3.92 \(\int \frac{1}{5+3 i \sinh (c+d x)} \, dx\)

Optimal. Leaf size=37 \[ \frac{x}{4}-\frac{i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \]

[Out]

x/4 - ((I/2)*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/d

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Rubi [A]  time = 0.0132772, antiderivative size = 37, 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 = {2657} \[ \frac{x}{4}-\frac{i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(5 + (3*I)*Sinh[c + d*x])^(-1),x]

[Out]

x/4 - ((I/2)*ArcTan[Cosh[c + d*x]/(3 + I*Sinh[c + d*x])])/d

Rule 2657

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

Rubi steps

\begin{align*} \int \frac{1}{5+3 i \sinh (c+d x)} \, dx &=\frac{x}{4}-\frac{i \tan ^{-1}\left (\frac{\cosh (c+d x)}{3+i \sinh (c+d x)}\right )}{2 d}\\ \end{align*}

Mathematica [B]  time = 0.0335293, size = 171, normalized size = 4.62 \[ -\frac{\log (5 \cosh (c+d x)-4 \sinh (c+d x))}{8 d}+\frac{\log (4 \sinh (c+d x)+5 \cosh (c+d x))}{8 d}-\frac{i \tan ^{-1}\left (\frac{2 \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )}{\cosh \left (\frac{1}{2} (c+d x)\right )-2 \sinh \left (\frac{1}{2} (c+d x)\right )}\right )}{4 d}+\frac{i \tan ^{-1}\left (\frac{2 \sinh \left (\frac{1}{2} (c+d x)\right )+\cosh \left (\frac{1}{2} (c+d x)\right )}{\sinh \left (\frac{1}{2} (c+d x)\right )+2 \cosh \left (\frac{1}{2} (c+d x)\right )}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + (3*I)*Sinh[c + d*x])^(-1),x]

[Out]

((-I/4)*ArcTan[(2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2])/(Cosh[(c + d*x)/2] - 2*Sinh[(c + d*x)/2])])/d + ((I/4
)*ArcTan[(Cosh[(c + d*x)/2] + 2*Sinh[(c + d*x)/2])/(2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2])])/d - Log[5*Cosh[
c + d*x] - 4*Sinh[c + d*x]]/(8*d) + Log[5*Cosh[c + d*x] + 4*Sinh[c + d*x]]/(8*d)

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Maple [A]  time = 0.023, size = 44, normalized size = 1.2 \begin{align*}{\frac{1}{4\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) +4-3\,i \right ) }-{\frac{1}{4\,d}\ln \left ( 5\,\tanh \left ( 1/2\,dx+c/2 \right ) -4-3\,i \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5+3*I*sinh(d*x+c)),x)

[Out]

1/4/d*ln(5*tanh(1/2*d*x+1/2*c)+4-3*I)-1/4/d*ln(5*tanh(1/2*d*x+1/2*c)-4-3*I)

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Maxima [A]  time = 1.76108, size = 49, normalized size = 1.32 \begin{align*} \frac{\log \left (-\frac{6 \,{\left (-i \, e^{\left (-d x - c\right )} + 3\right )}}{6 i \, e^{\left (-d x - c\right )} - 2}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.02601, size = 80, normalized size = 2.16 \begin{align*} \frac{\log \left (e^{\left (d x + c\right )} - \frac{1}{3} i\right ) - \log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(log(e^(d*x + c) - 1/3*I) - log(e^(d*x + c) - 3*I))/d

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Sympy [A]  time = 0.599206, size = 31, normalized size = 0.84 \begin{align*} \frac{- \frac{\log{\left (e^{d x} - 3 i e^{- c} \right )}}{4} + \frac{\log{\left (e^{d x} - \frac{i e^{- c}}{3} \right )}}{4}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c)),x)

[Out]

(-log(exp(d*x) - 3*I*exp(-c))/4 + log(exp(d*x) - I*exp(-c)/3)/4)/d

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Giac [A]  time = 1.24547, size = 42, normalized size = 1.14 \begin{align*} \frac{\log \left (3 \, e^{\left (d x + c\right )} - i\right )}{4 \, d} - \frac{\log \left (e^{\left (d x + c\right )} - 3 i\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5+3*I*sinh(d*x+c)),x, algorithm="giac")

[Out]

1/4*log(3*e^(d*x + c) - I)/d - 1/4*log(e^(d*x + c) - 3*I)/d