3.256 \(\int \frac{\cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=23 \[ -\frac{i \log (-\sinh (c+d x)+i)}{a d} \]

[Out]

((-I)*Log[I - Sinh[c + d*x]])/(a*d)

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Rubi [A]  time = 0.0273148, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2667, 31} \[ -\frac{i \log (-\sinh (c+d x)+i)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(a + I*a*Sinh[c + d*x]),x]

[Out]

((-I)*Log[I - Sinh[c + d*x]])/(a*d)

Rule 2667

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,i a \sinh (c+d x)\right )}{a d}\\ &=-\frac{i \log (i-\sinh (c+d x))}{a d}\\ \end{align*}

Mathematica [A]  time = 0.0145465, size = 23, normalized size = 1. \[ -\frac{i \log (-\sinh (c+d x)+i)}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]/(a + I*a*Sinh[c + d*x]),x]

[Out]

((-I)*Log[I - Sinh[c + d*x]])/(a*d)

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Maple [A]  time = 0.013, size = 23, normalized size = 1. \begin{align*}{\frac{-i\ln \left ( a+ia\sinh \left ( dx+c \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)/(a+I*a*sinh(d*x+c)),x)

[Out]

-I/d*ln(a+I*a*sinh(d*x+c))/a

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Maxima [A]  time = 1.12375, size = 27, normalized size = 1.17 \begin{align*} -\frac{i \, \log \left (i \, a \sinh \left (d x + c\right ) + a\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-I*log(I*a*sinh(d*x + c) + a)/(a*d)

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Fricas [A]  time = 2.18425, size = 57, normalized size = 2.48 \begin{align*} \frac{i \, d x - 2 i \, \log \left (e^{\left (d x + c\right )} - i\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.53724, size = 22, normalized size = 0.96 \begin{align*} \frac{i x}{a} - \frac{2 i \log{\left (e^{d x} - i e^{- c} \right )}}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+I*a*sinh(d*x+c)),x)

[Out]

I*x/a - 2*I*log(exp(d*x) - I*exp(-c))/(a*d)

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Giac [A]  time = 1.23608, size = 45, normalized size = 1.96 \begin{align*} \frac{i \,{\left (d x + c\right )}}{a d} - \frac{2 i \, \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

I*(d*x + c)/(a*d) - 2*I*log(I*e^(d*x + c) + 1)/(a*d)