3.2.89 \(\int (d+e x^2) \cot ^{-1}(a x) \log (c x^n) \, dx\) [189]

Optimal. Leaf size=182 \[ -\frac {5 e n x^2}{36 a}-d n x \cot ^{-1}(a x)-\frac {1}{9} e n x^3 \cot ^{-1}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac {d n \log \left (1+a^2 x^2\right )}{2 a}+\frac {e n \log \left (1+a^2 x^2\right )}{18 a^3}+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d-e\right ) n \text {Li}_2\left (-a^2 x^2\right )}{12 a^3} \]

[Out]

-5/36*e*n*x^2/a-d*n*x*arccot(a*x)-1/9*e*n*x^3*arccot(a*x)+1/6*e*x^2*ln(c*x^n)/a+d*x*arccot(a*x)*ln(c*x^n)+1/3*
e*x^3*arccot(a*x)*ln(c*x^n)-1/2*d*n*ln(a^2*x^2+1)/a+1/18*e*n*ln(a^2*x^2+1)/a^3+1/6*(3*a^2*d-e)*ln(c*x^n)*ln(a^
2*x^2+1)/a^3+1/12*(3*a^2*d-e)*n*polylog(2,-a^2*x^2)/a^3

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Rubi [A]
time = 0.11, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5033, 1607, 455, 45, 2435, 4931, 266, 4947, 272, 2438} \begin {gather*} \frac {n \left (3 a^2 d-e\right ) \text {PolyLog}\left (2,-a^2 x^2\right )}{12 a^3}-\frac {d n \log \left (a^2 x^2+1\right )}{2 a}+\frac {\left (3 a^2 d-e\right ) \log \left (a^2 x^2+1\right ) \log \left (c x^n\right )}{6 a^3}+\frac {e n \log \left (a^2 x^2+1\right )}{18 a^3}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {e x^2 \log \left (c x^n\right )}{6 a}-d n x \cot ^{-1}(a x)-\frac {1}{9} e n x^3 \cot ^{-1}(a x)-\frac {5 e n x^2}{36 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)*ArcCot[a*x]*Log[c*x^n],x]

[Out]

(-5*e*n*x^2)/(36*a) - d*n*x*ArcCot[a*x] - (e*n*x^3*ArcCot[a*x])/9 + (e*x^2*Log[c*x^n])/(6*a) + d*x*ArcCot[a*x]
*Log[c*x^n] + (e*x^3*ArcCot[a*x]*Log[c*x^n])/3 - (d*n*Log[1 + a^2*x^2])/(2*a) + (e*n*Log[1 + a^2*x^2])/(18*a^3
) + ((3*a^2*d - e)*Log[c*x^n]*Log[1 + a^2*x^2])/(6*a^3) + ((3*a^2*d - e)*n*PolyLog[2, -(a^2*x^2)])/(12*a^3)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 455

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2435

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(Px_.)*(F_)[(d_.)*((e_.) + (f_.)*(x_))], x_Symbol] :> With[{u = IntH
ide[Px*F[d*(e + f*x)], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist[1/x, u, x], x], x]] /; FreeQ[{a,
b, c, d, e, f, n}, x] && PolynomialQ[Px, x] && MemberQ[{ArcTan, ArcCot, ArcTanh, ArcCoth}, F]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4931

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Dist[b*c
*n*p, Int[x^n*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

Rule 4947

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

Rule 5033

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcCot[c*x], u, x] + Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rubi steps

\begin {align*} \int \left (d+e x^2\right ) \cot ^{-1}(a x) \log \left (c x^n\right ) \, dx &=\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}-n \int \left (\frac {e x}{6 a}+d \cot ^{-1}(a x)+\frac {1}{3} e x^2 \cot ^{-1}(a x)+\frac {\left (3 a^2 d-e\right ) \log \left (1+a^2 x^2\right )}{6 a^3 x}\right ) \, dx\\ &=-\frac {e n x^2}{12 a}+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}-(d n) \int \cot ^{-1}(a x) \, dx-\frac {\left (\left (3 a^2 d-e\right ) n\right ) \int \frac {\log \left (1+a^2 x^2\right )}{x} \, dx}{6 a^3}-\frac {1}{3} (e n) \int x^2 \cot ^{-1}(a x) \, dx\\ &=-\frac {e n x^2}{12 a}-d n x \cot ^{-1}(a x)-\frac {1}{9} e n x^3 \cot ^{-1}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d-e\right ) n \text {Li}_2\left (-a^2 x^2\right )}{12 a^3}-(a d n) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{9} (a e n) \int \frac {x^3}{1+a^2 x^2} \, dx\\ &=-\frac {e n x^2}{12 a}-d n x \cot ^{-1}(a x)-\frac {1}{9} e n x^3 \cot ^{-1}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac {d n \log \left (1+a^2 x^2\right )}{2 a}+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d-e\right ) n \text {Li}_2\left (-a^2 x^2\right )}{12 a^3}-\frac {1}{18} (a e n) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {e n x^2}{12 a}-d n x \cot ^{-1}(a x)-\frac {1}{9} e n x^3 \cot ^{-1}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac {d n \log \left (1+a^2 x^2\right )}{2 a}+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d-e\right ) n \text {Li}_2\left (-a^2 x^2\right )}{12 a^3}-\frac {1}{18} (a e n) \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {5 e n x^2}{36 a}-d n x \cot ^{-1}(a x)-\frac {1}{9} e n x^3 \cot ^{-1}(a x)+\frac {e x^2 \log \left (c x^n\right )}{6 a}+d x \cot ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cot ^{-1}(a x) \log \left (c x^n\right )-\frac {d n \log \left (1+a^2 x^2\right )}{2 a}+\frac {e n \log \left (1+a^2 x^2\right )}{18 a^3}+\frac {\left (3 a^2 d-e\right ) \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )}{6 a^3}+\frac {\left (3 a^2 d-e\right ) n \text {Li}_2\left (-a^2 x^2\right )}{12 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 178, normalized size = 0.98 \begin {gather*} \frac {-5 a^2 e n x^2+36 a^2 d n \log \left (\frac {1}{a \sqrt {1+\frac {1}{a^2 x^2}} x}\right )+6 a^2 e x^2 \log \left (c x^n\right )-4 a^3 x \cot ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+2 e n \log \left (1+a^2 x^2\right )+18 a^2 d \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )-6 e \log \left (c x^n\right ) \log \left (1+a^2 x^2\right )+\left (9 a^2 d n-3 e n\right ) \text {Li}_2\left (-a^2 x^2\right )}{36 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)*ArcCot[a*x]*Log[c*x^n],x]

[Out]

(-5*a^2*e*n*x^2 + 36*a^2*d*n*Log[1/(a*Sqrt[1 + 1/(a^2*x^2)]*x)] + 6*a^2*e*x^2*Log[c*x^n] - 4*a^3*x*ArcCot[a*x]
*(n*(9*d + e*x^2) - 3*(3*d + e*x^2)*Log[c*x^n]) + 2*e*n*Log[1 + a^2*x^2] + 18*a^2*d*Log[c*x^n]*Log[1 + a^2*x^2
] - 6*e*Log[c*x^n]*Log[1 + a^2*x^2] + (9*a^2*d*n - 3*e*n)*PolyLog[2, -(a^2*x^2)])/(36*a^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 14.17, size = 152337, normalized size = 837.02

method	result	size
risch	\(\text {Expression too large to display}\)	\(3017\)
default	\(\text {Expression too large to display}\)	\(152337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*arccot(a*x)*ln(c*x^n),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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^2+d)*arccot(a*x)*log(c*x^n),x, algorithm="maxima")

[Out]

1/36*(69984*a^4*n*e*integrate(1/11664*x^4*log(x)/(a^2*x^3 + x), x) + 209952*a^4*d*n*integrate(1/11664*x^2*log(
x)/(a^2*x^3 + x), x) + 1944*a^4*e*integrate(1/216*(2*a*x^4*arctan2(1, a*x) + x^3*log(a^2*x^2 + 1))/(a^2*x^2 +
1), x)*log(c) + 1944*a^4*d*integrate(1/216*(2*a*x^2*arctan2(1, a*x) + x*log(a^2*x^2 + 1))/(a^2*x^2 + 1), x)*lo
g(c) + 1944*a^4*e*integrate(1/216*(2*a*x^4*arctan2(1, a*x) + x^3*log(a^2*x^2 + 1))*log(x^n)/(a^2*x^2 + 1), x)
+ 1944*a^4*d*integrate(1/216*(2*a*x^2*arctan2(1, a*x) + x*log(a^2*x^2 + 1))*log(x^n)/(a^2*x^2 + 1), x) - 9*(21
6*a*integrate(1/216*x*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - arctan(a*x)^2/a - 2*arctan(a*x)*arctan(1/(a*x))/a)*
a^3*d*log(c) - 1944*a^3*e*integrate(1/216*(a*x^3*log(a^2*x^2 + 1) - 2*x^2*arctan2(1, a*x))/(a^2*x^2 + 1), x)*l
og(c) - 1944*a^3*e*integrate(1/216*(a*x^3*log(a^2*x^2 + 1) - 2*x^2*arctan2(1, a*x))*log(x^n)/(a^2*x^2 + 1), x)
- 1944*a^3*d*integrate(1/216*(a*x*log(a^2*x^2 + 1) - 2*arctan2(1, a*x))*log(x^n)/(a^2*x^2 + 1), x) - 2*(a^3*n
*arctan2(1, a*x)*e - 3*a^3*arctan2(1, a*x)*e*log(c))*x^3 - (a^2*n*e - 3*a^2*e*log(c))*x^2 - 18*(a^3*d*n*arctan
2(1, a*x) - a^3*d*arctan2(1, a*x)*log(c))*x + (9*a^2*d*log(c) - (9*a^2*d - e)*n - 3*e*log(c))*log(a^2*x^2 + 1)
+ 6*(a^3*x^3*arctan2(1, a*x)*e + 3*a^3*d*x*arctan2(1, a*x))*log(x^n))/a^3

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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^2+d)*arccot(a*x)*log(c*x^n),x, algorithm="fricas")

[Out]

integral((x^2*e + d)*arccot(a*x)*log(c*x^n), x)

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Sympy [A]
time = 52.74, size = 231, normalized size = 1.27 \begin {gather*} - d n \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: a = 0 \\\begin {cases} x \operatorname {acot}{\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {for}\: a \neq 0 \\\frac {\pi x}{2} & \text {otherwise} \end {cases} - \frac {\operatorname {Li}_{2}\left (a^{2} x^{2} e^{i \pi }\right )}{4 a} & \text {otherwise} \end {cases}\right ) + d \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: a = 0 \\x \operatorname {acot}{\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} + 1 \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} - \frac {e n x^{3} \operatorname {acot}{\left (a x \right )}}{9} + \frac {e x^{3} \log {\left (c x^{n} \right )} \operatorname {acot}{\left (a x \right )}}{3} - \frac {5 e n x^{2}}{36 a} + \frac {e n \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a = 0 \\- \frac {\operatorname {Li}_{2}\left (a^{2} x^{2} e^{i \pi }\right )}{2 a^{2}} & \text {otherwise} \end {cases}\right )}{6 a} + \frac {e n \left (\begin {cases} x^{2} & \text {for}\: a^{2} = 0 \\\frac {\log {\left (a^{2} x^{2} + 1 \right )}}{a^{2}} & \text {otherwise} \end {cases}\right )}{18 a} + \frac {e x^{2} \log {\left (c x^{n} \right )}}{6 a} - \frac {e \left (\begin {cases} x^{2} & \text {for}\: a^{2} = 0 \\\frac {\log {\left (a^{2} x^{2} + 1 \right )}}{a^{2}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{6 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*acot(a*x)*ln(c*x**n),x)

[Out]

-d*n*Piecewise((pi*x/2, Eq(a, 0)), (Piecewise((x*acot(a*x) + log(a**2*x**2 + 1)/(2*a), Ne(a, 0)), (pi*x/2, Tru
e)) - polylog(2, a**2*x**2*exp_polar(I*pi))/(4*a), True)) + d*Piecewise((pi*x/2, Eq(a, 0)), (x*acot(a*x) + log
(a**2*x**2 + 1)/(2*a), True))*log(c*x**n) - e*n*x**3*acot(a*x)/9 + e*x**3*log(c*x**n)*acot(a*x)/3 - 5*e*n*x**2
/(36*a) + e*n*Piecewise((x**2/2, Eq(a, 0)), (-polylog(2, a**2*x**2*exp_polar(I*pi))/(2*a**2), True))/(6*a) + e
*n*Piecewise((x**2, Eq(a**2, 0)), (log(a**2*x**2 + 1)/a**2, True))/(18*a) + e*x**2*log(c*x**n)/(6*a) - e*Piece
wise((x**2, Eq(a**2, 0)), (log(a**2*x**2 + 1)/a**2, True))*log(c*x**n)/(6*a)

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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^2+d)*arccot(a*x)*log(c*x^n),x, algorithm="giac")

[Out]

integrate((x^2*e + d)*arccot(a*x)*log(c*x^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*x^n)*acot(a*x)*(d + e*x^2),x)

[Out]

int(log(c*x^n)*acot(a*x)*(d + e*x^2), x)

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