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

Optimal. Leaf size=47 \[ \frac {4}{3} i \coth ^3(x)-4 i \coth (x)+\frac {3}{2} \tanh ^{-1}(\cosh (x))-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2768, 2748, 3767, 3768, 3770} \[ \frac {4}{3} i \coth ^3(x)-4 i \coth (x)+\frac {3}{2} \tanh ^{-1}(\cosh (x))-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{\sinh (x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(I + Sinh[x]),x]

[Out]

(3*ArcTanh[Cosh[x]])/2 - (4*I)*Coth[x] + ((4*I)/3)*Coth[x]^3 - (3*Coth[x]*Csch[x])/2 + (Coth[x]*Csch[x]^2)/(I
+ Sinh[x])

Rule 2748

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 2768

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] && NeQ[
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 3767

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 3768

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 3770

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^4(x)}{i+\sinh (x)} \, dx &=\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)}+\int \text {csch}^4(x) (-4 i+3 \sinh (x)) \, dx\\ &=\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)}-4 i \int \text {csch}^4(x) \, dx+3 \int \text {csch}^3(x) \, dx\\ &=-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)}-\frac {3}{2} \int \text {csch}(x) \, dx+4 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )\\ &=\frac {3}{2} \tanh ^{-1}(\cosh (x))-4 i \coth (x)+\frac {4}{3} i \coth ^3(x)-\frac {3}{2} \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{i+\sinh (x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 53, normalized size = 1.13 \[ \frac {1}{6} \text {sech}(x) \left (-16 i \sinh (x)+2 i \text {csch}^3(x)-3 \text {csch}^2(x)-8 i \text {csch}(x)+9 \sqrt {\cosh ^2(x)} \tanh ^{-1}\left (\sqrt {\cosh ^2(x)}\right )-9\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(I + Sinh[x]),x]

[Out]

(Sech[x]*(-9 + 9*ArcTanh[Sqrt[Cosh[x]^2]]*Sqrt[Cosh[x]^2] - (8*I)*Csch[x] - 3*Csch[x]^2 + (2*I)*Csch[x]^3 - (1
6*I)*Sinh[x]))/6

________________________________________________________________________________________

fricas [B]  time = 1.16, size = 178, normalized size = 3.79 \[ \frac {{\left (9 \, e^{\left (7 \, x\right )} + 9 i \, e^{\left (6 \, x\right )} - 27 \, e^{\left (5 \, x\right )} - 27 i \, e^{\left (4 \, x\right )} + 27 \, e^{\left (3 \, x\right )} + 27 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 9 i\right )} \log \left (e^{x} + 1\right ) - {\left (9 \, e^{\left (7 \, x\right )} + 9 i \, e^{\left (6 \, x\right )} - 27 \, e^{\left (5 \, x\right )} - 27 i \, e^{\left (4 \, x\right )} + 27 \, e^{\left (3 \, x\right )} + 27 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 9 i\right )} \log \left (e^{x} - 1\right ) - 18 \, e^{\left (6 \, x\right )} - 18 i \, e^{\left (5 \, x\right )} + 48 \, e^{\left (4 \, x\right )} + 48 i \, e^{\left (3 \, x\right )} - 78 \, e^{\left (2 \, x\right )} - 14 i \, e^{x} + 32}{6 \, e^{\left (7 \, x\right )} + 6 i \, e^{\left (6 \, x\right )} - 18 \, e^{\left (5 \, x\right )} - 18 i \, e^{\left (4 \, x\right )} + 18 \, e^{\left (3 \, x\right )} + 18 i \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 6 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((9*e^(7*x) + 9*I*e^(6*x) - 27*e^(5*x) - 27*I*e^(4*x) + 27*e^(3*x) + 27*I*e^(2*x) - 9*e^x - 9*I)*log(e^x + 1)
- (9*e^(7*x) + 9*I*e^(6*x) - 27*e^(5*x) - 27*I*e^(4*x) + 27*e^(3*x) + 27*I*e^(2*x) - 9*e^x - 9*I)*log(e^x - 1)
- 18*e^(6*x) - 18*I*e^(5*x) + 48*e^(4*x) + 48*I*e^(3*x) - 78*e^(2*x) - 14*I*e^x + 32)/(6*e^(7*x) + 6*I*e^(6*x
) - 18*e^(5*x) - 18*I*e^(4*x) + 18*e^(3*x) + 18*I*e^(2*x) - 6*e^x - 6*I)

________________________________________________________________________________________

giac [A]  time = 0.44, size = 58, normalized size = 1.23 \[ -\frac {2}{e^{x} + i} - \frac {3 \, e^{\left (5 \, x\right )} + 6 i \, e^{\left (4 \, x\right )} - 24 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} + 10 i}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} + \frac {3}{2} \, \log \left (e^{x} + 1\right ) - \frac {3}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(e^x + I) - 1/3*(3*e^(5*x) + 6*I*e^(4*x) - 24*I*e^(2*x) - 3*e^x + 10*I)/(e^(2*x) - 1)^3 + 3/2*log(e^x + 1)
- 3/2*log(abs(e^x - 1))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 71, normalized size = 1.51 \[ -\frac {7 i \tanh \left (\frac {x}{2}\right )}{8}+\frac {i \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{8}-\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}+\frac {i}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {7 i}{8 \tanh \left (\frac {x}{2}\right )}-\frac {1}{8 \tanh \left (\frac {x}{2}\right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.35, size = 105, normalized size = 2.23 \[ \frac {16 \, {\left (-7 i \, e^{\left (-x\right )} + 39 \, e^{\left (-2 \, x\right )} + 24 i \, e^{\left (-3 \, x\right )} - 24 \, e^{\left (-4 \, x\right )} - 9 i \, e^{\left (-5 \, x\right )} + 9 \, e^{\left (-6 \, x\right )} - 16\right )}}{48 \, e^{\left (-x\right )} + 144 i \, e^{\left (-2 \, x\right )} - 144 \, e^{\left (-3 \, x\right )} - 144 i \, e^{\left (-4 \, x\right )} + 144 \, e^{\left (-5 \, x\right )} + 48 i \, e^{\left (-6 \, x\right )} - 48 \, e^{\left (-7 \, x\right )} - 48 i} + \frac {3}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {3}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*(-7*I*e^(-x) + 39*e^(-2*x) + 24*I*e^(-3*x) - 24*e^(-4*x) - 9*I*e^(-5*x) + 9*e^(-6*x) - 16)/(48*e^(-x) + 144
*I*e^(-2*x) - 144*e^(-3*x) - 144*I*e^(-4*x) + 144*e^(-5*x) + 48*I*e^(-6*x) - 48*e^(-7*x) - 48*I) + 3/2*log(e^(
-x) + 1) - 3/2*log(e^(-x) - 1)

________________________________________________________________________________________

mupad [B]  time = 0.72, size = 85, normalized size = 1.81 \[ \frac {3\,\ln \left (3\,{\mathrm {e}}^x+3\right )}{2}-\frac {3\,\ln \left (3\,{\mathrm {e}}^x-3\right )}{2}-\frac {{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1}-\frac {2\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}-\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1}+\frac {4{}\mathrm {i}}{{\left ({\mathrm {e}}^{2\,x}-1\right )}^2}+\frac {8{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^4*(sinh(x) + 1i)),x)

[Out]

(3*log(3*exp(x) + 3))/2 - (3*log(3*exp(x) - 3))/2 - exp(x)/(exp(2*x) - 1) - (2*exp(x))/(exp(2*x) - 1)^2 - 2/(e
xp(x) + 1i) - 2i/(exp(2*x) - 1) + 4i/(exp(2*x) - 1)^2 + 8i/(3*(exp(2*x) - 1)^3)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________