3.110 \(\int x (a+b \tan ^{-1}(c x^3)) \, dx\)

Optimal. Leaf size=165 \[ \frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{\sqrt{3} b \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac{\sqrt{3} b \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{4 c^{2/3}} \]

[Out]

-(b*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + (x^2*(a + b*ArcTan[c*x^3]))/2 + (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2
/3)) - (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8
*c^(2/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))

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Rubi [A]  time = 0.393489, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5033, 295, 634, 618, 204, 628, 203} \[ \frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{\sqrt{3} b \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac{\sqrt{3} b \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{4 c^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTan[c*x^3]),x]

[Out]

-(b*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + (x^2*(a + b*ArcTan[c*x^3]))/2 + (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2
/3)) - (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8
*c^(2/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))

Rule 5033

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

Rule 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{1}{2} (3 b c) \int \frac{x^4}{1+c^2 x^6} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \int \frac{1}{1+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac{b \int \frac{-\frac{1}{2}+\frac{1}{2} \sqrt{3} \sqrt [3]{c} x}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac{b \int \frac{-\frac{1}{2}-\frac{1}{2} \sqrt{3} \sqrt [3]{c} x}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}\\ &=-\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{\left (\sqrt{3} b\right ) \int \frac{-\sqrt{3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{2/3}}+\frac{\left (\sqrt{3} b\right ) \int \frac{\sqrt{3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 c^{2/3}}-\frac{b \int \frac{1}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}-\frac{b \int \frac{1}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}\\ &=-\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{\sqrt{3} b \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac{\sqrt{3} b \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 \sqrt{3} c^{2/3}}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{4 \sqrt{3} c^{2/3}}\\ &=-\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{\sqrt{3} b \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac{\sqrt{3} b \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.0254029, size = 170, normalized size = 1.03 \[ \frac{a x^2}{2}-\frac{\sqrt{3} b \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac{\sqrt{3} b \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}-\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac{b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{4 c^{2/3}}+\frac{1}{2} b x^2 \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcTan[c*x^3]),x]

[Out]

(a*x^2)/2 - (b*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + (b*x^2*ArcTan[c*x^3])/2 + (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4
*c^(2/3)) - (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2
])/(8*c^(2/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))

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Maple [A]  time = 0.056, size = 154, normalized size = 0.9 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}\arctan \left ( c{x}^{3} \right ) }{2}}+{\frac{bc\sqrt{3}}{8} \left ({c}^{-2} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) }-{\frac{b}{4\,c}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}-{\frac{bc\sqrt{3}}{8} \left ({c}^{-2} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) }-{\frac{b}{4\,c}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}-{\frac{b}{2\,c}\arctan \left ({x{\frac{1}{\sqrt [6]{{c}^{-2}}}}} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arctan(c*x^3)),x)

[Out]

1/2*a*x^2+1/2*b*x^2*arctan(c*x^3)+1/8*b*c*3^(1/2)*(1/c^2)^(5/6)*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))-
1/4*b/c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))-1/8*b*c*3^(1/2)*(1/c^2)^(5/6)*ln(x^2-3^(1/2)*(1/c^2)^(
1/6)*x+(1/c^2)^(1/3))-1/4*b/c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))-1/2*b/c/(1/c^2)^(1/6)*arctan(x/(
1/c^2)^(1/6))

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Maxima [B]  time = 1.5567, size = 412, normalized size = 2.5 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{8} \,{\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c{\left (\frac{\sqrt{3} \log \left ({\left (c^{2}\right )}^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{\sqrt{3} \log \left ({\left (c^{2}\right )}^{\frac{1}{3}} x^{2} - \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{5}{6}}} - \frac{2 \, \log \left (\frac{{\left (c^{2}\right )}^{\frac{1}{3}} x - \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}{{\left (c^{2}\right )}^{\frac{1}{3}} x + \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}\right )}{{\left (c^{2}\right )}^{\frac{2}{3}} \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}} - \frac{{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (\frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x + \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} - \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x + \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} + \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}\right )}{c^{2} \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}} - \frac{{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (\frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x - \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} - \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} x - \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{6}} + \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}\right )}{c^{2} \sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}\right )}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctan(c*x^3)),x, algorithm="maxima")

[Out]

1/2*a*x^2 + 1/8*(4*x^2*arctan(c*x^3) + c*(sqrt(3)*log((c^2)^(1/3)*x^2 + sqrt(3)*(c^2)^(1/6)*x + 1)/(c^2)^(5/6)
- sqrt(3)*log((c^2)^(1/3)*x^2 - sqrt(3)*(c^2)^(1/6)*x + 1)/(c^2)^(5/6) - 2*log(((c^2)^(1/3)*x - sqrt(-(c^2)^(
1/3)))/((c^2)^(1/3)*x + sqrt(-(c^2)^(1/3))))/((c^2)^(2/3)*sqrt(-(c^2)^(1/3))) - (c^2)^(1/3)*log((2*(c^2)^(1/3)
*x + sqrt(3)*(c^2)^(1/6) - sqrt(-(c^2)^(1/3)))/(2*(c^2)^(1/3)*x + sqrt(3)*(c^2)^(1/6) + sqrt(-(c^2)^(1/3))))/(
c^2*sqrt(-(c^2)^(1/3))) - (c^2)^(1/3)*log((2*(c^2)^(1/3)*x - sqrt(3)*(c^2)^(1/6) - sqrt(-(c^2)^(1/3)))/(2*(c^2
)^(1/3)*x - sqrt(3)*(c^2)^(1/6) + sqrt(-(c^2)^(1/3))))/(c^2*sqrt(-(c^2)^(1/3)))))*b

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Fricas [B]  time = 2.53996, size = 971, normalized size = 5.88 \begin{align*} \frac{1}{2} \, b x^{2} \arctan \left (c x^{3}\right ) + \frac{1}{2} \, a x^{2} + \frac{1}{8} \, \sqrt{3} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} \log \left (b^{10} x^{2} + \sqrt{3} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{5}{6}} b^{5} c^{3} x + \left (\frac{b^{6}}{c^{4}}\right )^{\frac{2}{3}} b^{6} c^{2}\right ) - \frac{1}{8} \, \sqrt{3} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} \log \left (b^{10} x^{2} - \sqrt{3} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{5}{6}} b^{5} c^{3} x + \left (\frac{b^{6}}{c^{4}}\right )^{\frac{2}{3}} b^{6} c^{2}\right ) + \frac{1}{2} \, \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} b^{5} c x + \sqrt{3} b^{6} - 2 \, \sqrt{b^{10} x^{2} + \sqrt{3} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{5}{6}} b^{5} c^{3} x + \left (\frac{b^{6}}{c^{4}}\right )^{\frac{2}{3}} b^{6} c^{2}} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} c}{b^{6}}\right ) + \frac{1}{2} \, \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} b^{5} c x - \sqrt{3} b^{6} - 2 \, \sqrt{b^{10} x^{2} - \sqrt{3} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{5}{6}} b^{5} c^{3} x + \left (\frac{b^{6}}{c^{4}}\right )^{\frac{2}{3}} b^{6} c^{2}} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} c}{b^{6}}\right ) + \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} \arctan \left (-\frac{\left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} b^{5} c x - \sqrt{b^{10} x^{2} + \left (\frac{b^{6}}{c^{4}}\right )^{\frac{2}{3}} b^{6} c^{2}} \left (\frac{b^{6}}{c^{4}}\right )^{\frac{1}{6}} c}{b^{6}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctan(c*x^3)),x, algorithm="fricas")

[Out]

1/2*b*x^2*arctan(c*x^3) + 1/2*a*x^2 + 1/8*sqrt(3)*(b^6/c^4)^(1/6)*log(b^10*x^2 + sqrt(3)*(b^6/c^4)^(5/6)*b^5*c
^3*x + (b^6/c^4)^(2/3)*b^6*c^2) - 1/8*sqrt(3)*(b^6/c^4)^(1/6)*log(b^10*x^2 - sqrt(3)*(b^6/c^4)^(5/6)*b^5*c^3*x
+ (b^6/c^4)^(2/3)*b^6*c^2) + 1/2*(b^6/c^4)^(1/6)*arctan(-(2*(b^6/c^4)^(1/6)*b^5*c*x + sqrt(3)*b^6 - 2*sqrt(b^
10*x^2 + sqrt(3)*(b^6/c^4)^(5/6)*b^5*c^3*x + (b^6/c^4)^(2/3)*b^6*c^2)*(b^6/c^4)^(1/6)*c)/b^6) + 1/2*(b^6/c^4)^
(1/6)*arctan(-(2*(b^6/c^4)^(1/6)*b^5*c*x - sqrt(3)*b^6 - 2*sqrt(b^10*x^2 - sqrt(3)*(b^6/c^4)^(5/6)*b^5*c^3*x +
(b^6/c^4)^(2/3)*b^6*c^2)*(b^6/c^4)^(1/6)*c)/b^6) + (b^6/c^4)^(1/6)*arctan(-((b^6/c^4)^(1/6)*b^5*c*x - sqrt(b^
10*x^2 + (b^6/c^4)^(2/3)*b^6*c^2)*(b^6/c^4)^(1/6)*c)/b^6)

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Sympy [A]  time = 67.1106, size = 1620, normalized size = 9.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*atan(c*x**3)),x)

[Out]

Piecewise((a*x**2/2 - I*b*x**2*atanh(x**3*sqrt(x**(-6)))/2, Eq(c, -sqrt(-1/x**6)) | Eq(c, -sqrt(1/(x**6*(1/2 -
sqrt(3)*I/2)**3))) | Eq(c, -sqrt(1/(x**6*(1/2 + sqrt(3)*I/2)**3)))), (a*x**2/2, Eq(c, 0)), (-4*(-1)**(2/3)*a*
c**28*x**8*(c**(-2))**(41/3)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41
/3)) - 4*(-1)**(2/3)*a*c**26*x**2*(c**(-2))**(41/3)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/
3)*c**26*(c**(-2))**(41/3)) - 2*sqrt(3)*I*b*c**37*x**6*(c**(-2))**(37/2)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(
-2))**(1/6)) - sqrt(3)/3)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)
) - 2*sqrt(3)*I*b*c**37*x**6*(c**(-2))**(37/2)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) + sqrt(3)/3)/
(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 2*sqrt(3)*I*b*c**35*(c
**(-2))**(37/2)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) - sqrt(3)/3)/(-8*(-1)**(2/3)*c**28*x**6*(c**
(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 2*sqrt(3)*I*b*c**35*(c**(-2))**(37/2)*atan(2*(-1)**(5
/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) + sqrt(3)/3)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c
**26*(c**(-2))**(41/3)) - 4*(-1)**(2/3)*b*c**28*x**8*(c**(-2))**(41/3)*atan(c*x**3)/(-8*(-1)**(2/3)*c**28*x**6
*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 4*(-1)**(2/3)*b*c**26*x**2*(c**(-2))**(41/3)*ata
n(c*x**3)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 3*I*b*c**21*
x**6*(c**(-2))**(21/2)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(-8*(-1
)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) + 6*I*b*c**21*x**6*(c**(-2))**(
21/2)*log(2)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 3*I*b*c**
19*(c**(-2))**(21/2)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(-8*(-1)*
*(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) + 6*I*b*c**19*(c**(-2))**(21/2)*l
og(2)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) + 3*I*b*c**15*x**6
*(c**(-2))**(15/2)*log(4*x**2 - 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(-8*(-1)**(
2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 6*I*b*c**15*x**6*(c**(-2))**(15/2
)*log(2)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) + 3*I*b*c**13*(
c**(-2))**(15/2)*log(4*x**2 - 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(-8*(-1)**(2/
3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 6*I*b*c**13*(c**(-2))**(15/2)*log(2
)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 4*b*x**6*atan(c*x**3
)/(-8*(-1)**(2/3)*c**28*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**26*(c**(-2))**(41/3)) - 4*b*atan(c*x**3)/(-8
*(-1)**(2/3)*c**30*x**6*(c**(-2))**(41/3) - 8*(-1)**(2/3)*c**28*(c**(-2))**(41/3)), True))

________________________________________________________________________________________

Giac [A]  time = 1.30739, size = 212, normalized size = 1.28 \begin{align*} \frac{1}{8} \, b c^{5}{\left (\frac{\sqrt{3}{\left | c \right |}^{\frac{1}{3}} \log \left (x^{2} + \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{6}} - \frac{\sqrt{3}{\left | c \right |}^{\frac{1}{3}} \log \left (x^{2} - \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{c^{6}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left ({\left (2 \, x + \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}} - \frac{2 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left ({\left (2 \, x - \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}} - \frac{4 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left (x{\left | c \right |}^{\frac{1}{3}}\right )}{c^{6}}\right )} + \frac{1}{2} \, b x^{2} \arctan \left (c x^{3}\right ) + \frac{1}{2} \, a x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arctan(c*x^3)),x, algorithm="giac")

[Out]

1/8*b*c^5*(sqrt(3)*abs(c)^(1/3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^6 - sqrt(3)*abs(c)^(1/3)*
log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^6 - 2*abs(c)^(1/3)*arctan((2*x + sqrt(3)/abs(c)^(1/3))*ab
s(c)^(1/3))/c^6 - 2*abs(c)^(1/3)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/c^6 - 4*abs(c)^(1/3)*arctan
(x*abs(c)^(1/3))/c^6) + 1/2*b*x^2*arctan(c*x^3) + 1/2*a*x^2