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

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

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {1}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx \]

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*}

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Mathematica [A]  time = 32.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 (a+a \cosh (e+f x))^2} \, dx \]

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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fricas [A]  time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2} + {\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2}\right )} \cosh \left (f x + e\right )}, x\right ) \]

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

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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maple [A]  time = 0.81, 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 \[ -\frac {2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} - 6 \, d^{2} + 2 \, {\left (d^{2} f x e^{\left (2 \, e\right )} + c d f e^{\left (2 \, e\right )} - 3 \, d^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + {\left (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 \, {\left (3 \, c d f^{2} e^{e} + d^{2} f e^{e}\right )} x\right )} e^{\left (f x\right )}\right )}}{3 \, {\left (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} + {\left (a^{2} d^{4} f^{3} x^{4} e^{\left (3 \, e\right )} + 4 \, a^{2} c d^{3} f^{3} x^{3} e^{\left (3 \, e\right )} + 6 \, a^{2} c^{2} d^{2} f^{3} x^{2} e^{\left (3 \, e\right )} + 4 \, a^{2} c^{3} d f^{3} x e^{\left (3 \, e\right )} + a^{2} c^{4} f^{3} e^{\left (3 \, e\right )}\right )} e^{\left (3 \, f x\right )} + 3 \, {\left (a^{2} d^{4} f^{3} x^{4} e^{\left (2 \, e\right )} + 4 \, a^{2} c d^{3} f^{3} x^{3} e^{\left (2 \, e\right )} + 6 \, a^{2} c^{2} d^{2} f^{3} x^{2} e^{\left (2 \, e\right )} + 4 \, a^{2} c^{3} d f^{3} x e^{\left (2 \, e\right )} + a^{2} c^{4} f^{3} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 3 \, {\left (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}\right )} e^{\left (f x\right )}\right )}} - \int \frac {4 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} - 12 \, d^{3}\right )}}{3 \, {\left (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} + {\left (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}\right )} e^{\left (f x\right )}\right )}}\,{d x} \]

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

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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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{c^{2} \cosh ^{2}{\left (e + f x \right )} + 2 c^{2} \cosh {\left (e + f x \right )} + c^{2} + 2 c d x \cosh ^{2}{\left (e + f x \right )} + 4 c d x \cosh {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \cosh ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \cosh {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \]

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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