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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, 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 = {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 = 0.09, size = 12, normalized size = 1.00 \[ \log (a+\text {eps} \cosh (x)+i x) \]

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]]

________________________________________________________________________________________

fricas [B]  time = 0.45, size = 26, normalized size = 2.17 \[ -x + \log \left (\frac {\mathit {eps} e^{\left (2 \, x\right )} + 2 \, {\left (a + i \, x\right )} e^{x} + \mathit {eps}}{\mathit {eps}}\right ) \]

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)

________________________________________________________________________________________

giac [B]  time = 1.37, size = 23, normalized size = 1.92 \[ -x + \log \left (\mathit {eps} e^{\left (2 \, x\right )} + 2 \, a e^{x} + 2 i \, x e^{x} + \mathit {eps}\right ) \]

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)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 16, normalized size = 1.33 \[ \ln \left (i \mathit {eps} \cosh \relax (x )+i a -x \right ) \]

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))

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 13, normalized size = 1.08 \[ \log \left (i \, \mathit {eps} \cosh \relax (x) + i \, a - x\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.32, size = 13, normalized size = 1.08 \[ \ln \left (x-a\,1{}\mathrm {i}-\mathrm {eps}\,\mathrm {cosh}\relax (x)\,1{}\mathrm {i}\right ) \]

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)

________________________________________________________________________________________

sympy [B]  time = 0.39, size = 22, normalized size = 1.83 \[ - x + \log {\left (e^{2 x} + 1 + \frac {\left (2 a + 2 i x\right ) e^{x}}{eps} \right )} \]

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)

________________________________________________________________________________________