\(\int (a+b \arctan (c x^3)) \, dx\) [105]

Optimal result

Integrand size = 10, antiderivative size = 101 \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a x+b x \arctan \left (c x^3\right )+\frac {\sqrt {3} b \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+\frac {b \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (1-c^{2/3} x^2+c^{4/3} x^4\right )}{4 \sqrt [3]{c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4930, 281, 298, 31, 648, 631, 210, 642} \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a x+\frac {\sqrt {3} b \arctan \left (\frac {1-2 c^{2/3} x^2}{\sqrt {3}}\right )}{2 \sqrt [3]{c}}+b x \arctan \left (c x^3\right )+\frac {b \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac {b \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}} \]

[In]

Int[a + b*ArcTan[c*x^3],x]

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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 648

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 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])

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.30 \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a x+b x \arctan \left (c x^3\right )-\frac {b \left (-2 \sqrt {3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )-2 \sqrt {3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )-2 \log \left (1+c^{2/3} x^2\right )+\log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )\right )}{4 \sqrt [3]{c}} \]

[In]

Integrate[a + b*ArcTan[c*x^3],x]

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.97

method	result	size
default	\(a x +b x \arctan \left (c \,x^{3}\right )+\frac {b \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{4}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}\)	\(98\)
parts	\(a x +b x \arctan \left (c \,x^{3}\right )+\frac {b \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{4}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}\)	\(98\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.32 \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\left [\frac {4 \, b c x \arctan \left (c x^{3}\right ) + \sqrt {3} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (\frac {2 \, c^{2} x^{6} - 3 \, c^{\frac {2}{3}} x^{2} - \sqrt {3} {\left (2 \, c^{\frac {5}{3}} x^{4} + c x^{2} - c^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 1}{c^{2} x^{6} + 1}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} - c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {1}{3}}\right )}{4 \, c}, \frac {4 \, b c x \arctan \left (c x^{3}\right ) + 2 \, \sqrt {3} b c^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} {\left (2 \, c x^{2} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} - c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {1}{3}}\right )}{4 \, c}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 13.07 (sec) , antiderivative size = 755, normalized size of antiderivative = 7.48 \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a x + b \left (\begin {cases} 0 & \text {for}\: c = 0 \\- \infty i x & \text {for}\: c = - \frac {i}{x^{3}} \\\infty i x & \text {for}\: c = \frac {i}{x^{3}} \\- \frac {4 c^{4} x^{6} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{4 c x^{6} + \frac {4}{c}} + \frac {3 c^{4} x^{6} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {c^{4} x^{6} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {2 \sqrt {3} c^{4} x^{6} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{4 c x^{6} + \frac {4}{c}} + \frac {2 \sqrt {3} c^{4} x^{6} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {4 c^{4} x^{6} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (2 \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {4 c^{3} x^{6} \left (- \frac {1}{c^{2}}\right )^{\frac {7}{6}} \operatorname {atan}{\left (c x^{3} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {4 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{4 c x^{6} + \frac {4}{c}} + \frac {3 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {2 \sqrt {3} c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{4 c x^{6} + \frac {4}{c}} + \frac {2 \sqrt {3} c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {4 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {5}{3}} \log {\left (2 \right )}}{4 c x^{6} + \frac {4}{c}} + \frac {4 c x^{7} \operatorname {atan}{\left (c x^{3} \right )}}{4 c x^{6} + \frac {4}{c}} - \frac {4 c \left (- \frac {1}{c^{2}}\right )^{\frac {7}{6}} \operatorname {atan}{\left (c x^{3} \right )}}{4 c x^{6} + \frac {4}{c}} + \frac {4 x \operatorname {atan}{\left (c x^{3} \right )}}{4 c^{2} x^{6} + 4} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

a*x + b*Piecewise((0, Eq(c, 0)), (-oo*I*x, Eq(c, -I/x**3)), (oo*I*x, Eq(c, I/x**3)), (-4*c**4*x**6*(-1/c**2)**
(5/3)*log(x - (-1/c**2)**(1/6))/(4*c*x**6 + 4/c) + 3*c**4*x**6*(-1/c**2)**(5/3)*log(4*x**2 - 4*x*(-1/c**2)**(1
/6) + 4*(-1/c**2)**(1/3))/(4*c*x**6 + 4/c) - c**4*x**6*(-1/c**2)**(5/3)*log(4*x**2 + 4*x*(-1/c**2)**(1/6) + 4*
(-1/c**2)**(1/3))/(4*c*x**6 + 4/c) - 2*sqrt(3)*c**4*x**6*(-1/c**2)**(5/3)*atan(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)
) - sqrt(3)/3)/(4*c*x**6 + 4/c) + 2*sqrt(3)*c**4*x**6*(-1/c**2)**(5/3)*atan(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)) +
sqrt(3)/3)/(4*c*x**6 + 4/c) - 4*c**4*x**6*(-1/c**2)**(5/3)*log(2)/(4*c*x**6 + 4/c) - 4*c**3*x**6*(-1/c**2)**(
7/6)*atan(c*x**3)/(4*c*x**6 + 4/c) - 4*c**2*(-1/c**2)**(5/3)*log(x - (-1/c**2)**(1/6))/(4*c*x**6 + 4/c) + 3*c*
*2*(-1/c**2)**(5/3)*log(4*x**2 - 4*x*(-1/c**2)**(1/6) + 4*(-1/c**2)**(1/3))/(4*c*x**6 + 4/c) - c**2*(-1/c**2)*
*(5/3)*log(4*x**2 + 4*x*(-1/c**2)**(1/6) + 4*(-1/c**2)**(1/3))/(4*c*x**6 + 4/c) - 2*sqrt(3)*c**2*(-1/c**2)**(5
/3)*atan(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)) - sqrt(3)/3)/(4*c*x**6 + 4/c) + 2*sqrt(3)*c**2*(-1/c**2)**(5/3)*atan
(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)) + sqrt(3)/3)/(4*c*x**6 + 4/c) - 4*c**2*(-1/c**2)**(5/3)*log(2)/(4*c*x**6 + 4
/c) + 4*c*x**7*atan(c*x**3)/(4*c*x**6 + 4/c) - 4*c*(-1/c**2)**(7/6)*atan(c*x**3)/(4*c*x**6 + 4/c) + 4*x*atan(c
*x**3)/(4*c**2*x**6 + 4), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b + a x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.94 \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{2}} + \frac {{\left | c \right |}^{\frac {2}{3}} \log \left (x^{4} - \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{2}} - \frac {2 \, \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b + a x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.43 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.90 \[ \int \left (a+b \arctan \left (c x^3\right )\right ) \, dx=a\,x+b\,x\,\mathrm {atan}\left (c\,x^3\right )+\frac {b\,\ln \left (c^{2/3}\,x^2+1\right )}{2\,c^{1/3}}-\frac {\ln \left (2-4\,c^{2/3}\,x^2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}}-\frac {\ln \left (4\,c^{2/3}\,x^2-2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}} \]

[In]

int(a + b*atan(c*x^3),x)

[Out]

a*x + b*x*atan(c*x^3) + (b*log(c^(2/3)*x^2 + 1))/(2*c^(1/3)) - (log(3^(1/2)*2i - 4*c^(2/3)*x^2 + 2)*(b - 3^(1/
2)*b*1i))/(4*c^(1/3)) - (log(3^(1/2)*2i + 4*c^(2/3)*x^2 - 2)*(b + 3^(1/2)*b*1i))/(4*c^(1/3))