}-\frac {i {\mathrm e}^{-2 x}}{4}+\frac {{\mathrm e}^{-3 x}}{24}+\frac {i {\mathrm e}^{-4 x}}{64}-\frac {2}{{\mathrm e}^{x}-i}\)	\(65\)
parallelrisch	\(\frac {432+360 \left (i-i \cosh \left (x \right )-\sinh \left (x \right )\right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )+360 \left (-i+i \cosh \left (x \right )+\sinh \left (x \right )\right ) \ln \left (\coth \left (x \right )-\operatorname {csch}\left (x \right )+1\right )-80 i \sinh \left (2 x \right )-45 i \sinh \left (3 x \right )+2 i \sinh \left (4 x \right )+3 i \sinh \left (5 x \right )-168 i \sinh \left (x \right )-552 \cosh \left (x \right )+160 \cosh \left (2 x \right )-35 \cosh \left (3 x \right )-8 \cosh \left (4 x \right )+3 \cosh \left (5 x \right )}{192 i \sinh \left (x \right )-192 \cosh \left (x \right )+192}\)	\(123\)
default	\(-\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {15 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}+\frac {\frac {3}{2}+\frac {7 i}{8}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {-\frac {1}{2}+\frac {5 i}{8}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {-\frac {1}{3}-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {2 i}{\tanh \left (\frac {x}{2}\right )-i}-\frac {15 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8}+\frac {i}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {\frac {1}{3}-\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {-\frac {1}{2}-\frac {5 i}{8}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {-\frac {3}{2}+\frac {7 i}{8}}{\tanh \left (\frac {x}{2}\right )+1}\)	\(128\)

[In]

int(sinh(x)^4/(I+csch(x)),x,method=_RETURNVERBOSE)

[Out]

-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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.36 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {24 \, {\left (15 i \, x - 7 i\right )} e^{\left (5 \, x\right )} + 24 \, {\left (15 \, x + 23\right )} e^{\left (4 \, x\right )} + 3 i \, e^{\left (9 \, x\right )} - 5 \, e^{\left (8 \, x\right )} - 40 i \, e^{\left (7 \, x\right )} + 120 \, e^{\left (6 \, x\right )} - 120 i \, e^{\left (3 \, x\right )} + 40 \, e^{\left (2 \, x\right )} + 5 i \, e^{x} - 3}{192 \, {\left (e^{\left (5 \, x\right )} - i \, e^{\left (4 \, x\right )}\right )}} \]

[In]

integrate(sinh(x)^4/(I+csch(x)),x, algorithm="fricas")

[Out]

-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))

Sympy [F]

\[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh ^{4}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

integrate(sinh(x)**4/(I+csch(x)),x)

[Out]

Integral(sinh(x)**4/(csch(x) + I), x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {15}{8} i \, x - \frac {-5 i \, e^{\left (-x\right )} + 40 \, e^{\left (-2 \, x\right )} + 120 i \, e^{\left (-3 \, x\right )} + 552 \, e^{\left (-4 \, x\right )} - 3}{16 \, {\left (12 i \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )}\right )}} - \frac {7}{8} \, e^{\left (-x\right )} - \frac {1}{4} i \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {1}{64} i \, e^{\left (-4 \, x\right )} \]

[In]

integrate(sinh(x)^4/(I+csch(x)),x, algorithm="maxima")

[Out]

-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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=-\frac {15}{8} i \, x - \frac {{\left (552 \, e^{\left (4 \, x\right )} - 120 i \, e^{\left (3 \, x\right )} + 40 \, e^{\left (2 \, x\right )} + 5 i \, e^{x} - 3\right )} e^{\left (-4 \, x\right )}}{192 \, {\left (e^{x} - i\right )}} - \frac {1}{64} i \, e^{\left (4 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{4} i \, e^{\left (2 \, x\right )} - \frac {7}{8} \, e^{x} \]

[In]

integrate(sinh(x)^4/(I+csch(x)),x, algorithm="giac")

[Out]

-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

Mupad [B] (verification not implemented)

Time = 2.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int \frac {\sinh ^4(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {7\,{\mathrm {e}}^{-x}}{8}-\frac {{\mathrm {e}}^{-2\,x}\,1{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}}{4}-\frac {x\,15{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{3\,x}}{24}+\frac {{\mathrm {e}}^{-4\,x}\,1{}\mathrm {i}}{64}-\frac {{\mathrm {e}}^{4\,x}\,1{}\mathrm {i}}{64}-\frac {7\,{\mathrm {e}}^x}{8}-\frac {2}{{\mathrm {e}}^x-\mathrm {i}} \]

[In]

int(sinh(x)^4/(1/sinh(x) + 1i),x)

[Out]

(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)/6
4 - (exp(4*x)*1i)/64 - (7*exp(x))/8 - 2/(exp(x) - 1i)