\(\int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx\) [46]

Optimal result

Integrand size = 13, antiderivative size = 37 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3}{2} i \text {arctanh}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)} \]

[Out]

-3/2*I*arctanh(cosh(x))-2*coth(x)+3/2*I*coth(x)*csch(x)+coth(x)*csch(x)/(I+sinh(x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2847, 2827, 3853, 3855, 3852, 8} \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3}{2} i \text {arctanh}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{\sinh (x)+i} \]

[In]

Int[Csch[x]^3/(I + Sinh[x]),x]

[Out]

((-3*I)/2)*ArcTanh[Cosh[x]] - 2*Coth[x] + ((3*I)/2)*Coth[x]*Csch[x] + (Coth[x]*Csch[x])/(I + Sinh[x])

Rule 8

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

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2847

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b
^2)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(a*f*(b*c - a*d)*(a + b*Sin[e + f*x]))), x] + Dist[d/(a*(b*c -
a*d)), Int[(c + d*Sin[e + f*x])^n*(a*n - b*(n + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Ne
Q[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 0] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3852

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

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}+\int \text {csch}^3(x) (-3 i+2 \sinh (x)) \, dx \\ & = \frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}-3 i \int \text {csch}^3(x) \, dx+2 \int \text {csch}^2(x) \, dx \\ & = \frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)}+\frac {3}{2} i \int \text {csch}(x) \, dx-2 i \text {Subst}(\int 1 \, dx,x,-i \coth (x)) \\ & = -\frac {3}{2} i \text {arctanh}(\cosh (x))-2 \coth (x)+\frac {3}{2} i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}(x)}{i+\sinh (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\frac {1}{2} i \left (4 i+3 \text {csch}(x)-3 \text {arctanh}\left (\sqrt {\cosh ^2(x)}\right ) \sqrt {\cosh ^2(x)} \text {csch}(x)+2 i \text {csch}^2(x)+\text {csch}^3(x)\right ) \tanh (x) \]

[In]

Integrate[Csch[x]^3/(I + Sinh[x]),x]

[Out]

(I/2)*(4*I + 3*Csch[x] - 3*ArcTanh[Sqrt[Cosh[x]^2]]*Sqrt[Cosh[x]^2]*Csch[x] + (2*I)*Csch[x]^2 + Csch[x]^3)*Tan
h[x]

Maple [A] (verified)

Time = 3.50 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43

method	result	size
default	\(-\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {i \tanh \left (\frac {x}{2}\right )^{2}}{8}-\frac {2}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{8 \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}\)	\(53\)
risch	\(\frac {i \left (3 \,{\mathrm e}^{4 x}-5 \,{\mathrm e}^{2 x}+3 i {\mathrm e}^{3 x}+4-i {\mathrm e}^{x}\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2} \left ({\mathrm e}^{x}+i\right )}+\frac {3 i \ln \left ({\mathrm e}^{x}-1\right )}{2}-\frac {3 i \ln \left ({\mathrm e}^{x}+1\right )}{2}\)	\(62\)
parallelrisch	\(\frac {\left (12 i \tanh \left (\frac {x}{2}\right )-12\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )-i \tanh \left (\frac {x}{2}\right )^{3}-3 i \coth \left (\frac {x}{2}\right )-\coth \left (\frac {x}{2}\right )^{2}-3 \tanh \left (\frac {x}{2}\right )^{2}-24}{8 \tanh \left (\frac {x}{2}\right )+8 i}\)	\(62\)

[In]

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

[Out]

-1/2*tanh(1/2*x)-1/8*I*tanh(1/2*x)^2-2/(tanh(1/2*x)+I)+1/8*I/tanh(1/2*x)^2+3/2*I*ln(tanh(1/2*x))-1/2/tanh(1/2*
x)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (27) = 54\).

Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.41 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {3 \, {\left (i \, e^{\left (5 \, x\right )} - e^{\left (4 \, x\right )} - 2 i \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + 3 \, {\left (-i \, e^{\left (5 \, x\right )} + e^{\left (4 \, x\right )} + 2 i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} - i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 6 i \, e^{\left (4 \, x\right )} + 6 \, e^{\left (3 \, x\right )} + 10 i \, e^{\left (2 \, x\right )} - 2 \, e^{x} - 8 i}{2 \, {\left (e^{\left (5 \, x\right )} + i \, e^{\left (4 \, x\right )} - 2 \, e^{\left (3 \, x\right )} - 2 i \, e^{\left (2 \, x\right )} + e^{x} + i\right )}} \]

[In]

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

[Out]

-1/2*(3*(I*e^(5*x) - e^(4*x) - 2*I*e^(3*x) + 2*e^(2*x) + I*e^x - 1)*log(e^x + 1) + 3*(-I*e^(5*x) + e^(4*x) + 2
*I*e^(3*x) - 2*e^(2*x) - I*e^x + 1)*log(e^x - 1) - 6*I*e^(4*x) + 6*e^(3*x) + 10*I*e^(2*x) - 2*e^x - 8*I)/(e^(5
*x) + I*e^(4*x) - 2*e^(3*x) - 2*I*e^(2*x) + e^x + I)

Sympy [F]

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

[In]

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

[Out]

Integral(csch(x)**3/(sinh(x) + I), x)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (27) = 54\).

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.03 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {e^{\left (-x\right )} + 5 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i \, e^{\left (-4 \, x\right )} - 4 i}{e^{\left (-x\right )} + 2 i \, e^{\left (-2 \, x\right )} - 2 \, e^{\left (-3 \, x\right )} - i \, e^{\left (-4 \, x\right )} + e^{\left (-5 \, x\right )} - i} - \frac {3}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {3}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

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

[Out]

-(e^(-x) + 5*I*e^(-2*x) - 3*e^(-3*x) - 3*I*e^(-4*x) - 4*I)/(e^(-x) + 2*I*e^(-2*x) - 2*e^(-3*x) - I*e^(-4*x) +
e^(-5*x) - I) - 3/2*I*log(e^(-x) + 1) + 3/2*I*log(e^(-x) - 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.38 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=\frac {i \, e^{\left (3 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + i \, e^{x} + 2}{{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} + \frac {2 i}{e^{x} + i} - \frac {3}{2} i \, \log \left (e^{x} + 1\right ) + \frac {3}{2} i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

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

[Out]

(I*e^(3*x) - 2*e^(2*x) + I*e^x + 2)/(e^(2*x) - 1)^2 + 2*I/(e^x + I) - 3/2*I*log(e^x + 1) + 3/2*I*log(abs(e^x -
1))

Mupad [B] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.89 \[ \int \frac {\text {csch}^3(x)}{i+\sinh (x)} \, dx=-\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}-3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {\ln \left (-{\mathrm {e}}^x\,3{}\mathrm {i}+3{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {2{}\mathrm {i}}{{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {{\mathrm {e}}^x\,2{}\mathrm {i}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {-2+{\mathrm {e}}^x\,1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1} \]

[In]

int(1/(sinh(x)^3*(sinh(x) + 1i)),x)

[Out]

(log(3i - exp(x)*3i)*3i)/2 - (log(- exp(x)*3i - 3i)*3i)/2 + 2i/(exp(x) + 1i) + (exp(x)*2i)/(exp(4*x) - 2*exp(2
*x) + 1) + (exp(x)*1i - 2)/(exp(2*x) - 1)