3.15 \(\int \frac{\tanh ^3(e+f x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{\tanh ^3(e+f x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable[Tanh[e + f*x]^3/(c + d*x)^2, x]

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Rubi [A]  time = 0.0369673, 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 = {} \[ \int \frac{\tanh ^3(e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[e + f*x]^3/(c + d*x)^2,x]

[Out]

Defer[Int][Tanh[e + f*x]^3/(c + d*x)^2, x]

Rubi steps

\begin{align*} \int \frac{\tanh ^3(e+f x)}{(c+d x)^2} \, dx &=\int \frac{\tanh ^3(e+f x)}{(c+d x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 22.3992, size = 0, normalized size = 0. \[ \int \frac{\tanh ^3(e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Tanh[e + f*x]^3/(c + d*x)^2,x]

[Out]

Integrate[Tanh[e + f*x]^3/(c + d*x)^2, x]

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Maple [A]  time = 0.485, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tanh \left ( fx+e \right ) \right ) ^{3}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(f*x+e)^3/(d*x+c)^2,x)

[Out]

int(tanh(f*x+e)^3/(d*x+c)^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)^3/(d*x+c)^2,x, algorithm="maxima")

[Out]

-(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + 2*d^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) + c^2*f^2*e^(2*e) - c*d*f*e^(2*e) + d^2*e^(2*e) + (2*c*d*f^2*e^(2*e) - d^2*
f*e^(2*e))*x)*e^(2*f*x))/(d^4*f^2*x^3 + 3*c*d^3*f^2*x^2 + 3*c^2*d^2*f^2*x + c^3*d*f^2 + (d^4*f^2*x^3*e^(4*e) +
3*c*d^3*f^2*x^2*e^(4*e) + 3*c^2*d^2*f^2*x*e^(4*e) + c^3*d*f^2*e^(4*e))*e^(4*f*x) + 2*(d^4*f^2*x^3*e^(2*e) + 3
*c*d^3*f^2*x^2*e^(2*e) + 3*c^2*d^2*f^2*x*e^(2*e) + c^3*d*f^2*e^(2*e))*e^(2*f*x)) - integrate(2*(d^2*f^2*x^2 +
2*c*d*f^2*x + c^2*f^2 + 3*d^2)/(d^4*f^2*x^4 + 4*c*d^3*f^2*x^3 + 6*c^2*d^2*f^2*x^2 + 4*c^3*d*f^2*x + c^4*f^2 +
(d^4*f^2*x^4*e^(2*e) + 4*c*d^3*f^2*x^3*e^(2*e) + 6*c^2*d^2*f^2*x^2*e^(2*e) + 4*c^3*d*f^2*x*e^(2*e) + c^4*f^2*e
^(2*e))*e^(2*f*x)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\tanh \left (f x + e\right )^{3}}{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(tanh(f*x+e)^3/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(tanh(f*x + e)^3/(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{\tanh ^{3}{\left (e + f x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)**3/(d*x+c)**2,x)

[Out]

Integral(tanh(e + f*x)**3/(c + d*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(f*x+e)^3/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(tanh(f*x + e)^3/(d*x + c)^2, x)