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

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

[Out]

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 51.53, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) (a+b \cosh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d x + a^{2} c + {\left (b^{2} d x + b^{2} c\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (a b d x + a b c\right )} \cosh \left (f x + e\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.59, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (a +b \cosh \left (f x +e \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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