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

Optimal. Leaf size=101 \[ a x+\frac{b \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac{b \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}+b x \tan ^{-1}\left (c x^3\right ) \]

[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))

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Rubi [A]  time = 0.0980866, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5027, 275, 292, 31, 634, 617, 204, 628} \[ a x+\frac{b \log \left (c^{2/3} x^2+1\right )}{2 \sqrt [3]{c}}-\frac{b \log \left (c^{4/3} x^4-c^{2/3} x^2+1\right )}{4 \sqrt [3]{c}}+\frac{\sqrt{3} b \tan ^{-1}\left (\frac{1-2 c^{2/3} x^2}{\sqrt{3}}\right )}{2 \sqrt [3]{c}}+b x \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[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 5027

Int[ArcTan[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTan[c*x^n], x] - Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x],
x] /; FreeQ[{c, n}, x]

Rule 275

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 292

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 31

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

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 617

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 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]

Rubi steps

\begin{align*} \int \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=a x+b \int \tan ^{-1}\left (c x^3\right ) \, dx\\ &=a x+b x \tan ^{-1}\left (c x^3\right )-(3 b c) \int \frac{x^3}{1+c^2 x^6} \, dx\\ &=a x+b x \tan ^{-1}\left (c x^3\right )-\frac{1}{2} (3 b c) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x^3} \, dx,x,x^2\right )\\ &=a x+b x \tan ^{-1}\left (c x^3\right )+\frac{1}{2} \left (b \sqrt [3]{c}\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^{2/3} x} \, dx,x,x^2\right )-\frac{1}{2} \left (b \sqrt [3]{c}\right ) \operatorname{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 \tan ^{-1}\left (c x^3\right )+\frac{b \log \left (1+c^{2/3} x^2\right )}{2 \sqrt [3]{c}}-\frac{b \operatorname{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 ) \operatorname{Subst}\left (\int \frac{1}{1-c^{2/3} x+c^{4/3} x^2} \, dx,x,x^2\right )\\ &=a x+b x \tan ^{-1}\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) \operatorname{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 \tan ^{-1}\left (c x^3\right )+\frac{\sqrt{3} b \tan ^{-1}\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]  time = 0.0404928, size = 131, normalized size = 1.3 \[ a x-\frac{b \left (-2 \log \left (c^{2/3} x^2+1\right )+\log \left (c^{2/3} x^2-\sqrt{3} \sqrt [3]{c} x+1\right )+\log \left (c^{2/3} x^2+\sqrt{3} \sqrt [3]{c} x+1\right )-2 \sqrt{3} \tan ^{-1}\left (\sqrt{3}-2 \sqrt [3]{c} x\right )-2 \sqrt{3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt{3}\right )\right )}{4 \sqrt [3]{c}}+b x \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.023, size = 98, normalized size = 1. \begin{align*} ax+bx\arctan \left ( c{x}^{3} \right ) +{\frac{b}{2\,c}\ln \left ({x}^{2}+\sqrt [3]{{c}^{-2}} \right ){\frac{1}{\sqrt [3]{{c}^{-2}}}}}-{\frac{b}{4\,c}\ln \left ({x}^{4}-\sqrt [3]{{c}^{-2}}{x}^{2}+ \left ({c}^{-2} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{c}^{-2}}}}}-{\frac{b\sqrt{3}}{2\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{{x}^{2}}{\sqrt [3]{{c}^{-2}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{c}^{-2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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/(1/c^2)^(1/3)*x^2-1))

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Maxima [A]  time = 1.49389, size = 143, normalized size = 1.42 \begin{align*} -\frac{1}{4} \,{\left (c{\left (\frac{2 \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}}{\left (2 \, x^{2} - \frac{1}{c^{2}}^{\frac{1}{3}}\right )}\right )}{c^{2}} + \frac{{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (x^{4} - \frac{1}{c^{2}}^{\frac{1}{3}} x^{2} + \frac{1}{c^{2}}^{\frac{2}{3}}\right )}{c^{2}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{1}{3}} \log \left (x^{2} + \frac{1}{c^{2}}^{\frac{1}{3}}\right )}{c^{2}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.30625, size = 632, normalized size = 6.26 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

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Sympy [A]  time = 55.8924, size = 1703, normalized size = 16.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*Piecewise((-I*x*atanh(x**3*sqrt(x**(-6))), Eq(c, -sqrt(-1/x**6)) | Eq(c, -sqrt(1/(x**6*(1/2 - sqrt(3)*
I/2)**3))) | Eq(c, -sqrt(1/(x**6*(1/2 + sqrt(3)*I/2)**3)))), (0, Eq(c, 0)), (2*(-1)**(1/3)*sqrt(3)*c**39*x**6*
(c**(-2))**(73/3)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) - sqrt(3)/3)/(-4*(-1)**(2/3)*c**38*x**6*(c
**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) - 2*(-1)**(1/3)*sqrt(3)*c**39*x**6*(c**(-2))**(73/3)*
atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) + sqrt(3)/3)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) -
4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) - 3*(-1)**(1/3)*c**37*x**6*(c**(-2))**(70/3)*log(4*x**2 - 4*(-1)**(1/6)
*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(
2/3)*c**36*(c**(-2))**(71/3)) + 6*(-1)**(1/3)*c**37*x**6*(c**(-2))**(70/3)*log(2)/(-4*(-1)**(2/3)*c**38*x**6*(
c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) + 2*(-1)**(1/3)*sqrt(3)*c**37*(c**(-2))**(73/3)*atan
(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) - sqrt(3)/3)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-
1)**(2/3)*c**36*(c**(-2))**(71/3)) - 2*(-1)**(1/3)*sqrt(3)*c**37*(c**(-2))**(73/3)*atan(2*(-1)**(5/6)*sqrt(3)*
x/(3*(c**(-2))**(1/6)) + sqrt(3)/3)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2
))**(71/3)) - 3*(-1)**(1/3)*c**35*(c**(-2))**(70/3)*log(4*x**2 - 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1
/3)*(c**(-2))**(1/3))/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) +
6*(-1)**(1/3)*c**35*(c**(-2))**(70/3)*log(2)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**3
6*(c**(-2))**(71/3)) + 4*(-1)**(1/3)*c**33*x**6*(c**(-2))**(64/3)*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(-4*(-
1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) + (-1)**(1/3)*c**31*x**6*(c**(
-2))**(61/3)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(-4*(-1)**(2/3)*c
**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) - 2*(-1)**(1/3)*c**31*x**6*(c**(-2))**(61
/3)*log(2)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) + 4*(-1)**(1/
3)*c**31*(c**(-2))**(64/3)*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3)
- 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) + (-1)**(1/3)*c**29*(c**(-2))**(61/3)*log(4*x**2 + 4*(-1)**(1/6)*x*(c
**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*
c**36*(c**(-2))**(71/3)) - 2*(-1)**(1/3)*c**29*(c**(-2))**(61/3)*log(2)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**
(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) - 4*(-1)**(2/3)*c**26*x**7*(c**(-2))**(53/3)*atan(c*x**3)/(-4*
(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)) - 4*(-1)**(2/3)*c**24*x*(c**
(-2))**(53/3)*atan(c*x**3)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3
)) - 4*(-1)**(5/6)*c**14*(c**(-2))**(77/6)*atan(c*x**3)/(-4*(-1)**(2/3)*c**38*x**6*(c**(-2))**(71/3) - 4*(-1)*
*(2/3)*c**36*(c**(-2))**(71/3)) - 4*(-1)**(5/6)*x**6*(c**(-2))**(29/6)*atan(c*x**3)/(-4*(-1)**(2/3)*c**38*x**6
*(c**(-2))**(71/3) - 4*(-1)**(2/3)*c**36*(c**(-2))**(71/3)), True))

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Giac [A]  time = 1.1522, size = 128, normalized size = 1.27 \begin{align*} -\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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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