3.180 \(\int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx\)
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 = 0.06, 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 =
{} \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx \]
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 = 93.77, size = 0, normalized size = 0.00 \[ \int \frac {1}{(e+f x)^2 (a+b \sinh (c+d x))^3} \, dx \]
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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fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{3} f^{2} x^{2} + 2 \, a^{3} e f x + a^{3} e^{2} + {\left (b^{3} f^{2} x^{2} + 2 \, b^{3} e f x + b^{3} e^{2}\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (a b^{2} f^{2} x^{2} + 2 \, a b^{2} e f x + a b^{2} e^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, {\left (a^{2} b f^{2} x^{2} + 2 \, a^{2} b e f x + a^{2} b e^{2}\right )} \sinh \left (d x + c\right )}, x\right ) \]
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*e*f*x + a^3*e^2 + (b^3*f^2*x^2 + 2*b^3*e*f*x + b^3*e^2)*sinh(d*x + c)^3 + 3*(a
*b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e^2)*sinh(d*x + c)^2 + 3*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x + a^2*b*e^2)*sinh
(d*x + c)), x)
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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]
Timed out
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maple [A] time = 1.08, 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 \[ \text {result too large to display} \]
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*(d*e + f)*a^2*b*e^(3*c) - (d*e - 2*f)*b^3*e^(3*c) + (2*a^2*b*d*f*e^(3*c) - b
^3*d*f*e^(3*c))*x)*e^(3*d*x) + (2*(3*d*e + 2*f)*a^3*e^(2*c) - (3*d*e - 4*f)*a*b^2*e^(2*c) + 3*(2*a^3*d*f*e^(2*
c) - a*b^2*d*f*e^(2*c))*x)*e^(2*d*x) - (2*(5*d*e + f)*a^2*b*e^c + (d*e + 2*f)*b^3*e^c + (10*a^2*b*d*f*e^c + b^
3*d*f*e^c)*x)*e^(d*x))/(a^4*b^2*d^2*e^3 + 2*a^2*b^4*d^2*e^3 + b^6*d^2*e^3 + (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*e*f^2 + 2*a^2*b^4*d^2*e*f^2 + b^6*d^2*e*f^2)*x^2 + 3*(a^4*b^2*d^2*e^2*f
+ 2*a^2*b^4*d^2*e^2*f + b^6*d^2*e^2*f)*x + (a^4*b^2*d^2*e^3*e^(4*c) + 2*a^2*b^4*d^2*e^3*e^(4*c) + b^6*d^2*e^3
*e^(4*c) + (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*e*
f^2*e^(4*c) + 2*a^2*b^4*d^2*e*f^2*e^(4*c) + b^6*d^2*e*f^2*e^(4*c))*x^2 + 3*(a^4*b^2*d^2*e^2*f*e^(4*c) + 2*a^2*
b^4*d^2*e^2*f*e^(4*c) + b^6*d^2*e^2*f*e^(4*c))*x)*e^(4*d*x) + 4*(a^5*b*d^2*e^3*e^(3*c) + 2*a^3*b^3*d^2*e^3*e^(
3*c) + a*b^5*d^2*e^3*e^(3*c) + (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*e*f^2*e^(3*c) + 2*a^3*b^3*d^2*e*f^2*e^(3*c) + a*b^5*d^2*e*f^2*e^(3*c))*x^2 + 3*(a^5*b*d^2*e^2*
f*e^(3*c) + 2*a^3*b^3*d^2*e^2*f*e^(3*c) + a*b^5*d^2*e^2*f*e^(3*c))*x)*e^(3*d*x) + 2*(2*a^6*d^2*e^3*e^(2*c) + 3
*a^4*b^2*d^2*e^3*e^(2*c) - b^6*d^2*e^3*e^(2*c) + (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*e*f^2*e^(2*c) + 3*a^4*b^2*d^2*e*f^2*e^(2*c) - b^6*d^2*e*f^2*e^(2*c))*x^2 + 3*(
2*a^6*d^2*e^2*f*e^(2*c) + 3*a^4*b^2*d^2*e^2*f*e^(2*c) - b^6*d^2*e^2*f*e^(2*c))*x)*e^(2*d*x) - 4*(a^5*b*d^2*e^3
*e^c + 2*a^3*b^3*d^2*e^3*e^c + a*b^5*d^2*e^3*e^c + (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*e*f^2*e^c + 2*a^3*b^3*d^2*e*f^2*e^c + a*b^5*d^2*e*f^2*e^c)*x^2 + 3*(a^5*b*d^2*e^2*f*e^
c + 2*a^3*b^3*d^2*e^2*f*e^c + a*b^5*d^2*e^2*f*e^c)*x)*e^(d*x)) + integrate((6*a*b*d*f^2*x + 6*a*b*d*e*f - (2*(
d^2*e^2 + 3*d*e*f + 3*f^2)*a^2*e^c - (d^2*e^2 - 6*f^2)*b^2*e^c + (2*a^2*d^2*f^2*e^c - b^2*d^2*f^2*e^c)*x^2 - 2
*(b^2*d^2*e*f*e^c - (2*d^2*e*f + 3*d*f^2)*a^2*e^c)*x)*e^(d*x))/(a^4*b*d^2*e^4 + 2*a^2*b^3*d^2*e^4 + b^5*d^2*e^
4 + (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*e*f^3 + 2*a^2*b^3*d^2*e*f^3 + b^5*d^2
*e*f^3)*x^3 + 6*(a^4*b*d^2*e^2*f^2 + 2*a^2*b^3*d^2*e^2*f^2 + b^5*d^2*e^2*f^2)*x^2 + 4*(a^4*b*d^2*e^3*f + 2*a^2
*b^3*d^2*e^3*f + b^5*d^2*e^3*f)*x - (a^4*b*d^2*e^4*e^(2*c) + 2*a^2*b^3*d^2*e^4*e^(2*c) + b^5*d^2*e^4*e^(2*c) +
(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*e*f^3*e^(2*c) +
2*a^2*b^3*d^2*e*f^3*e^(2*c) + b^5*d^2*e*f^3*e^(2*c))*x^3 + 6*(a^4*b*d^2*e^2*f^2*e^(2*c) + 2*a^2*b^3*d^2*e^2*f^
2*e^(2*c) + b^5*d^2*e^2*f^2*e^(2*c))*x^2 + 4*(a^4*b*d^2*e^3*f*e^(2*c) + 2*a^2*b^3*d^2*e^3*f*e^(2*c) + b^5*d^2*
e^3*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^5*d^2*e^4*e^c + 2*a^3*b^2*d^2*e^4*e^c + a*b^4*d^2*e^4*e^c + (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*e*f^3*e^c + 2*a^3*b^2*d^2*e*f^3*e^c + a*b^4*d^
2*e*f^3*e^c)*x^3 + 6*(a^5*d^2*e^2*f^2*e^c + 2*a^3*b^2*d^2*e^2*f^2*e^c + a*b^4*d^2*e^2*f^2*e^c)*x^2 + 4*(a^5*d^
2*e^3*f*e^c + 2*a^3*b^2*d^2*e^3*f*e^c + a*b^4*d^2*e^3*f*e^c)*x)*e^(d*x)), x)
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \]
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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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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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