\(\int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx\) [63]

Optimal result

Integrand size = 13, antiderivative size = 46 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 x}{2}+4 i \cosh (x)-\frac {4}{3} i \cosh ^3(x)+\frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)} \]

[Out]

-3/2*x+4*I*cosh(x)-4/3*I*cosh(x)^3+3/2*cosh(x)*sinh(x)-cosh(x)*sinh(x)^2/(I+csch(x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2713, 2715, 8} \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 x}{2}-\frac {4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac {3}{2} \sinh (x) \cosh (x)-\frac {\sinh ^2(x) \cosh (x)}{\text {csch}(x)+i} \]

[In]

Int[Sinh[x]^3/(I + Csch[x]),x]

[Out]

(-3*x)/2 + (4*I)*Cosh[x] - ((4*I)/3)*Cosh[x]^3 + (3*Cosh[x]*Sinh[x])/2 - (Cosh[x]*Sinh[x]^2)/(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 ^2(x)}{i+\text {csch}(x)}+\int (-4 i+3 \text {csch}(x)) \sinh ^3(x) \, dx \\ & = -\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)}-4 i \int \sinh ^3(x) \, dx+3 \int \sinh ^2(x) \, dx \\ & = \frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)}+4 i \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (x)\right )-\frac {3 \int 1 \, dx}{2} \\ & = -\frac {3 x}{2}+4 i \cosh (x)-\frac {4}{3} i \cosh ^3(x)+\frac {3}{2} \cosh (x) \sinh (x)-\frac {\cosh (x) \sinh ^2(x)}{i+\text {csch}(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{12} \left (21 i \cosh (x)-i \cosh (3 x)+3 \left (-6 x+\frac {8 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )}+\sinh (2 x)\right )\right ) \]

[In]

Integrate[Sinh[x]^3/(I + Csch[x]),x]

[Out]

((21*I)*Cosh[x] - I*Cosh[3*x] + 3*(-6*x + (8*Sinh[x/2])/(Cosh[x/2] + I*Sinh[x/2]) + Sinh[2*x]))/12

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15

method	result	size
risch	\(-\frac {3 x}{2}-\frac {i {\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{8}+\frac {7 i {\mathrm e}^{x}}{8}+\frac {7 i {\mathrm e}^{-x}}{8}-\frac {{\mathrm e}^{-2 x}}{8}-\frac {i {\mathrm e}^{-3 x}}{24}+\frac {2 i}{{\mathrm e}^{x}-i}\)	\(53\)
default	\(\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {\frac {1}{2}+\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {\frac {1}{2}-\frac {3 i}{2}}{\tanh \left (\frac {x}{2}\right )-1}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {i}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {\frac {1}{2}+\frac {3 i}{2}}{\tanh \left (\frac {x}{2}\right )+1}+\frac {-\frac {1}{2}+\frac {i}{2}}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}+\frac {2}{\tanh \left (\frac {x}{2}\right )-i}\)	\(101\)
parallelrisch	\(\frac {\left (36 i \sinh \left (x \right )-36 \cosh \left (x \right )+36\right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )+\left (-36 i \sinh \left (x \right )+36 \cosh \left (x \right )-36\right ) \ln \left (\coth \left (x \right )-\operatorname {csch}\left (x \right )+1\right )+23 i \cosh \left (x \right )-20 i \cosh \left (2 x \right )+i \cosh \left (3 x \right )+i \cosh \left (4 x \right )-5 i-3 \sinh \left (3 x \right )+\sinh \left (4 x \right )-67 \sinh \left (x \right )-16 \sinh \left (2 x \right )}{24 i \sinh \left (x \right )-24 \cosh \left (x \right )+24}\)	\(106\)

[In]

int(sinh(x)^3/(I+csch(x)),x,method=_RETURNVERBOSE)

[Out]

-3/2*x-1/24*I*exp(x)^3+1/8*exp(x)^2+7/8*I*exp(x)+7/8*I/exp(x)-1/8/exp(x)^2-1/24*I/exp(x)^3+2*I/(exp(x)-I)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.46 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3 \, {\left (12 \, x - 7\right )} e^{\left (4 \, x\right )} + 3 \, {\left (-12 i \, x - 23 i\right )} e^{\left (3 \, x\right )} + i \, e^{\left (7 \, x\right )} - 2 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1}{24 \, {\left (e^{\left (4 \, x\right )} - i \, e^{\left (3 \, x\right )}\right )}} \]

[In]

integrate(sinh(x)^3/(I+csch(x)),x, algorithm="fricas")

[Out]

-1/24*(3*(12*x - 7)*e^(4*x) + 3*(-12*I*x - 23*I)*e^(3*x) + I*e^(7*x) - 2*e^(6*x) - 18*I*e^(5*x) - 18*e^(2*x) -
2*I*e^x + 1)/(e^(4*x) - I*e^(3*x))

Sympy [F]

\[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

integrate(sinh(x)**3/(I+csch(x)),x)

[Out]

Integral(sinh(x)**3/(csch(x) + I), x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.28 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3}{2} \, x + \frac {2 i \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} + 69 i \, e^{\left (-3 \, x\right )} + 1}{8 \, {\left (3 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}\right )}} + \frac {7}{8} i \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {1}{24} i \, e^{\left (-3 \, x\right )} \]

[In]

integrate(sinh(x)^3/(I+csch(x)),x, algorithm="maxima")

[Out]

-3/2*x + 1/8*(2*I*e^(-x) - 18*e^(-2*x) + 69*I*e^(-3*x) + 1)/(3*I*e^(-3*x) + 3*e^(-4*x)) + 7/8*I*e^(-x) - 1/8*e
^(-2*x) - 1/24*I*e^(-3*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {3}{2} \, x - \frac {{\left (-69 i \, e^{\left (3 \, x\right )} - 18 \, e^{\left (2 \, x\right )} - 2 i \, e^{x} + 1\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (e^{x} - i\right )}} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {7}{8} i \, e^{x} \]

[In]

integrate(sinh(x)^3/(I+csch(x)),x, algorithm="giac")

[Out]

-3/2*x - 1/24*(-69*I*e^(3*x) - 18*e^(2*x) - 2*I*e^x + 1)*e^(-3*x)/(e^x - I) - 1/24*I*e^(3*x) + 1/8*e^(2*x) + 7
/8*I*e^x

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {\sinh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8}+\frac {{\mathrm {e}}^{-x}\,7{}\mathrm {i}}{8}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,x}{2}-\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^x\,7{}\mathrm {i}}{8}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \]

[In]

int(sinh(x)^3/(1/sinh(x) + 1i),x)

[Out]

(exp(-x)*7i)/8 - (3*x)/2 - exp(-2*x)/8 + exp(2*x)/8 - (exp(-3*x)*1i)/24 - (exp(3*x)*1i)/24 + (exp(x)*7i)/8 + 2
i/(exp(x) - 1i)