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

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

[Out]

Unintegrable(1/(d*x+c)^2/(a+a*cosh(f*x+e)),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 = {} \[ \int \frac {1}{(c+d x)^2 (a+a \cosh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 9.61, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 (a+a \cosh (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a d^{2} x^{2} + 2 \, a c d x + a c^{2} + {\left (a d^{2} x^{2} + 2 \, a c d x + a c^{2}\right )} \cosh \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{2} \left (a +a \cosh \left (f x +e \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -4 \, d \int \frac {1}{a d^{3} f x^{3} + 3 \, a c d^{2} f x^{2} + 3 \, a c^{2} d f x + a c^{3} f + {\left (a d^{3} f x^{3} e^{e} + 3 \, a c d^{2} f x^{2} e^{e} + 3 \, a c^{2} d f x e^{e} + a c^{3} f e^{e}\right )} e^{\left (f x\right )}}\,{d x} - \frac {2}{a d^{2} f x^{2} + 2 \, a c d f x + a c^{2} f + {\left (a d^{2} f x^{2} e^{e} + 2 \, a c d f x e^{e} + a c^{2} f e^{e}\right )} e^{\left (f x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*d*integrate(1/(a*d^3*f*x^3 + 3*a*c*d^2*f*x^2 + 3*a*c^2*d*f*x + a*c^3*f + (a*d^3*f*x^3*e^e + 3*a*c*d^2*f*x^2
*e^e + 3*a*c^2*d*f*x*e^e + a*c^3*f*e^e)*e^(f*x)), x) - 2/(a*d^2*f*x^2 + 2*a*c*d*f*x + a*c^2*f + (a*d^2*f*x^2*e
^e + 2*a*c*d*f*x*e^e + a*c^2*f*e^e)*e^(f*x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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