3.2.97 \(\int \frac {\sinh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [197]

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\sinh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sinh(d*x+c)^2/(f*x+e)/(a+I*a*sinh(d*x+c)),x)

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Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sinh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 170.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{2}\left (d x +c \right )}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*f*integrate(1/(-I*a*d*f^2*x^2 - 2*I*a*d*f*x*e - I*a*d*e^2 + (a*d*f^2*x^2*e^c + 2*a*d*f*x*e^(c + 1) + a*d*
e^(c + 2))*e^(d*x)), x) - 1/2*I*e^(-c + d*e/f)*exp_integral_e(1, (f*x + e)*d/f)/(a*f) + 1/2*I*e^(c - d*e/f)*ex
p_integral_e(1, -(f*x + e)*d/f)/(a*f) - 2*I/(-I*a*d*f*x - I*a*d*e + (a*d*f*x*e^c + a*d*e^(c + 1))*e^(d*x)) + l
og(f*x + e)/(a*f)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-I*a*d*f*x - I*a*d*e + (a*d*f*x + a*d*e)*e^(d*x + c))*integral(1/2*(d*f*x + d*e + (-I*d*f*x - I*d*e)*e^(3*d*
x + 3*c) + (d*f*x + d*e)*e^(2*d*x + 2*c) + (-I*d*f*x - I*d*e - 4*I*f)*e^(d*x + c))/((a*d*f^2*x^2 + 2*a*d*f*x*e
+ a*d*e^2)*e^(2*d*x + 2*c) - (I*a*d*f^2*x^2 + 2*I*a*d*f*x*e + I*a*d*e^2)*e^(d*x + c)), x) - 2*I)/(-I*a*d*f*x
- I*a*d*e + (a*d*f*x + a*d*e)*e^(d*x + c))

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {2 i}{- i a d e - i a d f x + \left (a d e e^{c} + a d f x e^{c}\right ) e^{d x}} - \frac {i \left (\int \frac {i d e}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {4 f e^{c} e^{d x}}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {i d f x}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {d e e^{c} e^{d x}}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {d e e^{3 c} e^{3 d x}}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {i d e e^{2 c} e^{2 d x}}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {d f x e^{c} e^{d x}}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {d f x e^{3 c} e^{3 d x}}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx + \int \frac {i d f x e^{2 c} e^{2 d x}}{e^{2} e^{c} e^{2 d x} - i e^{2} e^{d x} + 2 e f x e^{c} e^{2 d x} - 2 i e f x e^{d x} + f^{2} x^{2} e^{c} e^{2 d x} - i f^{2} x^{2} e^{d x}}\, dx\right ) e^{- c}}{2 a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I/(-I*a*d*e - I*a*d*f*x + (a*d*e*exp(c) + a*d*f*x*exp(c))*exp(d*x)) - I*(Integral(I*d*e/(e**2*exp(c)*exp(2*
d*x) - I*e**2*exp(d*x) + 2*e*f*x*exp(c)*exp(2*d*x) - 2*I*e*f*x*exp(d*x) + f**2*x**2*exp(c)*exp(2*d*x) - I*f**2
*x**2*exp(d*x)), x) + Integral(4*f*exp(c)*exp(d*x)/(e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*x) + 2*e*f*x*exp(c)*
exp(2*d*x) - 2*I*e*f*x*exp(d*x) + f**2*x**2*exp(c)*exp(2*d*x) - I*f**2*x**2*exp(d*x)), x) + Integral(I*d*f*x/(
e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*x) + 2*e*f*x*exp(c)*exp(2*d*x) - 2*I*e*f*x*exp(d*x) + f**2*x**2*exp(c)*e
xp(2*d*x) - I*f**2*x**2*exp(d*x)), x) + Integral(d*e*exp(c)*exp(d*x)/(e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*x)
+ 2*e*f*x*exp(c)*exp(2*d*x) - 2*I*e*f*x*exp(d*x) + f**2*x**2*exp(c)*exp(2*d*x) - I*f**2*x**2*exp(d*x)), x) +
Integral(d*e*exp(3*c)*exp(3*d*x)/(e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*x) + 2*e*f*x*exp(c)*exp(2*d*x) - 2*I*e
*f*x*exp(d*x) + f**2*x**2*exp(c)*exp(2*d*x) - I*f**2*x**2*exp(d*x)), x) + Integral(I*d*e*exp(2*c)*exp(2*d*x)/(
e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*x) + 2*e*f*x*exp(c)*exp(2*d*x) - 2*I*e*f*x*exp(d*x) + f**2*x**2*exp(c)*e
xp(2*d*x) - I*f**2*x**2*exp(d*x)), x) + Integral(d*f*x*exp(c)*exp(d*x)/(e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*
x) + 2*e*f*x*exp(c)*exp(2*d*x) - 2*I*e*f*x*exp(d*x) + f**2*x**2*exp(c)*exp(2*d*x) - I*f**2*x**2*exp(d*x)), x)
+ Integral(d*f*x*exp(3*c)*exp(3*d*x)/(e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*x) + 2*e*f*x*exp(c)*exp(2*d*x) - 2
*I*e*f*x*exp(d*x) + f**2*x**2*exp(c)*exp(2*d*x) - I*f**2*x**2*exp(d*x)), x) + Integral(I*d*f*x*exp(2*c)*exp(2*
d*x)/(e**2*exp(c)*exp(2*d*x) - I*e**2*exp(d*x) + 2*e*f*x*exp(c)*exp(2*d*x) - 2*I*e*f*x*exp(d*x) + f**2*x**2*ex
p(c)*exp(2*d*x) - I*f**2*x**2*exp(d*x)), x))*exp(-c)/(2*a*d)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sinh(d*x + c)^2/((f*x + e)*(I*a*sinh(d*x + c) + a)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^2/((e + f*x)*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int(sinh(c + d*x)^2/((e + f*x)*(a + a*sinh(c + d*x)*1i)), x)

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