3.3.28 \(\int \frac {-1+i \text {eps} \sinh (x)}{i a-x+i \text {eps} \cosh (x)} \, dx\) [228]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {6816} \begin {gather*} \log (a+\text {eps} \cosh (x)+i x) \end {gather*}

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 6816

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*}

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Mathematica [A]
time = 0.07, size = 12, normalized size = 1.00 \begin {gather*} \log (a+i x+\text {eps} \cosh (x)) \end {gather*}

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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Mathics [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(31\) vs. \(2(12)=24\).
time = 2.10, size = 29, normalized size = 2.42 \begin {gather*} -x+\text {Log}\left [\frac {\text {eps} \left (1+E^{2 x}\right )+2 \left (a+I x\right ) E^x}{\text {eps}}\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-x + Log[(eps (1 + E ^ (2 x)) + 2 (a + I x) E ^ x) / eps]

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Maple [A]
time = 0.03, size = 16, normalized size = 1.33

method	result	size
derivativedivides	\(\ln \left (i a -x +i \mathit {eps} \cosh \left (x \right )\right )\)	\(16\)
default	\(\ln \left (i a -x +i \mathit {eps} \cosh \left (x \right )\right )\)	\(16\)
risch	\(-x +\ln \left (1+\frac {2 \left (i x +a \right ) {\mathrm e}^{x}}{\mathit {eps}}+{\mathrm e}^{2 x}\right )\)	\(25\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.25, size = 13, normalized size = 1.08 \begin {gather*} \log \left (i \, \mathit {eps} \cosh \left (x\right ) + i \, a - x\right ) \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).
time = 0.31, size = 26, normalized size = 2.17 \begin {gather*} -x + \log \left (\frac {\mathit {eps} e^{\left (2 \, x\right )} + 2 \, {\left (a + i \, x\right )} e^{x} + \mathit {eps}}{\mathit {eps}}\right ) \end {gather*}

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 + I*x)*e^x + eps)/eps)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\)
time = 0.17, size = 22, normalized size = 1.83 \begin {gather*} - x + \log {\left (e^{2 x} + 1 + \frac {\left (2 a + 2 i x\right ) e^{x}}{eps} \right )} \end {gather*}

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] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (10) = 20\).
time = 0.02, size = 27, normalized size = 2.25 \begin {gather*} -x+\ln \left (2 I a \mathrm {e}^{x}+I\cdot \mathrm {eps}\cdot \left (\mathrm {e}^{x}\right )^{2}+I\cdot \mathrm {eps}-2 x \mathrm {e}^{x}\right ) \end {gather*}

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

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Mupad [B]
time = 0.32, size = 13, normalized size = 1.08 \begin {gather*} \ln \left (x-a\,1{}\mathrm {i}-\mathrm {eps}\,\mathrm {cosh}\left (x\right )\,1{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - a*1i - eps*cosh(x)*1i)

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