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

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

[Out]

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

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Rubi [A]  time = 0.0552581, 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 = {} \[ \int \frac{(a+b \coth (e+f x))^2}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 42.6526, size = 0, normalized size = 0. \[ \int \frac{(a+b \coth (e+f x))^2}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.376, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm coth} \left (fx+e\right ) \right ) ^{2}}{dx+c}}\, 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),x)

[Out]

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

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

[Out]

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

[Out]

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

[Out]

Integral((a + b*coth(e + f*x))**2/(c + d*x), 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}}{d x + c}\,{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),x, algorithm="giac")

[Out]

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