3.3.3 \(\int \frac {\sinh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [203]

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\sinh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(sinh(d*x+c)^3/(f*x+e)/(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 = {} \begin {gather*} \int \frac {\sinh ^3(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]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]

[Out]

Defer[Int][Sinh[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\sinh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=\int \frac {\sinh ^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\\ \end {align*}

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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sinh[c + d*x]^3/((e + f*x)*(a + I*a*Sinh[c + d*x])),x]

[Out]

$Aborted

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{3}\left (d x +c \right )}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)

[Out]

int(sinh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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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)^3/(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(-1/4*(d*f*x + d*e - (-I*d*f*x - I*d*e)*e^(5*d
*x + 5*c) - (d*f*x + d*e)*e^(4*d*x + 4*c) + 4*(-I*d*f*x - I*d*e)*e^(3*d*x + 3*c) - 4*(d*f*x + d*e + 2*f)*e^(2*
d*x + 2*c) - (I*d*f*x + I*d*e)*e^(d*x + c))/((a*d*f^2*x^2 + 2*a*d*f*x*e + a*d*e^2)*e^(3*d*x + 3*c) - (I*a*d*f^
2*x^2 + 2*I*a*d*f*x*e + I*a*d*e^2)*e^(2*d*x + 2*c)), x) + 2)/(-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 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 \left (- \frac {i d e}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\right )\, dx + \int \left (- \frac {i d f x}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\right )\, dx + \int \frac {8 i f e^{2 c} e^{2 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\, dx + \int \left (- \frac {d e e^{c} e^{d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d e e^{3 c} e^{3 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\right )\, dx + \int \frac {d e e^{5 c} e^{5 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\, dx + \int \frac {4 i d e e^{2 c} e^{2 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\, dx + \int \frac {i d e e^{4 c} e^{4 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\, dx + \int \left (- \frac {d f x e^{c} e^{d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d f x e^{3 c} e^{3 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\right )\, dx + \int \frac {d f x e^{5 c} e^{5 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\, dx + \int \frac {4 i d f x e^{2 c} e^{2 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\, dx + \int \frac {i d f x e^{4 c} e^{4 d x}}{e^{2} e^{c} e^{3 d x} - i e^{2} e^{2 d x} + 2 e f x e^{c} e^{3 d x} - 2 i e f x e^{2 d x} + f^{2} x^{2} e^{c} e^{3 d x} - i f^{2} x^{2} e^{2 d x}}\, dx\right ) e^{- 2 c}}{4 a d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)

[Out]

2/(-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(3*d*
x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I*f*
*2*x**2*exp(2*d*x)), x) + Integral(-I*d*f*x/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3
*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x) + Integral(8*I*f*exp(
2*c)*exp(2*d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x)
+ f**2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x) + Integral(-d*e*exp(c)*exp(d*x)/(e**2*exp(c)*exp(
3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) -
I*f**2*x**2*exp(2*d*x)), x) + Integral(-4*d*e*exp(3*c)*exp(3*d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x)
+ 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x)
+ Integral(d*e*exp(5*c)*exp(5*d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) -
2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x) + Integral(4*I*d*e*exp(2*c)*e
xp(2*d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**
2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x) + Integral(I*d*e*exp(4*c)*exp(4*d*x)/(e**2*exp(c)*exp(3
*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I
*f**2*x**2*exp(2*d*x)), x) + Integral(-d*f*x*exp(c)*exp(d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x) + 2*e
*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x) + In
tegral(-4*d*f*x*exp(3*c)*exp(3*d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) -
2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x) + Integral(d*f*x*exp(5*c)*exp
(5*d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*
x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)), x) + Integral(4*I*d*f*x*exp(2*c)*exp(2*d*x)/(e**2*exp(c)*exp
(3*d*x) - I*e**2*exp(2*d*x) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) -
I*f**2*x**2*exp(2*d*x)), x) + Integral(I*d*f*x*exp(4*c)*exp(4*d*x)/(e**2*exp(c)*exp(3*d*x) - I*e**2*exp(2*d*x
) + 2*e*f*x*exp(c)*exp(3*d*x) - 2*I*e*f*x*exp(2*d*x) + f**2*x**2*exp(c)*exp(3*d*x) - I*f**2*x**2*exp(2*d*x)),
x))*exp(-2*c)/(4*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)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

integrate(sinh(d*x + c)^3/((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 )}^3}{\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)^3/((e + f*x)*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int(sinh(c + d*x)^3/((e + f*x)*(a + a*sinh(c + d*x)*1i)), x)

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