3.2.6 \(\int (d x)^m \text {PolyLog}(3,a x^2) \, dx\) [106]

Optimal. Leaf size=118 \[ -\frac {8 a (d x)^{3+m} \, _2F_1\left (1,\frac {3+m}{2};\frac {5+m}{2};a x^2\right )}{d^3 (1+m)^3 (3+m)}-\frac {4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac {2 (d x)^{1+m} \text {PolyLog}\left (2,a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {PolyLog}\left (3,a x^2\right )}{d (1+m)} \]

[Out]

-8*a*(d*x)^(3+m)*hypergeom([1, 3/2+1/2*m],[5/2+1/2*m],a*x^2)/d^3/(1+m)^3/(3+m)-4*(d*x)^(1+m)*ln(-a*x^2+1)/d/(1
+m)^3-2*(d*x)^(1+m)*polylog(2,a*x^2)/d/(1+m)^2+(d*x)^(1+m)*polylog(3,a*x^2)/d/(1+m)

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Rubi [A]
time = 0.05, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {6726, 2505, 16, 371} \begin {gather*} -\frac {8 a (d x)^{m+3} \, _2F_1\left (1,\frac {m+3}{2};\frac {m+5}{2};a x^2\right )}{d^3 (m+1)^3 (m+3)}-\frac {2 \text {Li}_2\left (a x^2\right ) (d x)^{m+1}}{d (m+1)^2}+\frac {\text {Li}_3\left (a x^2\right ) (d x)^{m+1}}{d (m+1)}-\frac {4 \log \left (1-a x^2\right ) (d x)^{m+1}}{d (m+1)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*PolyLog[3, a*x^2],x]

[Out]

(-8*a*(d*x)^(3 + m)*Hypergeometric2F1[1, (3 + m)/2, (5 + m)/2, a*x^2])/(d^3*(1 + m)^3*(3 + m)) - (4*(d*x)^(1 +
m)*Log[1 - a*x^2])/(d*(1 + m)^3) - (2*(d*x)^(1 + m)*PolyLog[2, a*x^2])/(d*(1 + m)^2) + ((d*x)^(1 + m)*PolyLog
[3, a*x^2])/(d*(1 + m))

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[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 2505

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

Rule 6726

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

Rubi steps

\begin {align*} \int (d x)^m \text {Li}_3\left (a x^2\right ) \, dx &=\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {2 \int (d x)^m \text {Li}_2\left (a x^2\right ) \, dx}{1+m}\\ &=-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {4 \int (d x)^m \log \left (1-a x^2\right ) \, dx}{(1+m)^2}\\ &=-\frac {4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {(8 a) \int \frac {x (d x)^{1+m}}{1-a x^2} \, dx}{d (1+m)^3}\\ &=-\frac {4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}-\frac {(8 a) \int \frac {(d x)^{2+m}}{1-a x^2} \, dx}{d^2 (1+m)^3}\\ &=-\frac {8 a (d x)^{3+m} \, _2F_1\left (1,\frac {3+m}{2};\frac {5+m}{2};a x^2\right )}{d^3 (1+m)^3 (3+m)}-\frac {4 (d x)^{1+m} \log \left (1-a x^2\right )}{d (1+m)^3}-\frac {2 (d x)^{1+m} \text {Li}_2\left (a x^2\right )}{d (1+m)^2}+\frac {(d x)^{1+m} \text {Li}_3\left (a x^2\right )}{d (1+m)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 0.06, size = 126, normalized size = 1.07 \begin {gather*} -\frac {2 x (d x)^m \Gamma \left (\frac {3+m}{2}\right ) \left (2 a (1+m) x^2 \Gamma \left (\frac {1+m}{2}\right ) \, _2\tilde {F}_1\left (1,\frac {3+m}{2};\frac {5+m}{2};a x^2\right )+4 \log \left (1-a x^2\right )+2 (1+m) \text {PolyLog}\left (2,a x^2\right )-\text {PolyLog}\left (3,a x^2\right )-2 m \text {PolyLog}\left (3,a x^2\right )-m^2 \text {PolyLog}\left (3,a x^2\right )\right )}{(1+m)^4 \Gamma \left (\frac {1+m}{2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*PolyLog[3, a*x^2],x]

[Out]

(-2*x*(d*x)^m*Gamma[(3 + m)/2]*(2*a*(1 + m)*x^2*Gamma[(1 + m)/2]*HypergeometricPFQRegularized[{1, (3 + m)/2},
{(5 + m)/2}, a*x^2] + 4*Log[1 - a*x^2] + 2*(1 + m)*PolyLog[2, a*x^2] - PolyLog[3, a*x^2] - 2*m*PolyLog[3, a*x^
2] - m^2*PolyLog[3, a*x^2]))/((1 + m)^4*Gamma[(1 + m)/2])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 5.
time = 0.23, size = 218, normalized size = 1.85

method	result	size
meijerg	\(-\frac {\left (d x \right )^{m} x^{-m} \left (-a \right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (24+8 m \right )}{\left (3+m \right ) \left (1+m \right )^{4} a}-\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (12+4 m \right ) \ln \left (-a \,x^{2}+1\right )}{\left (3+m \right ) \left (1+m \right )^{3} a}+\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (-6-2 m \right ) \polylog \left (2, a \,x^{2}\right )}{\left (3+m \right ) \left (1+m \right )^{2} a}+\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \polylog \left (3, a \,x^{2}\right )}{\left (1+m \right ) a}+\frac {2 x^{1+m} \left (-a \right )^{\frac {3}{2}+\frac {m}{2}} \left (-12-4 m \right ) \Phi \left (a \,x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{\left (3+m \right ) \left (1+m \right )^{3} a}\right )}{2}\)	\(218\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*polylog(3,a*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(d*x)^m*x^(-m)*(-a)^(-1/2-1/2*m)*(2/(3+m)*x^(1+m)*(-a)^(3/2+1/2*m)*(24+8*m)/(1+m)^4/a-2/(3+m)*x^(1+m)*(-a
)^(3/2+1/2*m)*(12+4*m)/(1+m)^3/a*ln(-a*x^2+1)+2/(3+m)*x^(1+m)*(-a)^(3/2+1/2*m)*(-6-2*m)/(1+m)^2/a*polylog(2,a*
x^2)+2*x^(1+m)*(-a)^(3/2+1/2*m)/(1+m)/a*polylog(3,a*x^2)+2/(3+m)*x^(1+m)*(-a)^(3/2+1/2*m)*(-12-4*m)/(1+m)^3/a*
LerchPhi(a*x^2,1,1/2+1/2*m))

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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((d*x)^m*polylog(3,a*x^2),x, algorithm="maxima")

[Out]

8*a*d^m*integrate(x^2*x^m/((m^3 + 3*m^2 + 3*m + 1)*a*x^2 - m^3 - 3*m^2 - 3*m - 1), x) - (2*d^m*(m + 1)*x*x^m*d
ilog(a*x^2) - (m^2 + 2*m + 1)*d^m*x*x^m*polylog(3, a*x^2) + 4*d^m*x*x^m*log(-a*x^2 + 1))/(m^3 + 3*m^2 + 3*m +
1)

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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((d*x)^m*polylog(3,a*x^2),x, algorithm="fricas")

[Out]

integral((d*x)^m*polylog(3, a*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{m} \operatorname {Li}_{3}\left (a x^{2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*polylog(3,a*x**2),x)

[Out]

Integral((d*x)**m*polylog(3, a*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((d*x)^m*polylog(3,a*x^2),x, algorithm="giac")

[Out]

integrate((d*x)^m*polylog(3, a*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {polylog}\left (3,a\,x^2\right )\,{\left (d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(polylog(3, a*x^2)*(d*x)^m,x)

[Out]

int(polylog(3, a*x^2)*(d*x)^m, x)

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