3.112 \(\int \frac{a+b \tan ^{-1}(c x^3)}{x^5} \, dx\)

Optimal. Leaf size=174 \[ -\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )+\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )-\frac{1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )+\frac{1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )-\frac{1}{8} b c^{4/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )-\frac{3 b c}{4 x} \]

[Out]

(-3*b*c)/(4*x) - (b*c^(4/3)*ArcTan[c^(1/3)*x])/4 - (a + b*ArcTan[c*x^3])/(4*x^4) + (b*c^(4/3)*ArcTan[Sqrt[3] -
2*c^(1/3)*x])/8 - (b*c^(4/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/8 - (Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^(1/3)*x
+ c^(2/3)*x^2])/16 + (Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16

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Rubi [A]  time = 0.412866, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5033, 325, 295, 634, 618, 204, 628, 203} \[ -\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )+\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )-\frac{1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )+\frac{1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )-\frac{1}{8} b c^{4/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )-\frac{3 b c}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^3])/x^5,x]

[Out]

(-3*b*c)/(4*x) - (b*c^(4/3)*ArcTan[c^(1/3)*x])/4 - (a + b*ArcTan[c*x^3])/(4*x^4) + (b*c^(4/3)*ArcTan[Sqrt[3] -
2*c^(1/3)*x])/8 - (b*c^(4/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/8 - (Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^(1/3)*x
+ c^(2/3)*x^2])/16 + (Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16

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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 \frac{a+b \tan ^{-1}\left (c x^3\right )}{x^5} \, dx &=-\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}+\frac{1}{4} (3 b c) \int \frac{1}{x^2 \left (1+c^2 x^6\right )} \, dx\\ &=-\frac{3 b c}{4 x}-\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac{1}{4} \left (3 b c^3\right ) \int \frac{x^4}{1+c^2 x^6} \, dx\\ &=-\frac{3 b c}{4 x}-\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac{1}{4} \left (b c^{5/3}\right ) \int \frac{1}{1+c^{2/3} x^2} \, dx-\frac{1}{4} \left (b c^{5/3}\right ) \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-\frac{1}{4} \left (b c^{5/3}\right ) \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\\ &=-\frac{3 b c}{4 x}-\frac{1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac{1}{16} \left (\sqrt{3} b c^{4/3}\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}{16} \left (\sqrt{3} b c^{4/3}\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}{16} \left (b c^{5/3}\right ) \int \frac{1}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx-\frac{1}{16} \left (b c^{5/3}\right ) \int \frac{1}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=-\frac{3 b c}{4 x}-\frac{1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )-\frac{\left (b c^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{8 \sqrt{3}}+\frac{\left (b c^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c} x}{\sqrt{3}}\right )}{8 \sqrt{3}}\\ &=-\frac{3 b c}{4 x}-\frac{1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac{a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}+\frac{1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )-\frac{1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )-\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0495834, size = 179, normalized size = 1.03 \[ -\frac{a}{4 x^4}-\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )+\frac{1}{16} \sqrt{3} b c^{4/3} \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )-\frac{1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )+\frac{1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )-\frac{1}{8} b c^{4/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )-\frac{b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac{3 b c}{4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x^3])/x^5,x]

[Out]

-a/(4*x^4) - (3*b*c)/(4*x) - (b*c^(4/3)*ArcTan[c^(1/3)*x])/4 - (b*ArcTan[c*x^3])/(4*x^4) + (b*c^(4/3)*ArcTan[S
qrt[3] - 2*c^(1/3)*x])/8 - (b*c^(4/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/8 - (Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^
(1/3)*x + c^(2/3)*x^2])/16 + (Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16

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Maple [A]  time = 0.059, size = 159, normalized size = 0.9 \begin{align*} -{\frac{a}{4\,{x}^{4}}}-{\frac{b\arctan \left ( c{x}^{3} \right ) }{4\,{x}^{4}}}+{\frac{b{c}^{3}\sqrt{3}}{16} \left ({c}^{-2} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) }-{\frac{bc}{8}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}-{\frac{b{c}^{3}\sqrt{3}}{16} \left ({c}^{-2} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{c}^{-2}}x+\sqrt [3]{{c}^{-2}} \right ) }-{\frac{bc}{8}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}-{\frac{bc}{4}\arctan \left ({x{\frac{1}{\sqrt [6]{{c}^{-2}}}}} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}-{\frac{3\,bc}{4\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^3))/x^5,x)

[Out]

-1/4*a/x^4-1/4*b/x^4*arctan(c*x^3)+1/16*b*c^3*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))-1/8*b*c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))-1/16*b*c^3*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))-1/8*b*c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))-1/4*b*c/(1/c^2)^(1/6)*arc
tan(x/(1/c^2)^(1/6))-3/4*b*c/x

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Maxima [B]  time = 1.5348, size = 425, normalized size = 2.44 \begin{align*} \frac{1}{16} \,{\left ({\left (c^{2}{\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 )} - \frac{12}{x}\right )} c - \frac{4 \, \arctan \left (c x^{3}\right )}{x^{4}}\right )} b - \frac{a}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x^5,x, algorithm="maxima")

[Out]

1/16*((c^2*(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) - sqr
t(-(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)))) - 12/x)*c - 4*arctan(c*x^3)/x^4)*b - 1/4*a/x^4

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Fricas [B]  time = 2.96316, size = 1385, normalized size = 7.96 \begin{align*} \frac{\sqrt{3} \left (b^{6} c^{8}\right )^{\frac{1}{6}} x^{4} \log \left (4 \, b^{10} c^{14} x^{2} + 4 \, \left (b^{6} c^{8}\right )^{\frac{2}{3}} b^{6} c^{8} + 4 \, \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{5}{6}} b^{5} c^{7} x\right ) - \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{1}{6}} x^{4} \log \left (4 \, b^{10} c^{14} x^{2} + 4 \, \left (b^{6} c^{8}\right )^{\frac{2}{3}} b^{6} c^{8} - 4 \, \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{5}{6}} b^{5} c^{7} x\right ) + \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{1}{6}} x^{4} \log \left (b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac{2}{3}} b^{6} c^{8} + \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{5}{6}} b^{5} c^{7} x\right ) - \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{1}{6}} x^{4} \log \left (b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac{2}{3}} b^{6} c^{8} - \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{5}{6}} b^{5} c^{7} x\right ) + 8 \, \left (b^{6} c^{8}\right )^{\frac{1}{6}} x^{4} \arctan \left (-\frac{\sqrt{3} b^{6} c^{8} + 2 \, \left (b^{6} c^{8}\right )^{\frac{1}{6}} b^{5} c^{7} x - 2 \, \sqrt{b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac{2}{3}} b^{6} c^{8} + \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{5}{6}} b^{5} c^{7} x} \left (b^{6} c^{8}\right )^{\frac{1}{6}}}{b^{6} c^{8}}\right ) + 8 \, \left (b^{6} c^{8}\right )^{\frac{1}{6}} x^{4} \arctan \left (\frac{\sqrt{3} b^{6} c^{8} - 2 \, \left (b^{6} c^{8}\right )^{\frac{1}{6}} b^{5} c^{7} x + 2 \, \sqrt{b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac{2}{3}} b^{6} c^{8} - \sqrt{3} \left (b^{6} c^{8}\right )^{\frac{5}{6}} b^{5} c^{7} x} \left (b^{6} c^{8}\right )^{\frac{1}{6}}}{b^{6} c^{8}}\right ) + 16 \, \left (b^{6} c^{8}\right )^{\frac{1}{6}} x^{4} \arctan \left (-\frac{\left (b^{6} c^{8}\right )^{\frac{1}{6}} b^{5} c^{7} x - \sqrt{b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac{2}{3}} b^{6} c^{8}} \left (b^{6} c^{8}\right )^{\frac{1}{6}}}{b^{6} c^{8}}\right ) - 24 \, b c x^{3} - 8 \, b \arctan \left (c x^{3}\right ) - 8 \, a}{32 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x^5,x, algorithm="fricas")

[Out]

1/32*(sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(4*b^10*c^14*x^2 + 4*(b^6*c^8)^(2/3)*b^6*c^8 + 4*sqrt(3)*(b^6*c^8)^(5/6)*
b^5*c^7*x) - sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(4*b^10*c^14*x^2 + 4*(b^6*c^8)^(2/3)*b^6*c^8 - 4*sqrt(3)*(b^6*c^8)
^(5/6)*b^5*c^7*x) + sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(b^10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8 + sqrt(3)*(b^6*c^8
)^(5/6)*b^5*c^7*x) - sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(b^10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8 - sqrt(3)*(b^6*c^
8)^(5/6)*b^5*c^7*x) + 8*(b^6*c^8)^(1/6)*x^4*arctan(-(sqrt(3)*b^6*c^8 + 2*(b^6*c^8)^(1/6)*b^5*c^7*x - 2*sqrt(b^
10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8 + sqrt(3)*(b^6*c^8)^(5/6)*b^5*c^7*x)*(b^6*c^8)^(1/6))/(b^6*c^8)) + 8*(b^
6*c^8)^(1/6)*x^4*arctan((sqrt(3)*b^6*c^8 - 2*(b^6*c^8)^(1/6)*b^5*c^7*x + 2*sqrt(b^10*c^14*x^2 + (b^6*c^8)^(2/3
)*b^6*c^8 - sqrt(3)*(b^6*c^8)^(5/6)*b^5*c^7*x)*(b^6*c^8)^(1/6))/(b^6*c^8)) + 16*(b^6*c^8)^(1/6)*x^4*arctan(-((
b^6*c^8)^(1/6)*b^5*c^7*x - sqrt(b^10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8)*(b^6*c^8)^(1/6))/(b^6*c^8)) - 24*b*c*
x^3 - 8*b*arctan(c*x^3) - 8*a)/x^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x**3))/x**5,x)

[Out]

Timed out

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Giac [A]  time = 1.71353, size = 217, normalized size = 1.25 \begin{align*} \frac{1}{16} \, b c^{3}{\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^{2}} - \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^{2}} - \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^{2}} - \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^{2}} - \frac{4 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left (x{\left | c \right |}^{\frac{1}{3}}\right )}{c^{2}}\right )} - \frac{3 \, b c x^{3} + b \arctan \left (c x^{3}\right ) + a}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x^5,x, algorithm="giac")

[Out]

1/16*b*c^3*(sqrt(3)*abs(c)^(1/3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - sqrt(3)*abs(c)^(1/3)
*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - 2*abs(c)^(1/3)*arctan((2*x + sqrt(3)/abs(c)^(1/3))*a
bs(c)^(1/3))/c^2 - 2*abs(c)^(1/3)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/c^2 - 4*abs(c)^(1/3)*arcta
n(x*abs(c)^(1/3))/c^2) - 1/4*(3*b*c*x^3 + b*arctan(c*x^3) + a)/x^4