3.50 \(\int \frac{(a+b \coth (e+f x))^3}{c+d x} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{(a+b \coth (e+f x))^3}{c+d x},x\right ) \]

[Out]

Unintegrable[(a + b*Coth[e + f*x])^3/(c + d*x), x]

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Rubi [A]  time = 0.054938, 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 = {} \[ \int \frac{(a+b \coth (e+f x))^3}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Coth[e + f*x])^3/(c + d*x),x]

[Out]

Defer[Int][(a + b*Coth[e + f*x])^3/(c + d*x), x]

Rubi steps

\begin{align*} \int \frac{(a+b \coth (e+f x))^3}{c+d x} \, dx &=\int \frac{(a+b \coth (e+f x))^3}{c+d x} \, dx\\ \end{align*}

Mathematica [A]  time = 62.5076, size = 0, normalized size = 0. \[ \int \frac{(a+b \coth (e+f x))^3}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Coth[e + f*x])^3/(c + d*x),x]

[Out]

Integrate[(a + b*Coth[e + f*x])^3/(c + d*x), x]

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Maple [A]  time = 0.599, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm coth} \left (fx+e\right ) \right ) ^{3}}{dx+c}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*coth(f*x+e))^3/(d*x+c),x)

[Out]

int((a+b*coth(f*x+e))^3/(d*x+c),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{3} \log \left (d x + c\right )}{d} + \frac{{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (d x + c\right )}{d} + \frac{6 \, a b^{2} d f x + 6 \, a b^{2} c f - b^{3} d -{\left (6 \, a b^{2} c f e^{\left (2 \, e\right )} +{\left (2 \, c f e^{\left (2 \, e\right )} - d e^{\left (2 \, e\right )}\right )} b^{3} + 2 \,{\left (3 \, a b^{2} 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 )}}{d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} +{\left (d^{2} f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, c d f^{2} x e^{\left (4 \, e\right )} + c^{2} f^{2} e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} - 2 \,{\left (d^{2} f^{2} x^{2} e^{\left (2 \, e\right )} + 2 \, c d f^{2} x e^{\left (2 \, e\right )} + c^{2} f^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}} - \int \frac{3 \, a^{2} b c^{2} f^{2} - 3 \, a b^{2} c d f +{\left (c^{2} f^{2} + d^{2}\right )} b^{3} +{\left (3 \, a^{2} b d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{2} +{\left (6 \, a^{2} b c d f^{2} + 2 \, b^{3} c d f^{2} - 3 \, a b^{2} d^{2} f\right )} x}{d^{3} f^{2} x^{3} + 3 \, c d^{2} f^{2} x^{2} + 3 \, c^{2} d f^{2} x + c^{3} f^{2} +{\left (d^{3} f^{2} x^{3} e^{e} + 3 \, c d^{2} f^{2} x^{2} e^{e} + 3 \, c^{2} d f^{2} x e^{e} + c^{3} f^{2} e^{e}\right )} e^{\left (f x\right )}}\,{d x} + \int -\frac{3 \, a^{2} b c^{2} f^{2} - 3 \, a b^{2} c d f +{\left (c^{2} f^{2} + d^{2}\right )} b^{3} +{\left (3 \, a^{2} b d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{2} +{\left (6 \, a^{2} b c d f^{2} + 2 \, b^{3} c d f^{2} - 3 \, a b^{2} d^{2} f\right )} x}{d^{3} f^{2} x^{3} + 3 \, c d^{2} f^{2} x^{2} + 3 \, c^{2} d f^{2} x + c^{3} f^{2} -{\left (d^{3} f^{2} x^{3} e^{e} + 3 \, c d^{2} f^{2} x^{2} e^{e} + 3 \, c^{2} d f^{2} x e^{e} + c^{3} f^{2} 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))^3/(d*x+c),x, algorithm="maxima")

[Out]

a^3*log(d*x + c)/d + (3*a^2*b + 3*a*b^2 + b^3)*log(d*x + c)/d + (6*a*b^2*d*f*x + 6*a*b^2*c*f - b^3*d - (6*a*b^
2*c*f*e^(2*e) + (2*c*f*e^(2*e) - d*e^(2*e))*b^3 + 2*(3*a*b^2*d*f*e^(2*e) + b^3*d*f*e^(2*e))*x)*e^(2*f*x))/(d^2
*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + (d^2*f^2*x^2*e^(4*e) + 2*c*d*f^2*x*e^(4*e) + c^2*f^2*e^(4*e))*e^(4*f*x) - 2
*(d^2*f^2*x^2*e^(2*e) + 2*c*d*f^2*x*e^(2*e) + c^2*f^2*e^(2*e))*e^(2*f*x)) - integrate((3*a^2*b*c^2*f^2 - 3*a*b
^2*c*d*f + (c^2*f^2 + d^2)*b^3 + (3*a^2*b*d^2*f^2 + b^3*d^2*f^2)*x^2 + (6*a^2*b*c*d*f^2 + 2*b^3*c*d*f^2 - 3*a*
b^2*d^2*f)*x)/(d^3*f^2*x^3 + 3*c*d^2*f^2*x^2 + 3*c^2*d*f^2*x + c^3*f^2 + (d^3*f^2*x^3*e^e + 3*c*d^2*f^2*x^2*e^
e + 3*c^2*d*f^2*x*e^e + c^3*f^2*e^e)*e^(f*x)), x) + integrate(-(3*a^2*b*c^2*f^2 - 3*a*b^2*c*d*f + (c^2*f^2 + d
^2)*b^3 + (3*a^2*b*d^2*f^2 + b^3*d^2*f^2)*x^2 + (6*a^2*b*c*d*f^2 + 2*b^3*c*d*f^2 - 3*a*b^2*d^2*f)*x)/(d^3*f^2*
x^3 + 3*c*d^2*f^2*x^2 + 3*c^2*d*f^2*x + c^3*f^2 - (d^3*f^2*x^3*e^e + 3*c*d^2*f^2*x^2*e^e + 3*c^2*d*f^2*x*e^e +
c^3*f^2*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^{3} \coth \left (f x + e\right )^{3} + 3 \, a b^{2} \coth \left (f x + e\right )^{2} + 3 \, a^{2} b \coth \left (f x + e\right ) + a^{3}}{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))^3/(d*x+c),x, algorithm="fricas")

[Out]

integral((b^3*coth(f*x + e)^3 + 3*a*b^2*coth(f*x + e)^2 + 3*a^2*b*coth(f*x + e) + a^3)/(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 )^{3}}{c + d x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(f*x+e))**3/(d*x+c),x)

[Out]

Integral((a + b*coth(e + f*x))**3/(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 )}^{3}}{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))^3/(d*x+c),x, algorithm="giac")

[Out]

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