3.228 \(\int \frac{-1+i \text{eps} \sinh (x)}{i a-x+i \text{eps} \cosh (x)} \, dx\)

Optimal. Leaf size=12 \[ \log (a+\text{eps} \cosh (x)+i x) \]

[Out]

Log[a + I*x + eps*Cosh[x]]

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Rubi [A]  time = 0.0325908, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {6684} \[ \log (a+\text{eps} \cosh (x)+i x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + I*eps*Sinh[x])/(I*a - x + I*eps*Cosh[x]),x]

[Out]

Log[a + I*x + eps*Cosh[x]]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin{align*} \int \frac{-1+i \text{eps} \sinh (x)}{i a-x+i \text{eps} \cosh (x)} \, dx &=\log (a+i x+\text{eps} \cosh (x))\\ \end{align*}

Mathematica [A]  time = 0.0827899, size = 12, normalized size = 1. \[ \log (a+\text{eps} \cosh (x)+i x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + I*eps*Sinh[x])/(I*a - x + I*eps*Cosh[x]),x]

[Out]

Log[a + I*x + eps*Cosh[x]]

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Maple [A]  time = 0.011, size = 16, normalized size = 1.3 \begin{align*} \ln \left ( ia-x+i{\it eps}\,\cosh \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+I*eps*sinh(x))/(I*a-x+I*eps*cosh(x)),x)

[Out]

ln(I*a-x+I*eps*cosh(x))

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Maxima [A]  time = 0.966939, size = 18, normalized size = 1.5 \begin{align*} \log \left (i \, \mathit{eps} \cosh \left (x\right ) + i \, a - x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+I*eps*sinh(x))/(I*a-x+I*eps*cosh(x)),x, algorithm="maxima")

[Out]

log(I*eps*cosh(x) + I*a - x)

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Fricas [B]  time = 1.72218, size = 74, normalized size = 6.17 \begin{align*} -x + \log \left (\frac{\mathit{eps} e^{\left (2 \, x\right )} +{\left (2 \, a + 2 i \, x\right )} e^{x} + \mathit{eps}}{\mathit{eps}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+I*eps*sinh(x))/(I*a-x+I*eps*cosh(x)),x, algorithm="fricas")

[Out]

-x + log((eps*e^(2*x) + (2*a + 2*I*x)*e^x + eps)/eps)

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Sympy [B]  time = 1.33109, size = 22, normalized size = 1.83 \begin{align*} - x + \log{\left (e^{2 x} + 1 + \frac{\left (2 a + 2 i x\right ) e^{x}}{eps} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+I*eps*sinh(x))/(I*a-x+I*eps*cosh(x)),x)

[Out]

-x + log(exp(2*x) + 1 + (2*a + 2*I*x)*exp(x)/eps)

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Giac [B]  time = 1.08743, size = 31, normalized size = 2.58 \begin{align*} -x + \log \left (\mathit{eps} e^{\left (2 \, x\right )} + 2 \, a e^{x} + 2 i \, x e^{x} + \mathit{eps}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+I*eps*sinh(x))/(I*a-x+I*eps*cosh(x)),x, algorithm="giac")

[Out]

-x + log(eps*e^(2*x) + 2*a*e^x + 2*I*x*e^x + eps)