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

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

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 30.76, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d x + a^{2} c + {\left (a^{2} d x + a^{2} c\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (a^{2} d x + a^{2} c\right )} \cosh \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (a +a \cosh \left (f x +e \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} - 2 \, d^{2} + {\left (d^{2} f x e^{\left (2 \, e\right )} + c d f e^{\left (2 \, e\right )} - 2 \, d^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + {\left (3 \, d^{2} f^{2} x^{2} e^{e} + 3 \, c^{2} f^{2} e^{e} + c d f e^{e} - 4 \, d^{2} e^{e} + {\left (6 \, c d f^{2} e^{e} + d^{2} f e^{e}\right )} x\right )} e^{\left (f x\right )}\right )}}{3 \, {\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x + a^{2} c^{3} f^{3} + {\left (a^{2} d^{3} f^{3} x^{3} e^{\left (3 \, e\right )} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{\left (3 \, e\right )} + 3 \, a^{2} c^{2} d f^{3} x e^{\left (3 \, e\right )} + a^{2} c^{3} f^{3} e^{\left (3 \, e\right )}\right )} e^{\left (3 \, f x\right )} + 3 \, {\left (a^{2} d^{3} f^{3} x^{3} e^{\left (2 \, e\right )} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{\left (2 \, e\right )} + 3 \, a^{2} c^{2} d f^{3} x e^{\left (2 \, e\right )} + a^{2} c^{3} f^{3} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 3 \, {\left (a^{2} d^{3} f^{3} x^{3} e^{e} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{e} + 3 \, a^{2} c^{2} d f^{3} x e^{e} + a^{2} c^{3} f^{3} e^{e}\right )} e^{\left (f x\right )}\right )}} - \int \frac {2 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} - 6 \, d^{3}\right )}}{3 \, {\left (a^{2} d^{4} f^{3} x^{4} + 4 \, a^{2} c d^{3} f^{3} x^{3} + 6 \, a^{2} c^{2} d^{2} f^{3} x^{2} + 4 \, a^{2} c^{3} d f^{3} x + a^{2} c^{4} f^{3} + {\left (a^{2} d^{4} f^{3} x^{4} e^{e} + 4 \, a^{2} c d^{3} f^{3} x^{3} e^{e} + 6 \, a^{2} c^{2} d^{2} f^{3} x^{2} e^{e} + 4 \, a^{2} c^{3} d f^{3} x e^{e} + a^{2} c^{4} f^{3} e^{e}\right )} e^{\left (f x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(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 - 2*d^2 + (d^2*f*x*e^(2*e) + c*d*f*e^(2*e) - 2*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 + c*d*f*e^e - 4*d^2*e^e + (6*c*d*f^2*e^e + d^2*f*e^e)*x)*e^(f*x))/(a^2*
d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3 + (a^2*d^3*f^3*x^3*e^(3*e) + 3*a^2*c*d^2*f
^3*x^2*e^(3*e) + 3*a^2*c^2*d*f^3*x*e^(3*e) + a^2*c^3*f^3*e^(3*e))*e^(3*f*x) + 3*(a^2*d^3*f^3*x^3*e^(2*e) + 3*a
^2*c*d^2*f^3*x^2*e^(2*e) + 3*a^2*c^2*d*f^3*x*e^(2*e) + a^2*c^3*f^3*e^(2*e))*e^(2*f*x) + 3*(a^2*d^3*f^3*x^3*e^e
+ 3*a^2*c*d^2*f^3*x^2*e^e + 3*a^2*c^2*d*f^3*x*e^e + a^2*c^3*f^3*e^e)*e^(f*x)) - integrate(2/3*(d^3*f^2*x^2 +
2*c*d^2*f^2*x + c^2*d*f^2 - 6*d^3)/(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^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)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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