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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{(a+b \tanh (e+f x))^3}{(c+d x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0531491, 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 = {} \[ \int \frac{(a+b \tanh (e+f x))^3}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 52.265, size = 0, normalized size = 0. \[ \int \frac{(a+b \tanh (e+f x))^3}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.563, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\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((a+b*tanh(f*x+e))^3/(d*x+c)^2,x)

[Out]

int((a+b*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*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*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{{\left (b \tanh \left (f x + e\right ) + a\right )}^{3}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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