3.48 \(\int (c+d x)^m (a+a \tanh (e+f x))^2 \, dx\)

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

[Out]

Unintegrable((d*x+c)^m*(a+a*tanh(f*x+e))^2,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 = {} \[ \int (c+d x)^m (a+a \tanh (e+f x))^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 34.65, size = 0, normalized size = 0.00 \[ \int (c+d x)^m (a+a \tanh (e+f x))^2 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} \tanh \left (f x + e\right )^{2} + 2 \, a^{2} \tanh \left (f x + e\right ) + a^{2}\right )} {\left (d x + c\right )}^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*(a+a*tanh(f*x+e))^2,x, algorithm="fricas")

[Out]

integral((a^2*tanh(f*x + e)^2 + 2*a^2*tanh(f*x + e) + a^2)*(d*x + c)^m, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \tanh \left (f x + e\right ) + a\right )}^{2} {\left (d x + c\right )}^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (a +a \tanh \left (f x +e \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} + \int \frac {2 \, {\left (d x + c\right )}^{m} a^{2} {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}}{e^{\left (f x + e\right )} + e^{\left (-f x - e\right )}} + \frac {{\left (d x + c\right )}^{m} a^{2} {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{2}}{{\left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int 2 \left (c + d x\right )^{m} \tanh {\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m} \tanh ^{2}{\left (e + f x \right )}\, dx + \int \left (c + d x\right )^{m}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**m*(a+a*tanh(f*x+e))**2,x)

[Out]

a**2*(Integral(2*(c + d*x)**m*tanh(e + f*x), x) + Integral((c + d*x)**m*tanh(e + f*x)**2, x) + Integral((c + d
*x)**m, x))

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