3.1.67 \(\int \frac {(a+b \tanh (e+f x))^3}{(c+d x)^2} \, dx\) [67]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(a+b \tanh (e+f x))^3}{(c+d x)^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(a+b \tanh (e+f x))^3}{(c+d x)^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 36.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \tanh (e+f x))^3}{(c+d x)^2} \, dx \end {gather*}

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 = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \tanh \left (f x +e \right )\right )^{3}}{\left (d x +c \right )^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*d^2*f^2 + 3*a*b^2*d^2*f^2 + b^3*d^2*f^2)*x^2*e^(4*e) + 2*(3*a^2*b*c*d*f^2 + 3*a*b^2*c*d*f^2 + b^3*c*d*f^
2)*x*e^(4*e) + (3*a^2*b*c^2*f^2 + 3*a*b^2*c^2*f^2 + b^3*c^2*f^2)*e^(4*e))*e^(4*f*x) + 2*((3*a^2*b*d^2*f^2 + 3*
a*b^2*d^2*f^2 + b^3*d^2*f^2)*x^2*e^(2*e) + (6*a^2*b*c*d*f^2 + 3*(2*c*d*f^2 - d^2*f)*a*b^2 + (2*c*d*f^2 - d^2*f
)*b^3)*x*e^(2*e) + (3*a^2*b*c^2*f^2 + 3*(c^2*f^2 - c*d*f)*a*b^2 + (c^2*f^2 - c*d*f + d^2)*b^3)*e^(2*e))*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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \tanh {\left (e + f x \right )}\right )^{3}}{\left (c + d x\right )^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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