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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a+b \sinh (e+f x))^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0615198, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(c+d x) (a+b \sinh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 53.3547, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+b \sinh (e+f x))^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.283, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+b\sinh \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+b*sinh(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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} d x + a^{2} c +{\left (b^{2} d x + b^{2} c\right )} \sinh \left (f x + e\right )^{2} + 2 \,{\left (a b d x + a b c\right )} \sinh \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x + c\right )}{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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