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

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

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 63.49, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \tanh (e+f x))^3}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \tanh \left (f x + e\right )^{3} + 3 \, a b^{2} \tanh \left (f x + e\right )^{2} + 3 \, a^{2} b \tanh \left (f x + e\right ) + a^{3}}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tanh \left (f x +e \right )\right )^{3}}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{3} \log \left (d x + c\right )}{d} + \frac {{\left (3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \log \left (d x + c\right )}{d} + \frac {6 \, a b^{2} d f x + 6 \, a b^{2} c f - b^{3} d + {\left (6 \, a b^{2} c f e^{\left (2 \, e\right )} + {\left (2 \, c f e^{\left (2 \, e\right )} - d e^{\left (2 \, e\right )}\right )} b^{3} + 2 \, {\left (3 \, a b^{2} d f e^{\left (2 \, e\right )} + b^{3} d f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}{d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} + {\left (d^{2} f^{2} x^{2} e^{\left (4 \, e\right )} + 2 \, c d f^{2} x e^{\left (4 \, e\right )} + c^{2} f^{2} e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left (d^{2} f^{2} x^{2} e^{\left (2 \, e\right )} + 2 \, c d f^{2} x e^{\left (2 \, e\right )} + c^{2} f^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}} - \int \frac {2 \, {\left (3 \, a^{2} b c^{2} f^{2} - 3 \, a b^{2} c d f + {\left (c^{2} f^{2} + d^{2}\right )} b^{3} + {\left (3 \, a^{2} b d^{2} f^{2} + b^{3} d^{2} f^{2}\right )} x^{2} + {\left (6 \, a^{2} b c d f^{2} + 2 \, b^{3} c d f^{2} - 3 \, a b^{2} d^{2} f\right )} x\right )}}{d^{3} f^{2} x^{3} + 3 \, c d^{2} f^{2} x^{2} + 3 \, c^{2} d f^{2} x + c^{3} f^{2} + {\left (d^{3} f^{2} x^{3} e^{\left (2 \, e\right )} + 3 \, c d^{2} f^{2} x^{2} e^{\left (2 \, e\right )} + 3 \, c^{2} d f^{2} x e^{\left (2 \, e\right )} + c^{3} f^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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