3.2.66 \(\int (e x)^m \coth ^2(a+2 \log (x)) \, dx\) [166]

Optimal. Leaf size=79 \[ \frac {(e x)^{1+m}}{e (1+m)}+\frac {(e x)^{1+m}}{e \left (1-e^{2 a} x^4\right )}-\frac {(e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};e^{2 a} x^4\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5657, 474, 470, 371} \begin {gather*} -\frac {(e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{4};\frac {m+5}{4};e^{2 a} x^4\right )}{e}+\frac {(e x)^{m+1}}{e \left (1-e^{2 a} x^4\right )}+\frac {(e x)^{m+1}}{e (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*Coth[a + 2*Log[x]]^2,x]

[Out]

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

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 474

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(-(b*c - a*
d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b^2*e*n*(p + 1))), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a +
b*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a
, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 5657

Int[Coth[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), 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 \coth ^2(a+2 \log (x)) \, dx &=\int (e x)^m \coth ^2(a+2 \log (x)) \, dx\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 77, normalized size = 0.97 \begin {gather*} -\frac {x (e x)^m \left (-1+4 \, _2F_1\left (1,\frac {1+m}{4};\frac {5+m}{4};x^4 (\cosh (2 a)+\sinh (2 a))\right )-4 \, _2F_1\left (2,\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*Coth[a + 2*Log[x]]^2,x]

[Out]

-((x*(e*x)^m*(-1 + 4*Hypergeometric2F1[1, (1 + m)/4, (5 + m)/4, x^4*(Cosh[2*a] + Sinh[2*a])] - 4*Hypergeometri
c2F1[2, (1 + m)/4, (5 + m)/4, x^4*(Cosh[2*a] + Sinh[2*a])]))/(1 + m))

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Maple [F]
time = 0.45, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\coth ^{2}\left (a +2 \ln \left (x \right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x)^m*coth(a+2*ln(x))^2,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*coth(a+2*log(x))^2,x, algorithm="maxima")

[Out]

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

[Out]

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

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (a+2\,\ln \left (x\right )\right )}^2\,{\left (e\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x))^2*(e*x)^m,x)

[Out]

int(coth(a + 2*log(x))^2*(e*x)^m, x)

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