3.1.60 \(\int \frac {1}{1-i \sinh (c+d x)} \, dx\) [60]

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

[Out]

-I*cosh(d*x+c)/d/(1-I*sinh(d*x+c))

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {2727} \begin {gather*} -\frac {i \cosh (c+d x)}{d (1-i \sinh (c+d x))} \end {gather*}

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 2727

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*}

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Mathematica [A]
time = 0.05, size = 42, normalized size = 1.56 \begin {gather*} \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 )} \end {gather*}

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 = 0.86, size = 20, normalized size = 0.74

method	result	size
risch	\(-\frac {2 i}{d \left ({\mathrm e}^{d x +c}+i\right )}\)	\(18\)
derivativedivides	\(\frac {2}{d \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\)	\(20\)
default	\(\frac {2}{d \left (i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-I*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 20, normalized size = 0.74 \begin {gather*} \frac {2}{d {\left (i \, e^{\left (-d x - c\right )} + 1\right )}} \end {gather*}

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 = 0.35, size = 16, normalized size = 0.59 \begin {gather*} -\frac {2 i}{d e^{\left (d x + c\right )} + i \, d} \end {gather*}

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 = 0.05, size = 17, normalized size = 0.63 \begin {gather*} - \frac {2 i}{d e^{c} e^{d x} + i d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I/(d*exp(c)*exp(d*x) + I*d)

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Giac [A]
time = 0.42, size = 15, normalized size = 0.56 \begin {gather*} -\frac {2 i}{d {\left (e^{\left (d x + c\right )} + i\right )}} \end {gather*}

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))

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Mupad [B]
time = 0.15, size = 17, normalized size = 0.63 \begin {gather*} -\frac {2{}\mathrm {i}}{d\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(sinh(c + d*x)*1i - 1),x)

[Out]

-2i/(d*(exp(c + d*x) + 1i))

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