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

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

[Out]

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

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Rubi [A]  time = 0.294441, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5033, 209, 634, 618, 204, 628, 203} \[ -\frac{a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )+\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )+\frac{1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )+\frac{1}{4} b c^{2/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^3])/x^3,x]

[Out]

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

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 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && 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 \frac{a+b \tan ^{-1}\left (c x^3\right )}{x^3} \, dx &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}+\frac{1}{2} (3 b c) \int \frac{1}{1+c^2 x^6} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}+\frac{1}{2} (b c) \int \frac{1}{1+c^{2/3} x^2} \, dx+\frac{1}{2} (b c) \int \frac{1-\frac{1}{2} \sqrt{3} \sqrt [3]{c} x}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac{1}{2} (b c) \int \frac{1+\frac{1}{2} \sqrt{3} \sqrt [3]{c} x}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=\frac{1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac{1}{8} \left (\sqrt{3} b c^{2/3}\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+\frac{1}{8} \left (\sqrt{3} b c^{2/3}\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+\frac{1}{8} (b c) \int \frac{1}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac{1}{8} (b c) \int \frac{1}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=\frac{1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac{\left (b c^{2/3}\right ) \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}}-\frac{\left (b c^{2/3}\right ) \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}}\\ &=\frac{1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac{1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )+\frac{1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )-\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0478968, size = 170, normalized size = 1.03 \[ -\frac{a}{2 x^2}-\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )+\frac{1}{8} \sqrt{3} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )+\frac{1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )+\frac{1}{4} b c^{2/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )-\frac{b \tan ^{-1}\left (c x^3\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x^3])/x^3,x]

[Out]

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

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Maple [A]  time = 0.056, size = 148, normalized size = 0.9 \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b\arctan \left ( c{x}^{3} \right ) }{2\,{x}^{2}}}+{\frac{bc\sqrt{3}}{8}\sqrt [6]{{c}^{-2}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) }+{\frac{bc}{4}\sqrt [6]{{c}^{-2}}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}+\sqrt{3} \right ) }-{\frac{bc\sqrt{3}}{8}\sqrt [6]{{c}^{-2}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) }+{\frac{bc}{4}\sqrt [6]{{c}^{-2}}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}-\sqrt{3} \right ) }+{\frac{bc}{2}\sqrt [6]{{c}^{-2}}\arctan \left ({x{\frac{1}{\sqrt [6]{{c}^{-2}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^3))/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)^(1/6)*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)^(1/6)*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 = 1.52117, size = 381, normalized size = 2.31 \begin{align*} \frac{1}{8} \,{\left ({\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{1}{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{1}{6}}} + \frac{\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 )}{\sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}} + \frac{\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 )}{\sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}} + \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 )}{\sqrt{-{\left (c^{2}\right )}^{\frac{1}{3}}}}\right )} c - \frac{4 \, \arctan \left (c x^{3}\right )}{x^{2}}\right )} b - \frac{a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.58416, size = 1224, normalized size = 7.42 \begin{align*} \frac{\sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} x^{2} \log \left (4 \, b^{2} c^{2} x^{2} + 4 \, \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} b c x + 4 \, \left (b^{6} c^{4}\right )^{\frac{1}{3}}\right ) - \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} x^{2} \log \left (4 \, b^{2} c^{2} x^{2} - 4 \, \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} b c x + 4 \, \left (b^{6} c^{4}\right )^{\frac{1}{3}}\right ) + \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} x^{2} \log \left (b^{2} c^{2} x^{2} + \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac{1}{3}}\right ) - \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} x^{2} \log \left (b^{2} c^{2} x^{2} - \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac{1}{3}}\right ) - 8 \, \left (b^{6} c^{4}\right )^{\frac{1}{6}} x^{2} \arctan \left (-\frac{\sqrt{3} b^{6} c^{4} + 2 \, \left (b^{6} c^{4}\right )^{\frac{5}{6}} b c x - 2 \, \left (b^{6} c^{4}\right )^{\frac{5}{6}} \sqrt{b^{2} c^{2} x^{2} + \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac{1}{3}}}}{b^{6} c^{4}}\right ) - 8 \, \left (b^{6} c^{4}\right )^{\frac{1}{6}} x^{2} \arctan \left (\frac{\sqrt{3} b^{6} c^{4} - 2 \, \left (b^{6} c^{4}\right )^{\frac{5}{6}} b c x + 2 \, \left (b^{6} c^{4}\right )^{\frac{5}{6}} \sqrt{b^{2} c^{2} x^{2} - \sqrt{3} \left (b^{6} c^{4}\right )^{\frac{1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac{1}{3}}}}{b^{6} c^{4}}\right ) - 16 \, \left (b^{6} c^{4}\right )^{\frac{1}{6}} x^{2} \arctan \left (-\frac{\left (b^{6} c^{4}\right )^{\frac{5}{6}} b c x - \left (b^{6} c^{4}\right )^{\frac{5}{6}} \sqrt{b^{2} c^{2} x^{2} + \left (b^{6} c^{4}\right )^{\frac{1}{3}}}}{b^{6} c^{4}}\right ) - 8 \, b \arctan \left (c x^{3}\right ) - 8 \, a}{16 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(sqrt(3)*(b^6*c^4)^(1/6)*x^2*log(4*b^2*c^2*x^2 + 4*sqrt(3)*(b^6*c^4)^(1/6)*b*c*x + 4*(b^6*c^4)^(1/3)) - s
qrt(3)*(b^6*c^4)^(1/6)*x^2*log(4*b^2*c^2*x^2 - 4*sqrt(3)*(b^6*c^4)^(1/6)*b*c*x + 4*(b^6*c^4)^(1/3)) + sqrt(3)*
(b^6*c^4)^(1/6)*x^2*log(b^2*c^2*x^2 + sqrt(3)*(b^6*c^4)^(1/6)*b*c*x + (b^6*c^4)^(1/3)) - sqrt(3)*(b^6*c^4)^(1/
6)*x^2*log(b^2*c^2*x^2 - sqrt(3)*(b^6*c^4)^(1/6)*b*c*x + (b^6*c^4)^(1/3)) - 8*(b^6*c^4)^(1/6)*x^2*arctan(-(sqr
t(3)*b^6*c^4 + 2*(b^6*c^4)^(5/6)*b*c*x - 2*(b^6*c^4)^(5/6)*sqrt(b^2*c^2*x^2 + sqrt(3)*(b^6*c^4)^(1/6)*b*c*x +
(b^6*c^4)^(1/3)))/(b^6*c^4)) - 8*(b^6*c^4)^(1/6)*x^2*arctan((sqrt(3)*b^6*c^4 - 2*(b^6*c^4)^(5/6)*b*c*x + 2*(b^
6*c^4)^(5/6)*sqrt(b^2*c^2*x^2 - sqrt(3)*(b^6*c^4)^(1/6)*b*c*x + (b^6*c^4)^(1/3)))/(b^6*c^4)) - 16*(b^6*c^4)^(1
/6)*x^2*arctan(-((b^6*c^4)^(5/6)*b*c*x - (b^6*c^4)^(5/6)*sqrt(b^2*c^2*x^2 + (b^6*c^4)^(1/3)))/(b^6*c^4)) - 8*b
*arctan(c*x^3) - 8*a)/x^2

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Sympy [A]  time = 124.381, size = 311, normalized size = 1.88 \begin{align*} \begin{cases} - \frac{a}{2 x^{2}} - \frac{\sqrt [6]{-1} \sqrt{3} b c^{7} \left (\frac{1}{c^{2}}\right )^{\frac{19}{6}} \operatorname{atan}{\left (\frac{2 \left (-1\right )^{\frac{5}{6}} \sqrt{3} x}{3 \sqrt [6]{\frac{1}{c^{2}}}} - \frac{\sqrt{3}}{3} \right )}}{4} - \frac{\sqrt [6]{-1} \sqrt{3} b c^{7} \left (\frac{1}{c^{2}}\right )^{\frac{19}{6}} \operatorname{atan}{\left (\frac{2 \left (-1\right )^{\frac{5}{6}} \sqrt{3} x}{3 \sqrt [6]{\frac{1}{c^{2}}}} + \frac{\sqrt{3}}{3} \right )}}{4} - \frac{b \operatorname{atan}{\left (c x^{3} \right )}}{2 x^{2}} + \frac{3 \sqrt [6]{-1} b \log{\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac{1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac{1}{c^{2}}} \right )}}{8 c^{7} \left (\frac{1}{c^{2}}\right )^{\frac{23}{6}}} - \frac{3 \sqrt [6]{-1} b \log{\left (4 x^{2} - 4 \sqrt [6]{-1} x \sqrt [6]{\frac{1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac{1}{c^{2}}} \right )}}{8 c^{11} \left (\frac{1}{c^{2}}\right )^{\frac{35}{6}}} - \frac{\left (-1\right )^{\frac{2}{3}} b \operatorname{atan}{\left (c x^{3} \right )}}{2 c^{24} \left (\frac{1}{c^{2}}\right )^{\frac{37}{3}}} & \text{for}\: c \neq 0 \\- \frac{a}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/(2*x**2) - (-1)**(1/6)*sqrt(3)*b*c**7*(c**(-2))**(19/6)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2)
)**(1/6)) - sqrt(3)/3)/4 - (-1)**(1/6)*sqrt(3)*b*c**7*(c**(-2))**(19/6)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-
2))**(1/6)) + sqrt(3)/3)/4 - b*atan(c*x**3)/(2*x**2) + 3*(-1)**(1/6)*b*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))*
*(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(8*c**7*(c**(-2))**(23/6)) - 3*(-1)**(1/6)*b*log(4*x**2 - 4*(-1)**(1/
6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(8*c**11*(c**(-2))**(35/6)) - (-1)**(2/3)*b*atan(c*x**
3)/(2*c**24*(c**(-2))**(37/3)), Ne(c, 0)), (-a/(2*x**2), True))

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Giac [A]  time = 1.24397, size = 185, normalized size = 1.12 \begin{align*} \frac{1}{8} \,{\left (\frac{\sqrt{3} \log \left (x^{2} + \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{{\left | c \right |}^{\frac{1}{3}}} - \frac{\sqrt{3} \log \left (x^{2} - \frac{\sqrt{3} x}{{\left | c \right |}^{\frac{1}{3}}} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{{\left | c \right |}^{\frac{1}{3}}} + \frac{2 \, \arctan \left ({\left (2 \, x + \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{{\left | c \right |}^{\frac{1}{3}}} + \frac{2 \, \arctan \left ({\left (2 \, x - \frac{\sqrt{3}}{{\left | c \right |}^{\frac{1}{3}}}\right )}{\left | c \right |}^{\frac{1}{3}}\right )}{{\left | c \right |}^{\frac{1}{3}}} + \frac{4 \, \arctan \left (x{\left | c \right |}^{\frac{1}{3}}\right )}{{\left | c \right |}^{\frac{1}{3}}}\right )} b c - \frac{b \arctan \left (c x^{3}\right ) + a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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