3.41 \(\int \frac{\sinh ^3(x)}{i+\sinh (x)} \, dx\)

Optimal. Leaf size=36 \[ -\frac{3 x}{2}-2 i \cosh (x)-\frac{\sinh ^2(x) \cosh (x)}{\sinh (x)+i}+\frac{3}{2} \sinh (x) \cosh (x) \]

[Out]

(-3*x)/2 - (2*I)*Cosh[x] + (3*Cosh[x]*Sinh[x])/2 - (Cosh[x]*Sinh[x]^2)/(I + Sinh[x])

________________________________________________________________________________________

Rubi [A]  time = 0.0463383, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2767, 2734} \[ -\frac{3 x}{2}-2 i \cosh (x)-\frac{\sinh ^2(x) \cosh (x)}{\sinh (x)+i}+\frac{3}{2} \sinh (x) \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(I + Sinh[x]),x]

[Out]

(-3*x)/2 - (2*I)*Cosh[x] + (3*Cosh[x]*Sinh[x])/2 - (Cosh[x]*Sinh[x]^2)/(I + Sinh[x])

Rule 2767

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

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin{align*} \int \frac{\sinh ^3(x)}{i+\sinh (x)} \, dx &=-\frac{\cosh (x) \sinh ^2(x)}{i+\sinh (x)}+\int \sinh (x) (-2 i+3 \sinh (x)) \, dx\\ &=-\frac{3 x}{2}-2 i \cosh (x)+\frac{3}{2} \cosh (x) \sinh (x)-\frac{\cosh (x) \sinh ^2(x)}{i+\sinh (x)}\\ \end{align*}

Mathematica [A]  time = 0.10594, size = 41, normalized size = 1.14 \[ \frac{1}{2} \cosh (x) \left (-\frac{3 \sinh ^{-1}(\sinh (x))}{\sqrt{\cosh ^2(x)}}+\frac{\sinh ^2(x)-i \sinh (x)+4}{\sinh (x)+i}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^3/(I + Sinh[x]),x]

[Out]

(Cosh[x]*((-3*ArcSinh[Sinh[x]])/Sqrt[Cosh[x]^2] + (4 - I*Sinh[x] + Sinh[x]^2)/(I + Sinh[x])))/2

________________________________________________________________________________________

Maple [B]  time = 0.034, size = 93, normalized size = 2.6 \begin{align*}{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{i \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3}{2}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{2} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{i \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{1}{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.16406, size = 61, normalized size = 1.69 \begin{align*} -\frac{3}{2} \, x - \frac{3 \, e^{\left (-x\right )} + 20 i \, e^{\left (-2 \, x\right )} + i}{8 \,{\left (-i \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}} - \frac{1}{2} i \, e^{\left (-x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x - 1/8*(3*e^(-x) + 20*I*e^(-2*x) + I)/(-I*e^(-2*x) + e^(-3*x)) - 1/2*I*e^(-x) - 1/8*e^(-2*x)

________________________________________________________________________________________

Fricas [A]  time = 2.03575, size = 153, normalized size = 4.25 \begin{align*} -\frac{4 \,{\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} -{\left (-12 i \, x - 20 i\right )} e^{\left (2 \, x\right )} - e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 \, e^{x} + i}{8 \,{\left (e^{\left (3 \, x\right )} + i \, e^{\left (2 \, x\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*(3*x - 1)*e^(3*x) - (-12*I*x - 20*I)*e^(2*x) - e^(5*x) + 3*I*e^(4*x) - 3*e^x + I)/(e^(3*x) + I*e^(2*x)
)

________________________________________________________________________________________

Sympy [A]  time = 0.310202, size = 41, normalized size = 1.14 \begin{align*} - \frac{3 x}{2} + \frac{e^{2 x}}{8} - \frac{i e^{x}}{2} - \frac{i e^{- x}}{2} - \frac{e^{- 2 x}}{8} - \frac{2 i}{e^{x} + i} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x/2 + exp(2*x)/8 - I*exp(x)/2 - I*exp(-x)/2 - exp(-2*x)/8 - 2*I/(exp(x) + I)

________________________________________________________________________________________

Giac [A]  time = 1.39074, size = 51, normalized size = 1.42 \begin{align*} -\frac{3}{2} \, x - \frac{{\left (20 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} + i\right )} e^{\left (-2 \, x\right )}}{8 \,{\left (e^{x} + i\right )}} + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{2} i \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x - 1/8*(20*I*e^(2*x) - 3*e^x + I)*e^(-2*x)/(e^x + I) + 1/8*e^(2*x) - 1/2*I*e^x