3.108 \(\int x^7 (a+b \tan ^{-1}(c x^3)) \, dx\)

Optimal. Leaf size=176 \[ \frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{\sqrt{3} b \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac{\sqrt{3} b \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}+\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac{b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{16 c^{8/3}}-\frac{3 b x^5}{40 c} \]

[Out]

(-3*b*x^5)/(40*c) + (b*ArcTan[c^(1/3)*x])/(8*c^(8/3)) + (x^8*(a + b*ArcTan[c*x^3]))/8 - (b*ArcTan[Sqrt[3] - 2*
c^(1/3)*x])/(16*c^(8/3)) + (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(16*c^(8/3)) + (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)
*x + c^(2/3)*x^2])/(32*c^(8/3)) - (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3))

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Rubi [A]  time = 0.430877, antiderivative size = 176, 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, 321, 295, 634, 618, 204, 628, 203} \[ \frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{\sqrt{3} b \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac{\sqrt{3} b \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}+\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac{b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{16 c^{8/3}}-\frac{3 b x^5}{40 c} \]

Antiderivative was successfully verified.

[In]

Int[x^7*(a + b*ArcTan[c*x^3]),x]

[Out]

(-3*b*x^5)/(40*c) + (b*ArcTan[c^(1/3)*x])/(8*c^(8/3)) + (x^8*(a + b*ArcTan[c*x^3]))/8 - (b*ArcTan[Sqrt[3] - 2*
c^(1/3)*x])/(16*c^(8/3)) + (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(16*c^(8/3)) + (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)
*x + c^(2/3)*x^2])/(32*c^(8/3)) - (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3))

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 x^7 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{1}{8} (3 b c) \int \frac{x^{10}}{1+c^2 x^6} \, dx\\ &=-\frac{3 b x^5}{40 c}+\frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{(3 b) \int \frac{x^4}{1+c^2 x^6} \, dx}{8 c}\\ &=-\frac{3 b x^5}{40 c}+\frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{b \int \frac{1}{1+c^{2/3} x^2} \, dx}{8 c^{7/3}}+\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}{8 c^{7/3}}+\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}{8 c^{7/3}}\\ &=-\frac{3 b x^5}{40 c}+\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac{1}{8} x^8 \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}{32 c^{8/3}}-\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}{32 c^{8/3}}+\frac{b \int \frac{1}{1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{7/3}}+\frac{b \int \frac{1}{1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{32 c^{7/3}}\\ &=-\frac{3 b x^5}{40 c}+\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac{1}{8} x^8 \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 )}{32 c^{8/3}}-\frac{\sqrt{3} b \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}+\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 )}{16 \sqrt{3} c^{8/3}}-\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 )}{16 \sqrt{3} c^{8/3}}\\ &=-\frac{3 b x^5}{40 c}+\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}+\frac{1}{8} x^8 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac{b \tan ^{-1}\left (\sqrt{3}+2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac{\sqrt{3} b \log \left (1-\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}-\frac{\sqrt{3} b \log \left (1+\sqrt{3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{32 c^{8/3}}\\ \end{align*}

Mathematica [A]  time = 0.0748782, size = 181, normalized size = 1.03 \[ \frac{a x^8}{8}+\frac{\sqrt{3} b \log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}-\frac{\sqrt{3} b \log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )}{32 c^{8/3}}+\frac{b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{8 c^{8/3}}-\frac{b \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )}{16 c^{8/3}}+\frac{b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )}{16 c^{8/3}}-\frac{3 b x^5}{40 c}+\frac{1}{8} b x^8 \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7*(a + b*ArcTan[c*x^3]),x]

[Out]

(-3*b*x^5)/(40*c) + (a*x^8)/8 + (b*ArcTan[c^(1/3)*x])/(8*c^(8/3)) + (b*x^8*ArcTan[c*x^3])/8 - (b*ArcTan[Sqrt[3
] - 2*c^(1/3)*x])/(16*c^(8/3)) + (b*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/(16*c^(8/3)) + (Sqrt[3]*b*Log[1 - Sqrt[3]*c
^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3)) - (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(32*c^(8/3))

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Maple [A]  time = 0.069, size = 170, normalized size = 1. \begin{align*}{\frac{{x}^{8}a}{8}}+{\frac{b{x}^{8}\arctan \left ( c{x}^{3} \right ) }{8}}-{\frac{3\,b{x}^{5}}{40\,c}}-{\frac{b\sqrt{3}}{32\,c} \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}{16\,{c}^{3}}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}+{\frac{b\sqrt{3}}{32\,c} \left ({c}^{-2} \right ) ^{{\frac{5}{6}}}\ln \left ( \sqrt{3}\sqrt [6]{{c}^{-2}}x-{x}^{2}-\sqrt [3]{{c}^{-2}} \right ) }+{\frac{b}{16\,{c}^{3}}\arctan \left ( 2\,{\frac{x}{\sqrt [6]{{c}^{-2}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{-2}}}}}+{\frac{b}{8\,{c}^{3}}\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^7*(a+b*arctan(c*x^3)),x)

[Out]

1/8*x^8*a+1/8*b*x^8*arctan(c*x^3)-3/40*b*x^5/c-1/32*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))+1/16*b/c^3/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))+1/32*b/c*3^(1/2)*(1/c^2)^(5/6)*ln(3^(
1/2)*(1/c^2)^(1/6)*x-x^2-(1/c^2)^(1/3))+1/16*b/c^3/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))+1/8*b/c^3/(
1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6))

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Maxima [B]  time = 1.50256, size = 432, normalized size = 2.45 \begin{align*} \frac{1}{8} \, a x^{8} + \frac{1}{160} \,{\left (20 \, x^{8} \arctan \left (c x^{3}\right ) -{\left (\frac{12 \, x^{5}}{c^{2}} + \frac{5 \,{\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 )}}{c^{2}}\right )} c\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^3)),x, algorithm="maxima")

[Out]

1/8*a*x^8 + 1/160*(20*x^8*arctan(c*x^3) - (12*x^5/c^2 + 5*(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)*l
og((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))))/c^2)*c)*b

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Fricas [B]  time = 3.25014, size = 1069, normalized size = 6.07 \begin{align*} \frac{20 \, b c x^{8} \arctan \left (c x^{3}\right ) + 20 \, a c x^{8} - 12 \, b x^{5} - 5 \, \sqrt{3} c \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} \log \left (\sqrt{3} b^{5} c^{13} x \left (\frac{b^{6}}{c^{16}}\right )^{\frac{5}{6}} + b^{6} c^{10} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{2}{3}} + b^{10} x^{2}\right ) + 5 \, \sqrt{3} c \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} \log \left (-\sqrt{3} b^{5} c^{13} x \left (\frac{b^{6}}{c^{16}}\right )^{\frac{5}{6}} + b^{6} c^{10} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{2}{3}} + b^{10} x^{2}\right ) - 20 \, c \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, b^{5} c^{3} x \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} + \sqrt{3} b^{6} - 2 \, \sqrt{\sqrt{3} b^{5} c^{13} x \left (\frac{b^{6}}{c^{16}}\right )^{\frac{5}{6}} + b^{6} c^{10} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{2}{3}} + b^{10} x^{2}} c^{3} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}}}{b^{6}}\right ) - 20 \, c \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, b^{5} c^{3} x \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} - \sqrt{3} b^{6} - 2 \, \sqrt{-\sqrt{3} b^{5} c^{13} x \left (\frac{b^{6}}{c^{16}}\right )^{\frac{5}{6}} + b^{6} c^{10} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{2}{3}} + b^{10} x^{2}} c^{3} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}}}{b^{6}}\right ) - 40 \, c \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} \arctan \left (-\frac{b^{5} c^{3} x \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}} - \sqrt{b^{6} c^{10} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{2}{3}} + b^{10} x^{2}} c^{3} \left (\frac{b^{6}}{c^{16}}\right )^{\frac{1}{6}}}{b^{6}}\right )}{160 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^3)),x, algorithm="fricas")

[Out]

1/160*(20*b*c*x^8*arctan(c*x^3) + 20*a*c*x^8 - 12*b*x^5 - 5*sqrt(3)*c*(b^6/c^16)^(1/6)*log(sqrt(3)*b^5*c^13*x*
(b^6/c^16)^(5/6) + b^6*c^10*(b^6/c^16)^(2/3) + b^10*x^2) + 5*sqrt(3)*c*(b^6/c^16)^(1/6)*log(-sqrt(3)*b^5*c^13*
x*(b^6/c^16)^(5/6) + b^6*c^10*(b^6/c^16)^(2/3) + b^10*x^2) - 20*c*(b^6/c^16)^(1/6)*arctan(-(2*b^5*c^3*x*(b^6/c
^16)^(1/6) + sqrt(3)*b^6 - 2*sqrt(sqrt(3)*b^5*c^13*x*(b^6/c^16)^(5/6) + b^6*c^10*(b^6/c^16)^(2/3) + b^10*x^2)*
c^3*(b^6/c^16)^(1/6))/b^6) - 20*c*(b^6/c^16)^(1/6)*arctan(-(2*b^5*c^3*x*(b^6/c^16)^(1/6) - sqrt(3)*b^6 - 2*sqr
t(-sqrt(3)*b^5*c^13*x*(b^6/c^16)^(5/6) + b^6*c^10*(b^6/c^16)^(2/3) + b^10*x^2)*c^3*(b^6/c^16)^(1/6))/b^6) - 40
*c*(b^6/c^16)^(1/6)*arctan(-(b^5*c^3*x*(b^6/c^16)^(1/6) - sqrt(b^6*c^10*(b^6/c^16)^(2/3) + b^10*x^2)*c^3*(b^6/
c^16)^(1/6))/b^6))/c

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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(x**7*(a+b*atan(c*x**3)),x)

[Out]

Timed out

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Giac [A]  time = 1.39375, size = 231, normalized size = 1.31 \begin{align*} -\frac{1}{32} \, b c^{15}{\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^{18}} - \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^{18}} - \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^{18}} - \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^{18}} - \frac{4 \,{\left | c \right |}^{\frac{1}{3}} \arctan \left (x{\left | c \right |}^{\frac{1}{3}}\right )}{c^{18}}\right )} + \frac{5 \, b c x^{8} \arctan \left (c x^{3}\right ) + 5 \, a c x^{8} - 3 \, b x^{5}}{40 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arctan(c*x^3)),x, algorithm="giac")

[Out]

-1/32*b*c^15*(sqrt(3)*abs(c)^(1/3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^18 - sqrt(3)*abs(c)^(1
/3)*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^18 - 2*abs(c)^(1/3)*arctan((2*x + sqrt(3)/abs(c)^(1/3
))*abs(c)^(1/3))/c^18 - 2*abs(c)^(1/3)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/c^18 - 4*abs(c)^(1/3)
*arctan(x*abs(c)^(1/3))/c^18) + 1/40*(5*b*c*x^8*arctan(c*x^3) + 5*a*c*x^8 - 3*b*x^5)/c