3.2.60 \(\int (e x)^m \tanh (a+2 \log (x)) \, dx\) [160]

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

[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 [A]
time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5656, 470, 371} \begin {gather*} \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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Tanh[a + 2*Log[x]],x]

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 5656

Int[((e_.)*(x_))^(m_.)*Tanh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[(e*x)^m*((-1 + E^(2*a*d)*x^
(2*b*d))^p/(1 + E^(2*a*d)*x^(2*b*d))^p), x] /; FreeQ[{a, b, d, e, m, p}, 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 = 0.07, size = 47, normalized size = 0.78 \begin {gather*} -\frac {x (e x)^m \left (-1+2 \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};-x^4 (\cosh (2 a)+\sinh (2 a))\right )\right )}{1+m} \end {gather*}

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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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \tanh \left (a +2 \ln \left (x \right )\right )\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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((x*e)^m*tanh(a + 2*log(x)), x)

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Fricas [F]
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((e*x)^m*tanh(a+2*log(x)),x, algorithm="fricas")

[Out]

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

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

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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Giac [F]
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((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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \mathrm {tanh}\left (a+2\,\ln \left (x\right )\right )\,{\left (e\,x\right )}^m \,d x \end {gather*}

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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