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

Optimal. Leaf size=46 \[ -\frac{3 x}{2}-\frac{4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac{3}{2} \sinh (x) \cosh (x)-\frac{\sinh ^2(x) \cosh (x)}{\text{csch}(x)+i} \]

[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])

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Rubi [A]  time = 0.0711676, antiderivative size = 46, normalized size of antiderivative = 1., 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 = {3819, 3787, 2633, 2635, 8} \[ -\frac{3 x}{2}-\frac{4}{3} i \cosh ^3(x)+4 i \cosh (x)+\frac{3}{2} \sinh (x) \cosh (x)-\frac{\sinh ^2(x) \cosh (x)}{\text{csch}(x)+i} \]

Antiderivative was successfully verified.

[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 3819

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]

Rule 3787

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 2633

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \frac{\sinh ^3(x)}{i+\text{csch}(x)} \, dx &=-\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 \operatorname{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]  time = 0.127716, size = 56, normalized size = 1.22 \[ \frac{1}{12} \left (21 i \cosh (x)-i \cosh (3 x)+3 \left (-6 x+\sinh (2 x)+\frac{8 \sinh \left (\frac{x}{2}\right )}{\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )}\right )\right ) \]

Antiderivative was successfully verified.

[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

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Maple [B]  time = 0.046, size = 137, normalized size = 3. \begin{align*}{-{\frac{i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{{\frac{3\,i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+2\, \left ( \tanh \left ( x/2 \right ) -i \right ) ^{-1}+{{\frac{i}{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{{\frac{3\,i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05446, size = 80, normalized size = 1.74 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.74752, size = 194, normalized size = 4.22 \begin{align*} -\frac{3 \,{\left (12 \, x - 7\right )} e^{\left (4 \, x\right )} -{\left (36 i \, x + 69 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*(12*x - 7)*e^(4*x) - (36*I*x + 69*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))

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

[Out]

Timed out

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Giac [A]  time = 1.19581, size = 68, normalized size = 1.48 \begin{align*} -\frac{3}{2} \, x + \frac{i \,{\left (69 \, e^{\left (3 \, x\right )} - 18 i \, e^{\left (2 \, x\right )} + 2 \, e^{x} + i\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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