3.51 \(\int \frac {\sinh (x)}{(i+\sinh (x))^2} \, dx\)

Optimal. Leaf size=31 \[ -\frac {2 i \cosh (x)}{3 (\sinh (x)+i)}-\frac {\cosh (x)}{3 (\sinh (x)+i)^2} \]

[Out]

-1/3*cosh(x)/(I+sinh(x))^2-2/3*I*cosh(x)/(I+sinh(x))

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2750, 2648} \[ -\frac {2 i \cosh (x)}{3 (\sinh (x)+i)}-\frac {\cosh (x)}{3 (\sinh (x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(I + Sinh[x])^2,x]

[Out]

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

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2750

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((b
*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(a*f*(2*m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)
), Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 -
b^2, 0] && LtQ[m, -2^(-1)]

Rubi steps

\begin {align*} \int \frac {\sinh (x)}{(i+\sinh (x))^2} \, dx &=-\frac {\cosh (x)}{3 (i+\sinh (x))^2}+\frac {2}{3} \int \frac {1}{i+\sinh (x)} \, dx\\ &=-\frac {\cosh (x)}{3 (i+\sinh (x))^2}-\frac {2 i \cosh (x)}{3 (i+\sinh (x))}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 0.71 \[ \frac {(1-2 i \sinh (x)) \cosh (x)}{3 (\sinh (x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(I + Sinh[x])^2,x]

[Out]

(Cosh[x]*(1 - (2*I)*Sinh[x]))/(3*(I + Sinh[x])^2)

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fricas [A]  time = 0.49, size = 34, normalized size = 1.10 \[ -\frac {6 \, e^{\left (2 \, x\right )} + 6 i \, e^{x} - 4}{3 \, e^{\left (3 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} - 9 \, e^{x} - 3 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6*e^(2*x) + 6*I*e^x - 4)/(3*e^(3*x) + 9*I*e^(2*x) - 9*e^x - 3*I)

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giac [A]  time = 0.20, size = 20, normalized size = 0.65 \[ -\frac {6 \, e^{\left (2 \, x\right )} + 6 i \, e^{x} - 4}{3 \, {\left (e^{x} + i\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(6*e^(2*x) + 6*I*e^x - 4)/(e^x + I)^3

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maple [A]  time = 0.05, size = 25, normalized size = 0.81 \[ \frac {2}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}-\frac {4 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(I+sinh(x))^2,x)

[Out]

2/(tanh(1/2*x)+I)^2-4/3*I/(tanh(1/2*x)+I)^3

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maxima [B]  time = 0.31, size = 81, normalized size = 2.61 \[ -\frac {6 i \, e^{\left (-x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} + \frac {6 \, e^{\left (-2 \, x\right )}}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} - \frac {4}{9 \, e^{\left (-x\right )} + 9 i \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-3 \, x\right )} - 3 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*I*e^(-x)/(9*e^(-x) + 9*I*e^(-2*x) - 3*e^(-3*x) - 3*I) + 6*e^(-2*x)/(9*e^(-x) + 9*I*e^(-2*x) - 3*e^(-3*x) -
3*I) - 4/(9*e^(-x) + 9*I*e^(-2*x) - 3*e^(-3*x) - 3*I)

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mupad [B]  time = 0.52, size = 25, normalized size = 0.81 \[ -\frac {2\,\left (3\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,3{}\mathrm {i}+2{}\mathrm {i}\right )}{3\,{\left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(sinh(x) + 1i)^2,x)

[Out]

-(2*(3*exp(x) - exp(2*x)*3i + 2i))/(3*(exp(x)*1i - 1)^3)

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sympy [A]  time = 0.13, size = 37, normalized size = 1.19 \[ \frac {- 6 e^{2 x} - 6 i e^{x} + 4}{3 e^{3 x} + 9 i e^{2 x} - 9 e^{x} - 3 i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-6*exp(2*x) - 6*I*exp(x) + 4)/(3*exp(3*x) + 9*I*exp(2*x) - 9*exp(x) - 3*I)

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