3.2.15 \(\int x^5 (a+b \text {ArcTan}(c x^3))^2 \, dx\) [115]

Optimal. Leaf size=90 \[ -\frac {a b x^3}{3 c}-\frac {b^2 x^3 \text {ArcTan}\left (c x^3\right )}{3 c}+\frac {\left (a+b \text {ArcTan}\left (c x^3\right )\right )^2}{6 c^2}+\frac {1}{6} x^6 \left (a+b \text {ArcTan}\left (c x^3\right )\right )^2+\frac {b^2 \log \left (1+c^2 x^6\right )}{6 c^2} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4948, 4946, 5036, 4930, 266, 5004} \begin {gather*} \frac {\left (a+b \text {ArcTan}\left (c x^3\right )\right )^2}{6 c^2}+\frac {1}{6} x^6 \left (a+b \text {ArcTan}\left (c x^3\right )\right )^2-\frac {a b x^3}{3 c}-\frac {b^2 x^3 \text {ArcTan}\left (c x^3\right )}{3 c}+\frac {b^2 \log \left (c^2 x^6+1\right )}{6 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcTan[c*x^3])^2,x]

[Out]

-1/3*(a*b*x^3)/c - (b^2*x^3*ArcTan[c*x^3])/(3*c) + (a + b*ArcTan[c*x^3])^2/(6*c^2) + (x^6*(a + b*ArcTan[c*x^3]
)^2)/6 + (b^2*Log[1 + c^2*x^6])/(6*c^2)

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 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[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 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[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 4948

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

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[d*(f^2/e), Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/
(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rubi steps

\begin {align*} \int x^5 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^5 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2+\frac {1}{2} b x^5 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac {1}{4} b^2 x^5 \log ^2\left (1+i c x^3\right )\right ) \, dx\\ &=\frac {1}{4} \int x^5 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2 \, dx+\frac {1}{2} b \int x^5 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right ) \, dx-\frac {1}{4} b^2 \int x^5 \log ^2\left (1+i c x^3\right ) \, dx\\ &=\frac {1}{12} \text {Subst}\left (\int x (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )+\frac {1}{6} b \text {Subst}\left (\int x (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^3\right )-\frac {1}{12} b^2 \text {Subst}\left (\int x \log ^2(1+i c x) \, dx,x,x^3\right )\\ &=-\frac {1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac {1}{12} \text {Subst}\left (\int \left (-\frac {i (2 a+i b \log (1-i c x))^2}{c}+\frac {i (1-i c x) (2 a+i b \log (1-i c x))^2}{c}\right ) \, dx,x,x^3\right )-\frac {1}{12} b^2 \text {Subst}\left (\int \left (\frac {i \log ^2(1+i c x)}{c}-\frac {i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,x^3\right )-\frac {1}{12} (i b c) \text {Subst}\left (\int \frac {x^2 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^3\right )+\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {x^2 \log (1+i c x)}{1-i c x} \, dx,x,x^3\right )\\ &=-\frac {1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac {i \text {Subst}\left (\int (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )}{12 c}+\frac {i \text {Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^2 \, dx,x,x^3\right )}{12 c}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^3\right )}{12 c}+\frac {\left (i b^2\right ) \text {Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,x^3\right )}{12 c}-\frac {1}{12} (i b c) \text {Subst}\left (\int \left (\frac {-2 i a+b \log (1-i c x)}{c^2}-\frac {i x (-2 i a+b \log (1-i c x))}{c}+\frac {i (-2 i a+b \log (1-i c x))}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )+\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log (1+i c x)}{c^2}+\frac {i x \log (1+i c x)}{c}-\frac {i \log (1+i c x)}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac {1}{12} b \text {Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )-\frac {1}{12} b^2 \text {Subst}\left (\int x \log (1+i c x) \, dx,x,x^3\right )+\frac {\text {Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-i c x^3\right )}{12 c^2}-\frac {\text {Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-i c x^3\right )}{12 c^2}-\frac {b^2 \text {Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^3\right )}{12 c^2}-\frac {(i b) \text {Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^3\right )}{12 c}+\frac {b \text {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^3\right )}{12 c}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \log (1+i c x) \, dx,x,x^3\right )}{12 c}+\frac {b^2 \text {Subst}\left (\int \frac {\log (1+i c x)}{i+c x} \, dx,x,x^3\right )}{12 c}\\ &=-\frac {a b x^3}{6 c}+\frac {1}{24} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac {\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 c^2}-\frac {\left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 c^2}-\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^3\right )\right )}{12 c^2}-\frac {1}{24} b^2 x^6 \log \left (1+i c x^3\right )+\frac {b^2 \log \left (\frac {1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 c^2}-\frac {1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac {b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^2}+\frac {b^2 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{24 c^2}+\frac {(i b) \text {Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-i c x^3\right )}{12 c^2}-\frac {(i b) \text {Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-i c x^3\right )}{6 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{12 c^2}-\frac {b^2 \text {Subst}\left (\int x \log (x) \, dx,x,1+i c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1+i c x^3\right )}{6 c^2}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \log (1-i c x) \, dx,x,x^3\right )}{12 c}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^3\right )}{12 c}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^3\right )}{12 c}-\frac {1}{24} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1-i c x} \, dx,x,x^3\right )+\frac {1}{24} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {x^2}{1+i c x} \, dx,x,x^3\right )\\ &=-\frac {a b x^3}{2 c}-\frac {i b^2 x^3}{4 c}+\frac {b^2 \left (1-i c x^3\right )^2}{48 c^2}+\frac {b^2 \left (1+i c x^3\right )^2}{48 c^2}+\frac {1}{24} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac {i b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{24 c^2}+\frac {\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 c^2}-\frac {\left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 c^2}-\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^3\right )\right )}{12 c^2}-\frac {1}{24} b^2 x^6 \log \left (1+i c x^3\right )+\frac {b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{4 c^2}-\frac {b^2 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{24 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 c^2}-\frac {1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac {b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^2}+\frac {b^2 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{24 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-i c x^3\right )}{12 c^2}-\frac {b^2 \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+i c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{12 c^2}+\frac {b^2 \text {Subst}\left (\int \log (x) \, dx,x,1-i c x^3\right )}{6 c^2}+\frac {1}{24} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {i x}{c}+\frac {i}{c^2 (-i+c x)}\right ) \, dx,x,x^3\right )-\frac {1}{24} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}+\frac {i x}{c}-\frac {i}{c^2 (i+c x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {a b x^3}{2 c}+\frac {b^2 x^6}{24}+\frac {b^2 \left (1-i c x^3\right )^2}{48 c^2}+\frac {b^2 \left (1+i c x^3\right )^2}{48 c^2}-\frac {b^2 \log \left (i-c x^3\right )}{24 c^2}+\frac {b^2 \left (1-i c x^3\right ) \log \left (1-i c x^3\right )}{4 c^2}+\frac {1}{24} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right )+\frac {i b \left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )}{24 c^2}+\frac {\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{12 c^2}-\frac {\left (1-i c x^3\right )^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 c^2}-\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^3\right )\right )}{12 c^2}-\frac {1}{24} b^2 x^6 \log \left (1+i c x^3\right )+\frac {b^2 \left (1+i c x^3\right ) \log \left (1+i c x^3\right )}{4 c^2}-\frac {b^2 \left (1+i c x^3\right )^2 \log \left (1+i c x^3\right )}{24 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 c^2}-\frac {1}{12} b x^6 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )-\frac {b^2 \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{12 c^2}+\frac {b^2 \left (1+i c x^3\right )^2 \log ^2\left (1+i c x^3\right )}{24 c^2}-\frac {b^2 \log \left (i+c x^3\right )}{24 c^2}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1-i c x^3\right )\right )}{12 c^2}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1+i c x^3\right )\right )}{12 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 85, normalized size = 0.94 \begin {gather*} \frac {a c x^3 \left (-2 b+a c x^3\right )+2 b \left (a-b c x^3+a c^2 x^6\right ) \text {ArcTan}\left (c x^3\right )+b^2 \left (1+c^2 x^6\right ) \text {ArcTan}\left (c x^3\right )^2+b^2 \log \left (1+c^2 x^6\right )}{6 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTan[c*x^3])^2,x]

[Out]

(a*c*x^3*(-2*b + a*c*x^3) + 2*b*(a - b*c*x^3 + a*c^2*x^6)*ArcTan[c*x^3] + b^2*(1 + c^2*x^6)*ArcTan[c*x^3]^2 +
b^2*Log[1 + c^2*x^6])/(6*c^2)

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Maple [A]
time = 0.14, size = 113, normalized size = 1.26

method	result	size
default	\(\frac {x^{6} a^{2}}{6}+\frac {b^{2} x^{6} \arctan \left (c \,x^{3}\right )^{2}}{6}-\frac {b^{2} x^{3} \arctan \left (c \,x^{3}\right )}{3 c}+\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{6 c^{2}}+\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{6 c^{2}}+\frac {a b \,x^{6} \arctan \left (c \,x^{3}\right )}{3}-\frac {a b \,x^{3}}{3 c}+\frac {a b \arctan \left (c \,x^{3}\right )}{3 c^{2}}\)	\(113\)
risch	\(-\frac {b^{2} \left (c^{2} x^{6}+1\right ) \ln \left (i c \,x^{3}+1\right )^{2}}{24 c^{2}}-\frac {i b \left (4 a^{2} c^{2} x^{6}+2 i x^{6} b \ln \left (-i c \,x^{3}+1\right ) a \,c^{2}-4 a b c \,x^{3}+b^{2}+2 i b \ln \left (-i c \,x^{3}+1\right ) a \right ) \ln \left (i c \,x^{3}+1\right )}{24 c^{2} a}+\frac {i b^{3} \ln \left (c^{2} x^{6}+1\right )}{48 c^{2} a}-\frac {b^{2} x^{6} \ln \left (-i c \,x^{3}+1\right )^{2}}{24}+\frac {x^{6} a^{2}}{6}+\frac {i a b \,x^{6} \ln \left (-i c \,x^{3}+1\right )}{6}-\frac {a b \,x^{3}}{3 c}+\frac {a b \arctan \left (c \,x^{3}\right )}{3 c^{2}}-\frac {i b^{2} x^{3} \ln \left (-i c \,x^{3}+1\right )}{6 c}-\frac {b^{3} \arctan \left (c \,x^{3}\right )}{24 c^{2} a}-\frac {b^{2} \ln \left (-i c \,x^{3}+1\right )^{2}}{24 c^{2}}+\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{6 c^{2}}+\frac {b^{2}}{6 c^{2}}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctan(c*x^3))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.54, size = 126, normalized size = 1.40 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} \arctan \left (c x^{3}\right )^{2} + \frac {1}{6} \, a^{2} x^{6} + \frac {1}{3} \, {\left (x^{6} \arctan \left (c x^{3}\right ) - c {\left (\frac {x^{3}}{c^{2}} - \frac {\arctan \left (c x^{3}\right )}{c^{3}}\right )}\right )} a b - \frac {1}{6} \, {\left (2 \, c {\left (\frac {x^{3}}{c^{2}} - \frac {\arctan \left (c x^{3}\right )}{c^{3}}\right )} \arctan \left (c x^{3}\right ) + \frac {\arctan \left (c x^{3}\right )^{2} - \log \left (6 \, c^{5} x^{6} + 6 \, c^{3}\right )}{c^{2}}\right )} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.03, size = 91, normalized size = 1.01 \begin {gather*} \frac {a^{2} c^{2} x^{6} - 2 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} + b^{2}\right )} \arctan \left (c x^{3}\right )^{2} + b^{2} \log \left (c^{2} x^{6} + 1\right ) + 2 \, {\left (a b c^{2} x^{6} - b^{2} c x^{3} + a b\right )} \arctan \left (c x^{3}\right )}{6 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3))^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (78) = 156\).
time = 50.50, size = 194, normalized size = 2.16 \begin {gather*} \begin {cases} \frac {a^{2} x^{6}}{6} + \frac {a b x^{6} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {a b x^{3}}{3 c} + \frac {a b \operatorname {atan}{\left (c x^{3} \right )}}{3 c^{2}} + \frac {b^{2} x^{6} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6} - \frac {b^{2} x^{3} \operatorname {atan}{\left (c x^{3} \right )}}{3 c} - \frac {b^{2} \sqrt {- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{3 c} + \frac {b^{2} \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{3 c^{2}} + \frac {b^{2} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{3 c^{2}} + \frac {b^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{6}}{6} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*atan(c*x**3))**2,x)

[Out]

Piecewise((a**2*x**6/6 + a*b*x**6*atan(c*x**3)/3 - a*b*x**3/(3*c) + a*b*atan(c*x**3)/(3*c**2) + b**2*x**6*atan
(c*x**3)**2/6 - b**2*x**3*atan(c*x**3)/(3*c) - b**2*sqrt(-1/c**2)*atan(c*x**3)/(3*c) + b**2*log(x - (-1/c**2)*
*(1/6))/(3*c**2) + b**2*log(4*x**2 + 4*x*(-1/c**2)**(1/6) + 4*(-1/c**2)**(1/3))/(3*c**2) + b**2*atan(c*x**3)**
2/(6*c**2), Ne(c, 0)), (a**2*x**6/6, True))

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Giac [A]
time = 0.42, size = 100, normalized size = 1.11 \begin {gather*} \frac {a^{2} c x^{6} + \frac {2 \, {\left (c^{2} x^{6} \arctan \left (c x^{3}\right ) - c x^{3} + \arctan \left (c x^{3}\right )\right )} a b}{c} + \frac {{\left (c^{2} x^{6} \arctan \left (c x^{3}\right )^{2} - 2 \, c x^{3} \arctan \left (c x^{3}\right ) + \arctan \left (c x^{3}\right )^{2} + \log \left (c^{2} x^{6} + 1\right )\right )} b^{2}}{c}}{6 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x^3))^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.72, size = 112, normalized size = 1.24 \begin {gather*} \frac {a^2\,x^6}{6}+\frac {b^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6\,c^2}+\frac {b^2\,x^6\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6}+\frac {b^2\,\ln \left (c^2\,x^6+1\right )}{6\,c^2}-\frac {b^2\,x^3\,\mathrm {atan}\left (c\,x^3\right )}{3\,c}-\frac {a\,b\,x^3}{3\,c}+\frac {a\,b\,\mathrm {atan}\left (c\,x^3\right )}{3\,c^2}+\frac {a\,b\,x^6\,\mathrm {atan}\left (c\,x^3\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*atan(c*x^3))^2,x)

[Out]

(a^2*x^6)/6 + (b^2*atan(c*x^3)^2)/(6*c^2) + (b^2*x^6*atan(c*x^3)^2)/6 + (b^2*log(c^2*x^6 + 1))/(6*c^2) - (b^2*
x^3*atan(c*x^3))/(3*c) - (a*b*x^3)/(3*c) + (a*b*atan(c*x^3))/(3*c^2) + (a*b*x^6*atan(c*x^3))/3

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