3.179 \(\int \frac{1}{(e+f x) (a+b \sinh (c+d x))^3} \, dx\)
Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(e+f x) (a+b \sinh (c+d x))^3},x\right ) \]
[Out]
Unintegrable[1/((e + f*x)*(a + b*Sinh[c + d*x])^3), x]
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Rubi [A] time = 0.0622738, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{1}{(e+f x) (a+b \sinh (c+d x))^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((e + f*x)*(a + b*Sinh[c + d*x])^3),x]
[Out]
Defer[Int][1/((e + f*x)*(a + b*Sinh[c + d*x])^3), x]
Rubi steps
\begin{align*} \int \frac{1}{(e+f x) (a+b \sinh (c+d x))^3} \, dx &=\int \frac{1}{(e+f x) (a+b \sinh (c+d x))^3} \, dx\\ \end{align*}
Mathematica [A] time = 104.981, size = 0, normalized size = 0. \[ \int \frac{1}{(e+f x) (a+b \sinh (c+d x))^3} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[1/((e + f*x)*(a + b*Sinh[c + d*x])^3),x]
[Out]
Integrate[1/((e + f*x)*(a + b*Sinh[c + d*x])^3), x]
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Maple [A] time = 0.61, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) \left ( a+b\sinh \left ( dx+c \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(f*x+e)/(a+b*sinh(d*x+c))^3,x)
[Out]
int(1/(f*x+e)/(a+b*sinh(d*x+c))^3,x)
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(f*x+e)/(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 - 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 + f)*a^3*e^(2*c) - (3*d*e - 2*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) - ((10*d*e + f)*a^2*b*e^c + (d*e + 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^2 + 2*a^2*b^4*d^2*e^2 + b^6*d^2*e^2 + (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 + 2*(a^4*b^2*d^2*e*f + 2*a^2*b^4*d^2*e*f + b^6*d^2*e*f)*x + (a^4*b^2*d^2*e^2*e^(4*c) + 2*a^2*b^
4*d^2*e^2*e^(4*c) + b^6*d^2*e^2*e^(4*c) + (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 + 2*(a^4*b^2*d^2*e*f*e^(4*c) + 2*a^2*b^4*d^2*e*f*e^(4*c) + b^6*d^2*e*f*e^(4*c))*x)*e^(4*d*x) + 4*(
a^5*b*d^2*e^2*e^(3*c) + 2*a^3*b^3*d^2*e^2*e^(3*c) + a*b^5*d^2*e^2*e^(3*c) + (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 + 2*(a^5*b*d^2*e*f*e^(3*c) + 2*a^3*b^3*d^2*e*f*e^(3*c) + a*b^5*d
^2*e*f*e^(3*c))*x)*e^(3*d*x) + 2*(2*a^6*d^2*e^2*e^(2*c) + 3*a^4*b^2*d^2*e^2*e^(2*c) - b^6*d^2*e^2*e^(2*c) + (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 + 2*(2*a^6*d^2*e*f*e^(2*c) + 3*a^4
*b^2*d^2*e*f*e^(2*c) - b^6*d^2*e*f*e^(2*c))*x)*e^(2*d*x) - 4*(a^5*b*d^2*e^2*e^c + 2*a^3*b^3*d^2*e^2*e^c + a*b^
5*d^2*e^2*e^c + (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 + 2*(a^5*b*d^2*e*f*e^c + 2
*a^3*b^3*d^2*e*f*e^c + a*b^5*d^2*e*f*e^c)*x)*e^(d*x)) + integrate((3*a*b*d*f^2*x + 3*a*b*d*e*f - ((2*d^2*e^2 +
3*d*e*f + 2*f^2)*a^2*e^c - (d^2*e^2 - 2*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 - (4*d^2*e*f + 3*d*f^2)*a^2*e^c)*x)*e^(d*x))/(a^4*b*d^2*e^3 + 2*a^2*b^3*d^2*e^3 + b^5*d^2*e^3 + (a^4*
b*d^2*f^3 + 2*a^2*b^3*d^2*f^3 + b^5*d^2*f^3)*x^3 + 3*(a^4*b*d^2*e*f^2 + 2*a^2*b^3*d^2*e*f^2 + b^5*d^2*e*f^2)*x
^2 + 3*(a^4*b*d^2*e^2*f + 2*a^2*b^3*d^2*e^2*f + b^5*d^2*e^2*f)*x - (a^4*b*d^2*e^3*e^(2*c) + 2*a^2*b^3*d^2*e^3*
e^(2*c) + b^5*d^2*e^3*e^(2*c) + (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
+ 3*(a^4*b*d^2*e*f^2*e^(2*c) + 2*a^2*b^3*d^2*e*f^2*e^(2*c) + b^5*d^2*e*f^2*e^(2*c))*x^2 + 3*(a^4*b*d^2*e^2*f*e
^(2*c) + 2*a^2*b^3*d^2*e^2*f*e^(2*c) + b^5*d^2*e^2*f*e^(2*c))*x)*e^(2*d*x) - 2*(a^5*d^2*e^3*e^c + 2*a^3*b^2*d^
2*e^3*e^c + a*b^4*d^2*e^3*e^c + (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 + 3*(a^5*d^2
*e*f^2*e^c + 2*a^3*b^2*d^2*e*f^2*e^c + a*b^4*d^2*e*f^2*e^c)*x^2 + 3*(a^5*d^2*e^2*f*e^c + 2*a^3*b^2*d^2*e^2*f*e
^c + a*b^4*d^2*e^2*f*e^c)*x)*e^(d*x)), x)
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{3} f x + a^{3} e +{\left (b^{3} f x + b^{3} e\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (a b^{2} f x + a b^{2} e\right )} \sinh \left (d x + c\right )^{2} + 3 \,{\left (a^{2} b f x + a^{2} b e\right )} \sinh \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="fricas")
[Out]
integral(1/(a^3*f*x + a^3*e + (b^3*f*x + b^3*e)*sinh(d*x + c)^3 + 3*(a*b^2*f*x + a*b^2*e)*sinh(d*x + c)^2 + 3*
(a^2*b*f*x + a^2*b*e)*sinh(d*x + c)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(f*x+e)/(a+b*sinh(d*x+c))**3,x)
[Out]
Timed out
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (f x + e\right )}{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="giac")
[Out]
integrate(1/((f*x + e)*(b*sinh(d*x + c) + a)^3), x)