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

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

[Out]

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

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Rubi [A]  time = 0.0525684, 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 = {} \[ \int \frac{(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 30.1259, size = 0, normalized size = 0. \[ \int \frac{(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.589, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm coth} \left (fx+e\right ) \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2}}{d^{2} x + c d} - \frac{2 \, a b c f +{\left (c f - 2 \, d\right )} b^{2} +{\left (2 \, a b d f + b^{2} d f\right )} x -{\left (2 \, a b c f e^{\left (2 \, e\right )} + b^{2} c f e^{\left (2 \, e\right )} +{\left (2 \, a b d f e^{\left (2 \, e\right )} + b^{2} d f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}{d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f -{\left (d^{3} f x^{2} e^{\left (2 \, e\right )} + 2 \, c d^{2} f x e^{\left (2 \, e\right )} + c^{2} d f e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}} - \int \frac{2 \,{\left (a b d f x + a b c f - b^{2} d\right )}}{d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f +{\left (d^{3} f x^{3} e^{e} + 3 \, c d^{2} f x^{2} e^{e} + 3 \, c^{2} d f x e^{e} + c^{3} f e^{e}\right )} e^{\left (f x\right )}}\,{d x} + \int -\frac{2 \,{\left (a b d f x + a b c f - b^{2} d\right )}}{d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f -{\left (d^{3} f x^{3} e^{e} + 3 \, c d^{2} f x^{2} e^{e} + 3 \, c^{2} d f x e^{e} + c^{3} f e^{e}\right )} e^{\left (f x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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