3.262 \(\int \frac{\cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\)

Optimal. Leaf size=22 \[ \frac{x}{a}-\frac{i \cosh (c+d x)}{a d} \]

[Out]

x/a - (I*Cosh[c + d*x])/(a*d)

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Rubi [A]  time = 0.0431233, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2682, 8} \[ \frac{x}{a}-\frac{i \cosh (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^2/(a + I*a*Sinh[c + d*x]),x]

[Out]

x/a - (I*Cosh[c + d*x])/(a*d)

Rule 2682

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\cosh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac{i \cosh (c+d x)}{a d}+\frac{\int 1 \, dx}{a}\\ &=\frac{x}{a}-\frac{i \cosh (c+d x)}{a d}\\ \end{align*}

Mathematica [B]  time = 0.1447, size = 139, normalized size = 6.32 \[ \frac{\cosh ^3(c+d x) \left (-i \sqrt{1+i \sinh (c+d x)} \sinh (c+d x)+\sqrt{1+i \sinh (c+d x)}-2 \sqrt{1-i \sinh (c+d x)} \sin ^{-1}\left (\frac{\sqrt{1-i \sinh (c+d x)}}{\sqrt{2}}\right )\right )}{a d \sqrt{1+i \sinh (c+d x)} (\sinh (c+d x)-i) (\sinh (c+d x)+i)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^2/(a + I*a*Sinh[c + d*x]),x]

[Out]

(Cosh[c + d*x]^3*(-2*ArcSin[Sqrt[1 - I*Sinh[c + d*x]]/Sqrt[2]]*Sqrt[1 - I*Sinh[c + d*x]] + Sqrt[1 + I*Sinh[c +
d*x]] - I*Sqrt[1 + I*Sinh[c + d*x]]*Sinh[c + d*x]))/(a*d*Sqrt[1 + I*Sinh[c + d*x]]*(-I + Sinh[c + d*x])*(I +
Sinh[c + d*x])^2)

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Maple [B]  time = 0.044, size = 85, normalized size = 3.9 \begin{align*}{\frac{-i}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{i}{da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x)

[Out]

-I/d/a/(tanh(1/2*d*x+1/2*c)+1)+1/d/a*ln(tanh(1/2*d*x+1/2*c)+1)+I/d/a/(tanh(1/2*d*x+1/2*c)-1)-1/d/a*ln(tanh(1/2
*d*x+1/2*c)-1)

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Maxima [B]  time = 1.06201, size = 59, normalized size = 2.68 \begin{align*} \frac{d x + c}{a d} - \frac{i \, e^{\left (d x + c\right )}}{2 \, a d} - \frac{i \, e^{\left (-d x - c\right )}}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.11257, size = 92, normalized size = 4.18 \begin{align*} \frac{{\left (2 \, d x e^{\left (d x + c\right )} - i \, e^{\left (2 \, d x + 2 \, c\right )} - i\right )} e^{\left (-d x - c\right )}}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.433829, size = 82, normalized size = 3.73 \begin{align*} \begin{cases} \frac{\left (- 2 i a d e^{2 c} e^{d x} - 2 i a d e^{- d x}\right ) e^{- c}}{4 a^{2} d^{2}} & \text{for}\: 4 a^{2} d^{2} e^{c} \neq 0 \\x \left (- \frac{\left (i e^{2 c} - 2 e^{c} - i\right ) e^{- c}}{2 a} - \frac{1}{a}\right ) & \text{otherwise} \end{cases} + \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)

[Out]

Piecewise(((-2*I*a*d*exp(2*c)*exp(d*x) - 2*I*a*d*exp(-d*x))*exp(-c)/(4*a**2*d**2), Ne(4*a**2*d**2*exp(c), 0)),
(x*(-(I*exp(2*c) - 2*exp(c) - I)*exp(-c)/(2*a) - 1/a), True)) + x/a

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Giac [A]  time = 1.22428, size = 47, normalized size = 2.14 \begin{align*} \frac{2 \, d x + 2 \, c - i \, e^{\left (d x + c\right )} - i \, e^{\left (-d x - c\right )}}{2 \, a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

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