3.56 \(\int \frac{1}{1+i \sinh (c+d x)} \, dx\)

Optimal. Leaf size=27 \[ \frac{i \cosh (c+d x)}{d (1+i \sinh (c+d x))} \]

[Out]

(I*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*x]))

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Rubi [A]  time = 0.0113079, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2648} \[ \frac{i \cosh (c+d x)}{d (1+i \sinh (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(1 + I*Sinh[c + d*x])^(-1),x]

[Out]

(I*Cosh[c + d*x])/(d*(1 + I*Sinh[c + d*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]

Rubi steps

\begin{align*} \int \frac{1}{1+i \sinh (c+d x)} \, dx &=\frac{i \cosh (c+d x)}{d (1+i \sinh (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0608458, size = 42, normalized size = 1.56 \[ \frac{2 \sinh \left (\frac{1}{2} (c+d x)\right )}{d \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + I*Sinh[c + d*x])^(-1),x]

[Out]

(2*Sinh[(c + d*x)/2])/(d*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]))

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Maple [A]  time = 0.017, size = 20, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{d \left ( -i+\tanh \left ( 1/2\,dx+c/2 \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+I*sinh(d*x+c)),x)

[Out]

2/d/(-I+tanh(1/2*d*x+1/2*c))

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Maxima [A]  time = 1.18467, size = 27, normalized size = 1. \begin{align*} -\frac{2}{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(1/(1+I*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-2/(d*(I*e^(-d*x - c) - 1))

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Fricas [A]  time = 1.96137, size = 36, normalized size = 1.33 \begin{align*} \frac{2 i}{d e^{\left (d x + c\right )} - i \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c)),x, algorithm="fricas")

[Out]

2*I/(d*e^(d*x + c) - I*d)

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Sympy [A]  time = 0.273093, size = 19, normalized size = 0.7 \begin{align*} \frac{2 i e^{c}}{d \left (i e^{c} + e^{- d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c)),x)

[Out]

2*I*exp(c)/(d*(I*exp(c) + exp(-d*x)))

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Giac [A]  time = 1.3587, size = 20, normalized size = 0.74 \begin{align*} \frac{2 i}{d{\left (e^{\left (d x + c\right )} - i\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+I*sinh(d*x+c)),x, algorithm="giac")

[Out]

2*I/(d*(e^(d*x + c) - I))