3.220 \(\int \frac{\text{csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=87 \[ \frac{2 i \coth (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{\coth (c+d x) \text{csch}(c+d x)}{d (a+i a \sinh (c+d x))} \]

[Out]

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

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Rubi [A]  time = 0.12336, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2768, 2748, 3768, 3770, 3767, 8} \[ \frac{2 i \coth (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac{3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{\coth (c+d x) \text{csch}(c+d x)}{d (a+i a \sinh (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]

[Out]

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

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 8

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

Rubi steps

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

Mathematica [A]  time = 0.41949, size = 90, normalized size = 1.03 \[ \frac{4 i \tanh (c+d x)+4 i \text{csch}(2 (c+d x))-3 \text{sech}(c+d x)+\text{csch}^2(c+d x) (-\text{sech}(c+d x))+3 \sqrt{\cosh ^2(c+d x)} \text{sech}(c+d x) \tanh ^{-1}\left (\sqrt{\cosh ^2(c+d x)}\right )}{2 a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]

[Out]

((4*I)*Csch[2*(c + d*x)] - 3*Sech[c + d*x] + 3*ArcTanh[Sqrt[Cosh[c + d*x]^2]]*Sqrt[Cosh[c + d*x]^2]*Sech[c + d
*x] - Csch[c + d*x]^2*Sech[c + d*x] + (4*I)*Tanh[c + d*x])/(2*a*d)

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Maple [A]  time = 0.049, size = 119, normalized size = 1.4 \begin{align*}{\frac{2\,i}{da} \left ( -i+\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{{\frac{i}{2}}}{da}\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{{\frac{i}{2}}}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{3}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 1.15247, size = 213, normalized size = 2.45 \begin{align*} -\frac{8 \,{\left (-i \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4\right )}}{{\left (8 \, a e^{\left (-d x - c\right )} - 16 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 16 \, a e^{\left (-3 \, d x - 3 \, c\right )} + 8 i \, a e^{\left (-4 \, d x - 4 \, c\right )} + 8 \, a e^{\left (-5 \, d x - 5 \, c\right )} + 8 i \, a\right )} d} + \frac{3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a d} - \frac{3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*(-I*e^(-d*x - c) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d*x - 4*c) + 4)/((8*a*e^(-d*x - c) -
16*I*a*e^(-2*d*x - 2*c) - 16*a*e^(-3*d*x - 3*c) + 8*I*a*e^(-4*d*x - 4*c) + 8*a*e^(-5*d*x - 5*c) + 8*I*a)*d) +
3/2*log(e^(-d*x - c) + 1)/(a*d) - 3/2*log(e^(-d*x - c) - 1)/(a*d)

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Fricas [B]  time = 2.6035, size = 640, normalized size = 7.36 \begin{align*} \frac{{\left (3 \, e^{\left (5 \, d x + 5 \, c\right )} - 3 i \, e^{\left (4 \, d x + 4 \, c\right )} - 6 \, e^{\left (3 \, d x + 3 \, c\right )} + 6 i \, e^{\left (2 \, d x + 2 \, c\right )} + 3 \, e^{\left (d x + c\right )} - 3 i\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) -{\left (3 \, e^{\left (5 \, d x + 5 \, c\right )} - 3 i \, e^{\left (4 \, d x + 4 \, c\right )} - 6 \, e^{\left (3 \, d x + 3 \, c\right )} + 6 i \, e^{\left (2 \, d x + 2 \, c\right )} + 3 \, e^{\left (d x + c\right )} - 3 i\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 6 \, e^{\left (4 \, d x + 4 \, c\right )} + 6 i \, e^{\left (3 \, d x + 3 \, c\right )} + 10 \, e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, e^{\left (d x + c\right )} - 8}{2 \, a d e^{\left (5 \, d x + 5 \, c\right )} - 2 i \, a d e^{\left (4 \, d x + 4 \, c\right )} - 4 \, a d e^{\left (3 \, d x + 3 \, c\right )} + 4 i \, a d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a d e^{\left (d x + c\right )} - 2 i \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.21453, size = 142, normalized size = 1.63 \begin{align*} \frac{3 \, \log \left (e^{\left (d x + c\right )} + 1\right )}{2 \, a d} - \frac{3 \, \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, a d} - \frac{e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (d x + c\right )} + 2 i}{a d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} - \frac{2 i}{a d{\left (i \, e^{\left (d x + c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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