3.160 \(\int (e x)^m \tanh (a+2 \log (x)) \, dx\)

Optimal. Leaf size=60 \[ \frac {(e x)^{m+1}}{e (m+1)}-\frac {2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};-e^{2 a} x^4\right )}{e (m+1)} \]

[Out]

(e*x)^(1+m)/e/(1+m)-2*(e*x)^(1+m)*hypergeom([1, 1/4+1/4*m],[5/4+1/4*m],-exp(2*a)*x^4)/e/(1+m)

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Rubi [F]  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 = {} \[ \int (e x)^m \tanh (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e*x)^m*Tanh[a + 2*Log[x]],x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 47, normalized size = 0.78 \[ -\frac {x (e x)^m \left (2 \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};-x^4 (\cosh (2 a)+\sinh (2 a))\right )-1\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*Tanh[a + 2*Log[x]],x]

[Out]

-((x*(e*x)^m*(-1 + 2*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, -(x^4*(Cosh[2*a] + Sinh[2*a]))]))/(1 + m))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \tanh \left (a + 2 \, \log \relax (x)\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x)^m*tanh(a + 2*log(x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*tanh(a + 2*log(x)), x)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \tanh \left (a +2 \ln \relax (x )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*tanh(a+2*ln(x)),x)

[Out]

int((e*x)^m*tanh(a+2*ln(x)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \tanh \left (a + 2 \, \log \relax (x)\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*tanh(a + 2*log(x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {tanh}\left (a+2\,\ln \relax (x)\right )\,{\left (e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + 2*log(x))*(e*x)^m,x)

[Out]

int(tanh(a + 2*log(x))*(e*x)^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \tanh {\left (a + 2 \log {\relax (x )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*tanh(a+2*ln(x)),x)

[Out]

Integral((e*x)**m*tanh(a + 2*log(x)), x)

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