3.1.55 \(\int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx\) [55]

Optimal. Leaf size=64 \[ -5 i \tanh ^{-1}(\cosh (x))-12 \coth (x)+4 \coth ^3(x)+5 i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))^2}-\frac {10 i \coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))} \]

[Out]

-5*I*arctanh(cosh(x))-12*coth(x)+4*coth(x)^3+5*I*coth(x)*csch(x)+1/3*coth(x)*csch(x)^2/(I+sinh(x))^2-10/3*I*co
th(x)*csch(x)^2/(I+sinh(x))

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Rubi [A]
time = 0.09, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2845, 3057, 2827, 3852, 3853, 3855} \begin {gather*} 4 \coth ^3(x)-12 \coth (x)-5 i \tanh ^{-1}(\cosh (x))+5 i \coth (x) \text {csch}(x)-\frac {10 i \coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)}+\frac {\coth (x) \text {csch}^2(x)}{3 (\sinh (x)+i)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(I + Sinh[x])^2,x]

[Out]

(-5*I)*ArcTanh[Cosh[x]] - 12*Coth[x] + 4*Coth[x]^3 + (5*I)*Coth[x]*Csch[x] + (Coth[x]*Csch[x]^2)/(3*(I + Sinh[
x])^2) - (((10*I)/3)*Coth[x]*Csch[x]^2)/(I + Sinh[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 2845

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

Rule 3057

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*
x])^(n + 1)/(a*f*(2*m + 1)*(b*c - a*d))), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] &&  !GtQ[n, 0] && IntegerQ[2*m] && (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*} \int \frac {\text {csch}^4(x)}{(i+\sinh (x))^2} \, dx &=\frac {\coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))^2}-\frac {1}{3} \int \frac {\text {csch}^4(x) (6 i-4 \sinh (x))}{i+\sinh (x)} \, dx\\ &=\frac {\coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))^2}-\frac {10 i \coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))}+\frac {1}{3} \int \text {csch}^4(x) (-36-30 i \sinh (x)) \, dx\\ &=\frac {\coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))^2}-\frac {10 i \coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))}-10 i \int \text {csch}^3(x) \, dx-12 \int \text {csch}^4(x) \, dx\\ &=5 i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))^2}-\frac {10 i \coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))}+5 i \int \text {csch}(x) \, dx-12 i \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )\\ &=-5 i \tanh ^{-1}(\cosh (x))-12 \coth (x)+4 \coth ^3(x)+5 i \coth (x) \text {csch}(x)+\frac {\coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))^2}-\frac {10 i \coth (x) \text {csch}^2(x)}{3 (i+\sinh (x))}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(64)=128\).
time = 1.42, size = 143, normalized size = 2.23 \begin {gather*} \frac {1}{24} \left (-44 \coth \left (\frac {x}{2}\right )+6 i \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{2} \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+2 \left (-60 i \log \left (\cosh \left (\frac {x}{2}\right )\right )+60 i \log \left (\sinh \left (\frac {x}{2}\right )\right )+3 i \text {sech}^2\left (\frac {x}{2}\right )-4 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {4}{i+\sinh (x)}+\frac {8 \sinh \left (\frac {x}{2}\right ) (14 i+13 \sinh (x))}{\left (i \cosh \left (\frac {x}{2}\right )+\sinh \left (\frac {x}{2}\right )\right )^3}-22 \tanh \left (\frac {x}{2}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(I + Sinh[x])^2,x]

[Out]

(-44*Coth[x/2] + (6*I)*Csch[x/2]^2 + (Csch[x/2]^4*Sinh[x])/2 + 2*((-60*I)*Log[Cosh[x/2]] + (60*I)*Log[Sinh[x/2
]] + (3*I)*Sech[x/2]^2 - 4*Csch[x]^3*Sinh[x/2]^4 - 4/(I + Sinh[x]) + (8*Sinh[x/2]*(14*I + 13*Sinh[x]))/(I*Cosh
[x/2] + Sinh[x/2])^3 - 22*Tanh[x/2]))/24

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Maple [A]
time = 0.74, size = 92, normalized size = 1.44

method	result	size
risch	\(\frac {2 i \left (45 i {\mathrm e}^{7 x}+15 \,{\mathrm e}^{8 x}-135 i {\mathrm e}^{5 x}-85 \,{\mathrm e}^{6 x}+155 i {\mathrm e}^{3 x}+153 \,{\mathrm e}^{4 x}-57 i {\mathrm e}^{x}-99 \,{\mathrm e}^{2 x}+24\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{3} \left ({\mathrm e}^{x}+i\right )^{3}}+5 i \ln \left ({\mathrm e}^{x}-1\right )-5 i \ln \left ({\mathrm e}^{x}+1\right )\)	\(88\)
default	\(-\frac {15 \tanh \left (\frac {x}{2}\right )}{8}+\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24}-\frac {i \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{4}+\frac {i}{4 \tanh \left (\frac {x}{2}\right )^{2}}+5 i \ln \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{24 \tanh \left (\frac {x}{2}\right )^{3}}-\frac {15}{8 \tanh \left (\frac {x}{2}\right )}+\frac {2 i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {10}{\tanh \left (\frac {x}{2}\right )+i}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(I+sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

-15/8*tanh(1/2*x)+1/24*tanh(1/2*x)^3-1/4*I*tanh(1/2*x)^2+1/4*I/tanh(1/2*x)^2+5*I*ln(tanh(1/2*x))+1/24/tanh(1/2
*x)^3-15/8/tanh(1/2*x)+2*I/(tanh(1/2*x)+I)^2+4/3/(tanh(1/2*x)+I)^3-10/(tanh(1/2*x)+I)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (50) = 100\).
time = 0.29, size = 127, normalized size = 1.98 \begin {gather*} -\frac {2 \, {\left (57 \, e^{\left (-x\right )} + 99 i \, e^{\left (-2 \, x\right )} - 155 \, e^{\left (-3 \, x\right )} - 153 i \, e^{\left (-4 \, x\right )} + 135 \, e^{\left (-5 \, x\right )} + 85 i \, e^{\left (-6 \, x\right )} - 45 \, e^{\left (-7 \, x\right )} - 15 i \, e^{\left (-8 \, x\right )} - 24 i\right )}}{3 \, {\left (3 \, e^{\left (-x\right )} + 6 i \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-3 \, x\right )} - 12 i \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )} + 10 i \, e^{\left (-6 \, x\right )} - 6 \, e^{\left (-7 \, x\right )} - 3 i \, e^{\left (-8 \, x\right )} + e^{\left (-9 \, x\right )} - i\right )}} - 5 i \, \log \left (e^{\left (-x\right )} + 1\right ) + 5 i \, \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(I+sinh(x))^2,x, algorithm="maxima")

[Out]

-2/3*(57*e^(-x) + 99*I*e^(-2*x) - 155*e^(-3*x) - 153*I*e^(-4*x) + 135*e^(-5*x) + 85*I*e^(-6*x) - 45*e^(-7*x) -
15*I*e^(-8*x) - 24*I)/(3*e^(-x) + 6*I*e^(-2*x) - 10*e^(-3*x) - 12*I*e^(-4*x) + 12*e^(-5*x) + 10*I*e^(-6*x) -
6*e^(-7*x) - 3*I*e^(-8*x) + e^(-9*x) - I) - 5*I*log(e^(-x) + 1) + 5*I*log(e^(-x) - 1)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (50) = 100\).
time = 0.42, size = 226, normalized size = 3.53 \begin {gather*} -\frac {15 \, {\left (i \, e^{\left (9 \, x\right )} - 3 \, e^{\left (8 \, x\right )} - 6 i \, e^{\left (7 \, x\right )} + 10 \, e^{\left (6 \, x\right )} + 12 i \, e^{\left (5 \, x\right )} - 12 \, e^{\left (4 \, x\right )} - 10 i \, e^{\left (3 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 3 i \, e^{x} - 1\right )} \log \left (e^{x} + 1\right ) + 15 \, {\left (-i \, e^{\left (9 \, x\right )} + 3 \, e^{\left (8 \, x\right )} + 6 i \, e^{\left (7 \, x\right )} - 10 \, e^{\left (6 \, x\right )} - 12 i \, e^{\left (5 \, x\right )} + 12 \, e^{\left (4 \, x\right )} + 10 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 3 i \, e^{x} + 1\right )} \log \left (e^{x} - 1\right ) - 30 i \, e^{\left (8 \, x\right )} + 90 \, e^{\left (7 \, x\right )} + 170 i \, e^{\left (6 \, x\right )} - 270 \, e^{\left (5 \, x\right )} - 306 i \, e^{\left (4 \, x\right )} + 310 \, e^{\left (3 \, x\right )} + 198 i \, e^{\left (2 \, x\right )} - 114 \, e^{x} - 48 i}{3 \, {\left (e^{\left (9 \, x\right )} + 3 i \, e^{\left (8 \, x\right )} - 6 \, e^{\left (7 \, x\right )} - 10 i \, e^{\left (6 \, x\right )} + 12 \, e^{\left (5 \, x\right )} + 12 i \, e^{\left (4 \, x\right )} - 10 \, e^{\left (3 \, x\right )} - 6 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} + i\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(I+sinh(x))^2,x, algorithm="fricas")

[Out]

-1/3*(15*(I*e^(9*x) - 3*e^(8*x) - 6*I*e^(7*x) + 10*e^(6*x) + 12*I*e^(5*x) - 12*e^(4*x) - 10*I*e^(3*x) + 6*e^(2
*x) + 3*I*e^x - 1)*log(e^x + 1) + 15*(-I*e^(9*x) + 3*e^(8*x) + 6*I*e^(7*x) - 10*e^(6*x) - 12*I*e^(5*x) + 12*e^
(4*x) + 10*I*e^(3*x) - 6*e^(2*x) - 3*I*e^x + 1)*log(e^x - 1) - 30*I*e^(8*x) + 90*e^(7*x) + 170*I*e^(6*x) - 270
*e^(5*x) - 306*I*e^(4*x) + 310*e^(3*x) + 198*I*e^(2*x) - 114*e^x - 48*I)/(e^(9*x) + 3*I*e^(8*x) - 6*e^(7*x) -
10*I*e^(6*x) + 12*e^(5*x) + 12*I*e^(4*x) - 10*e^(3*x) - 6*I*e^(2*x) + 3*e^x + I)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**4/(I+sinh(x))**2,x)

[Out]

Timed out

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Giac [A]
time = 0.42, size = 84, normalized size = 1.31 \begin {gather*} -\frac {2 \, {\left (-15 i \, e^{\left (8 \, x\right )} + 45 \, e^{\left (7 \, x\right )} + 85 i \, e^{\left (6 \, x\right )} - 135 \, e^{\left (5 \, x\right )} - 153 i \, e^{\left (4 \, x\right )} + 155 \, e^{\left (3 \, x\right )} + 99 i \, e^{\left (2 \, x\right )} - 57 \, e^{x} - 24 i\right )}}{3 \, {\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )} - e^{x} - i\right )}^{3}} - 5 i \, \log \left (e^{x} + 1\right ) + 5 i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(I+sinh(x))^2,x, algorithm="giac")

[Out]

-2/3*(-15*I*e^(8*x) + 45*e^(7*x) + 85*I*e^(6*x) - 135*e^(5*x) - 153*I*e^(4*x) + 155*e^(3*x) + 99*I*e^(2*x) - 5
7*e^x - 24*I)/(e^(3*x) + I*e^(2*x) - e^x - I)^3 - 5*I*log(e^x + 1) + 5*I*log(abs(e^x - 1))

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Mupad [B]
time = 1.09, size = 189, normalized size = 2.95 \begin {gather*} -\ln \left (-{\mathrm {e}}^x\,10{}\mathrm {i}-10{}\mathrm {i}\right )\,5{}\mathrm {i}+\ln \left (-{\mathrm {e}}^x\,10{}\mathrm {i}+10{}\mathrm {i}\right )\,5{}\mathrm {i}-\frac {\frac {16\,{\mathrm {e}}^x}{3}-\frac {{\mathrm {e}}^{2\,x}\,32{}\mathrm {i}}{3}+\frac {16}{3}{}\mathrm {i}}{12\,{\mathrm {e}}^{5\,x}-10\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}\,12{}\mathrm {i}-{\mathrm {e}}^{2\,x}\,6{}\mathrm {i}-{\mathrm {e}}^{6\,x}\,10{}\mathrm {i}-6\,{\mathrm {e}}^{7\,x}+{\mathrm {e}}^{8\,x}\,3{}\mathrm {i}+{\mathrm {e}}^{9\,x}+3\,{\mathrm {e}}^x+1{}\mathrm {i}}+\frac {\frac {20\,{\mathrm {e}}^{2\,x}}{3}-\frac {44}{3}+\frac {{\mathrm {e}}^x\,16{}\mathrm {i}}{3}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1-{\mathrm {e}}^{3\,x}\,4{}\mathrm {i}+{\mathrm {e}}^{5\,x}\,2{}\mathrm {i}+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {10\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,10{}\mathrm {i}+\frac {20}{3}{}\mathrm {i}}{{\mathrm {e}}^{2\,x}\,1{}\mathrm {i}+{\mathrm {e}}^{3\,x}-{\mathrm {e}}^x-\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^4*(sinh(x) + 1i)^2),x)

[Out]

log(10i - exp(x)*10i)*5i - log(- exp(x)*10i - 10i)*5i - ((16*exp(x))/3 - (exp(2*x)*32i)/3 + 16i/3)/(exp(4*x)*1
2i - 10*exp(3*x) - exp(2*x)*6i + 12*exp(5*x) - exp(6*x)*10i - 6*exp(7*x) + exp(8*x)*3i + exp(9*x) + 3*exp(x) +
1i) + ((20*exp(2*x))/3 + (exp(x)*16i)/3 - 44/3)/(3*exp(2*x) - exp(3*x)*4i - 3*exp(4*x) + exp(5*x)*2i + exp(6*
x) + exp(x)*2i - 1) - (10*exp(x) - exp(2*x)*10i + 20i/3)/(exp(2*x)*1i + exp(3*x) - exp(x) - 1i)

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