3.120 \(\int \frac{1}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x)^2 (a \cosh (e+f x)+a)^2},x\right ) \]

[Out]

Unintegrable[1/((c + d*x)^2*(a + a*Cosh[e + f*x])^2), x]

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Rubi [A]  time = 0.0524199, 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 = {} \[ \int \frac{1}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 31.2858, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.465, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2} \left ( a+a\cosh \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-2/3*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 - 6*d^2 + 2*(d^2*f*x*e^(2*e) + c*d*f*e^(2*e) - 3*d^2*e^(2*e))*e^(2*f
*x) + (3*d^2*f^2*x^2*e^e + 3*c^2*f^2*e^e + 2*c*d*f*e^e - 12*d^2*e^e + 2*(3*c*d*f^2*e^e + d^2*f*e^e)*x)*e^(f*x)
)/(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3 + (a^2*d^4*
f^3*x^4*e^(3*e) + 4*a^2*c*d^3*f^3*x^3*e^(3*e) + 6*a^2*c^2*d^2*f^3*x^2*e^(3*e) + 4*a^2*c^3*d*f^3*x*e^(3*e) + a^
2*c^4*f^3*e^(3*e))*e^(3*f*x) + 3*(a^2*d^4*f^3*x^4*e^(2*e) + 4*a^2*c*d^3*f^3*x^3*e^(2*e) + 6*a^2*c^2*d^2*f^3*x^
2*e^(2*e) + 4*a^2*c^3*d*f^3*x*e^(2*e) + a^2*c^4*f^3*e^(2*e))*e^(2*f*x) + 3*(a^2*d^4*f^3*x^4*e^e + 4*a^2*c*d^3*
f^3*x^3*e^e + 6*a^2*c^2*d^2*f^3*x^2*e^e + 4*a^2*c^3*d*f^3*x*e^e + a^2*c^4*f^3*e^e)*e^(f*x)) - integrate(4/3*(d
^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 12*d^3)/(a^2*d^5*f^3*x^5 + 5*a^2*c*d^4*f^3*x^4 + 10*a^2*c^2*d^3*f^3*x
^3 + 10*a^2*c^3*d^2*f^3*x^2 + 5*a^2*c^4*d*f^3*x + a^2*c^5*f^3 + (a^2*d^5*f^3*x^5*e^e + 5*a^2*c*d^4*f^3*x^4*e^e
+ 10*a^2*c^2*d^3*f^3*x^3*e^e + 10*a^2*c^3*d^2*f^3*x^2*e^e + 5*a^2*c^4*d*f^3*x*e^e + a^2*c^5*f^3*e^e)*e^(f*x))
, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2} +{\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \,{\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \cosh \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*cosh(f*x+e))^2,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}^{2}{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*cosh(f*x+e))^2,x, algorithm="giac")

[Out]

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