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

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

[Out]

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

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Rubi [A]  time = 0.0345955, 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 = {} \[ \int \frac{\tanh ^2(e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[e + f*x]^2/(c + d*x)^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 19.143, size = 0, normalized size = 0. \[ \int \frac{\tanh ^2(e+f x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \tanh \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(tanh(f*x+e)^2/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, d \int \frac{1}{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^{\left (2 \, e\right )} + 3 \, c d^{2} f x^{2} e^{\left (2 \, e\right )} + 3 \, c^{2} d f x e^{\left (2 \, e\right )} + c^{3} f e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}\,{d x} - \frac{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 )} - 2 \, d}{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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*d*integrate(1/(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^(2*e) + 3*c*d^2*f*x^2*e^(2*e)
+ 3*c^2*d*f*x*e^(2*e) + c^3*f*e^(2*e))*e^(2*f*x)), x) - (d*f*x + c*f + (d*f*x*e^(2*e) + c*f*e^(2*e))*e^(2*f*x)
- 2*d)/(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))

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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 )^{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(tanh(f*x+e)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

[Out]

Integral(tanh(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{\tanh \left (f x + e\right )^{2}}{{\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)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

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