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

Optimal. Leaf size=32 \[ \text {Int}\left (\frac {\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 41.75, size = 0, normalized size = 0.00 \[ \int \frac {\sinh (c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ \frac {{\left (-i \, a d f^{2} x^{2} - 2 i \, a d e f x - i \, a d e^{2} + {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm integral}\left (-\frac {d f x + d e - {\left (-i \, d f x - i \, d e\right )} e^{\left (d x + c\right )} + 4 \, f}{-i \, a d f^{3} x^{3} - 3 i \, a d e f^{2} x^{2} - 3 i \, a d e^{2} f x - i \, a d e^{3} + {\left (a d f^{3} x^{3} + 3 \, a d e f^{2} x^{2} + 3 \, a d e^{2} f x + a d e^{3}\right )} e^{\left (d x + c\right )}}, x\right ) - 2}{-i \, a d f^{2} x^{2} - 2 i \, a d e f x - i \, a d e^{2} + {\left (a d f^{2} x^{2} + 2 \, a d e f x + a d e^{2}\right )} e^{\left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +i a \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -4 \, f \int \frac {1}{-i \, a d f^{3} x^{3} - 3 i \, a d e f^{2} x^{2} - 3 i \, a d e^{2} f x - i \, a d e^{3} + {\left (a d f^{3} x^{3} e^{c} + 3 \, a d e f^{2} x^{2} e^{c} + 3 \, a d e^{2} f x e^{c} + a d e^{3} e^{c}\right )} e^{\left (d x\right )}}\,{d x} + \frac {2 \, d f x + 2 \, d e + {\left (2 i \, d f x e^{c} + 2 i \, d e e^{c}\right )} e^{\left (d x\right )} - 4 \, f}{2 \, {\left (-i \, a d f^{3} x^{2} - 2 i \, a d e f^{2} x - i \, a d e^{2} f + {\left (a d f^{3} x^{2} e^{c} + 2 \, a d e f^{2} x e^{c} + a d e^{2} f e^{c}\right )} e^{\left (d x\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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