3.2.20 \(\int \frac {1}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx\) [120]

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

[Out]

Unintegrable(1/(d*x+c)^2/(a+a*cosh(f*x+e))^2,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}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 21.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx \end {gather*}

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

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 = 0.00, 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/(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 - 3*d^2)*e^(2*e))*e^(2*f*x) +
(3*d^2*f^2*x^2*e^e + 2*(3*c*d*f^2 + d^2*f)*x*e^e + (3*c^2*f^2 + 2*c*d*f - 12*d^2)*e^e)*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 = 0.00, 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/(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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c^{2} \cosh ^{2}{\left (e + f x \right )} + 2 c^{2} \cosh {\left (e + f x \right )} + c^{2} + 2 c d x \cosh ^{2}{\left (e + f x \right )} + 4 c d x \cosh {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \cosh ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \cosh {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \end {gather*}

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]

Integral(1/(c**2*cosh(e + f*x)**2 + 2*c**2*cosh(e + f*x) + c**2 + 2*c*d*x*cosh(e + f*x)**2 + 4*c*d*x*cosh(e +
f*x) + 2*c*d*x + d**2*x**2*cosh(e + f*x)**2 + 2*d**2*x**2*cosh(e + f*x) + d**2*x**2), x)/a**2

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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/(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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