3.2.80 \(\int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx\) [180]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3},x\right ) \]

[Out]

Unintegrable(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x)

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Rubi [A]
time = 0.04, 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 {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3),x]

[Out]

Defer[Int][1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3), x]

Rubi steps

\begin {align*} \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx &=\int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 65.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3),x]

[Out]

Integrate[1/((e + f*x)^2*(a + b*Sinh[c + d*x])^3), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (f x +e \right )^{2} \left (a +b \sinh \left (d x +c \right )\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x)

[Out]

int(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,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(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x, algorithm="maxima")

[Out]

(3*a*b^2*d*f*x + 3*a*b^2*d*e + (2*a^2*b*f*e^(3*c) + 2*b^3*f*e^(3*c) + (2*a^2*b*d*f*e^(3*c) - b^3*d*f*e^(3*c))*
x + (2*a^2*b*d*e^(3*c) - b^3*d*e^(3*c))*e)*e^(3*d*x) + (4*a^3*f*e^(2*c) + 4*a*b^2*f*e^(2*c) + 3*(2*a^3*d*f*e^(
2*c) - a*b^2*d*f*e^(2*c))*x + 3*(2*a^3*d*e^(2*c) - a*b^2*d*e^(2*c))*e)*e^(2*d*x) - (2*a^2*b*f*e^c + 2*b^3*f*e^
c + (10*a^2*b*d*f*e^c + b^3*d*f*e^c)*x + (10*a^2*b*d*e^c + b^3*d*e^c)*e)*e^(d*x))/((a^4*b^2*d^2*f^3 + 2*a^2*b^
4*d^2*f^3 + b^6*d^2*f^3)*x^3 + 3*(a^4*b^2*d^2*f^2 + 2*a^2*b^4*d^2*f^2 + b^6*d^2*f^2)*x^2*e + 3*(a^4*b^2*d^2*f
+ 2*a^2*b^4*d^2*f + b^6*d^2*f)*x*e^2 + (a^4*b^2*d^2 + 2*a^2*b^4*d^2 + b^6*d^2)*e^3 + ((a^4*b^2*d^2*f^3*e^(4*c)
+ 2*a^2*b^4*d^2*f^3*e^(4*c) + b^6*d^2*f^3*e^(4*c))*x^3 + 3*(a^4*b^2*d^2*f^2*e^(4*c) + 2*a^2*b^4*d^2*f^2*e^(4*
c) + b^6*d^2*f^2*e^(4*c))*x^2*e + 3*(a^4*b^2*d^2*f*e^(4*c) + 2*a^2*b^4*d^2*f*e^(4*c) + b^6*d^2*f*e^(4*c))*x*e^
2 + (a^4*b^2*d^2*e^(4*c) + 2*a^2*b^4*d^2*e^(4*c) + b^6*d^2*e^(4*c))*e^3)*e^(4*d*x) + 4*((a^5*b*d^2*f^3*e^(3*c)
+ 2*a^3*b^3*d^2*f^3*e^(3*c) + a*b^5*d^2*f^3*e^(3*c))*x^3 + 3*(a^5*b*d^2*f^2*e^(3*c) + 2*a^3*b^3*d^2*f^2*e^(3*
c) + a*b^5*d^2*f^2*e^(3*c))*x^2*e + 3*(a^5*b*d^2*f*e^(3*c) + 2*a^3*b^3*d^2*f*e^(3*c) + a*b^5*d^2*f*e^(3*c))*x*
e^2 + (a^5*b*d^2*e^(3*c) + 2*a^3*b^3*d^2*e^(3*c) + a*b^5*d^2*e^(3*c))*e^3)*e^(3*d*x) + 2*((2*a^6*d^2*f^3*e^(2*
c) + 3*a^4*b^2*d^2*f^3*e^(2*c) - b^6*d^2*f^3*e^(2*c))*x^3 + 3*(2*a^6*d^2*f^2*e^(2*c) + 3*a^4*b^2*d^2*f^2*e^(2*
c) - b^6*d^2*f^2*e^(2*c))*x^2*e + 3*(2*a^6*d^2*f*e^(2*c) + 3*a^4*b^2*d^2*f*e^(2*c) - b^6*d^2*f*e^(2*c))*x*e^2
+ (2*a^6*d^2*e^(2*c) + 3*a^4*b^2*d^2*e^(2*c) - b^6*d^2*e^(2*c))*e^3)*e^(2*d*x) - 4*((a^5*b*d^2*f^3*e^c + 2*a^3
*b^3*d^2*f^3*e^c + a*b^5*d^2*f^3*e^c)*x^3 + 3*(a^5*b*d^2*f^2*e^c + 2*a^3*b^3*d^2*f^2*e^c + a*b^5*d^2*f^2*e^c)*
x^2*e + 3*(a^5*b*d^2*f*e^c + 2*a^3*b^3*d^2*f*e^c + a*b^5*d^2*f*e^c)*x*e^2 + (a^5*b*d^2*e^c + 2*a^3*b^3*d^2*e^c
+ a*b^5*d^2*e^c)*e^3)*e^(d*x)) + integrate((6*a*b*d*f^2*x + 6*a*b*d*f*e - (6*a^2*d*f*e^(c + 1) + 6*a^2*f^2*e^
c + 6*b^2*f^2*e^c + (2*a^2*d^2*f^2*e^c - b^2*d^2*f^2*e^c)*x^2 + 2*(3*a^2*d*f^2*e^c + (2*a^2*d^2*f*e^c - b^2*d^
2*f*e^c)*e)*x + (2*a^2*d^2*e^c - b^2*d^2*e^c)*e^2)*e^(d*x))/((a^4*b*d^2*f^4 + 2*a^2*b^3*d^2*f^4 + b^5*d^2*f^4)
*x^4 + 4*(a^4*b*d^2*f^3 + 2*a^2*b^3*d^2*f^3 + b^5*d^2*f^3)*x^3*e + 6*(a^4*b*d^2*f^2 + 2*a^2*b^3*d^2*f^2 + b^5*
d^2*f^2)*x^2*e^2 + 4*(a^4*b*d^2*f + 2*a^2*b^3*d^2*f + b^5*d^2*f)*x*e^3 + (a^4*b*d^2 + 2*a^2*b^3*d^2 + b^5*d^2)
*e^4 - ((a^4*b*d^2*f^4*e^(2*c) + 2*a^2*b^3*d^2*f^4*e^(2*c) + b^5*d^2*f^4*e^(2*c))*x^4 + 4*(a^4*b*d^2*f^3*e^(2*
c) + 2*a^2*b^3*d^2*f^3*e^(2*c) + b^5*d^2*f^3*e^(2*c))*x^3*e + 6*(a^4*b*d^2*f^2*e^(2*c) + 2*a^2*b^3*d^2*f^2*e^(
2*c) + b^5*d^2*f^2*e^(2*c))*x^2*e^2 + 4*(a^4*b*d^2*f*e^(2*c) + 2*a^2*b^3*d^2*f*e^(2*c) + b^5*d^2*f*e^(2*c))*x*
e^3 + (a^4*b*d^2*e^(2*c) + 2*a^2*b^3*d^2*e^(2*c) + b^5*d^2*e^(2*c))*e^4)*e^(2*d*x) - 2*((a^5*d^2*f^4*e^c + 2*a
^3*b^2*d^2*f^4*e^c + a*b^4*d^2*f^4*e^c)*x^4 + 4*(a^5*d^2*f^3*e^c + 2*a^3*b^2*d^2*f^3*e^c + a*b^4*d^2*f^3*e^c)*
x^3*e + 6*(a^5*d^2*f^2*e^c + 2*a^3*b^2*d^2*f^2*e^c + a*b^4*d^2*f^2*e^c)*x^2*e^2 + 4*(a^5*d^2*f*e^c + 2*a^3*b^2
*d^2*f*e^c + a*b^4*d^2*f*e^c)*x*e^3 + (a^5*d^2*e^c + 2*a^3*b^2*d^2*e^c + a*b^4*d^2*e^c)*e^4)*e^(d*x)), x)

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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(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(1/(a^3*f^2*x^2 + 2*a^3*f*x*e + a^3*e^2 + (b^3*f^2*x^2 + 2*b^3*f*x*e + b^3*e^2)*sinh(d*x + c)^3 + 3*(a
*b^2*f^2*x^2 + 2*a*b^2*f*x*e + a*b^2*e^2)*sinh(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*f*x*e + a^2*b*e^2)*sinh
(d*x + c)), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(f*x+e)**2/(a+b*sinh(d*x+c))**3,x)

[Out]

Timed out

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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(1/(f*x+e)^2/(a+b*sinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate(1/((f*x + e)^2*(b*sinh(d*x + c) + a)^3), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e + f*x)^2*(a + b*sinh(c + d*x))^3),x)

[Out]

int(1/((e + f*x)^2*(a + b*sinh(c + d*x))^3), x)

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