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

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

[Out]

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

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Rubi [A]  time = 0.0599392, 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 = {} \[ \int \frac{1}{(c+d x)^2 (a+b \coth (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 22.4706, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x)^2 (a+b \coth (e+f x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.156, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) ^{2} \left ( a+b{\rm coth} \left (fx+e\right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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