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

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

[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)

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Rubi [A]  time = 0.10, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, 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 31

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

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 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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.05, size = 131, normalized size = 1.30 \[ a x-\frac {b \left (-2 \log \left (c^{2/3} x^2+1\right )+\log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )+\log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )-2 \sqrt {3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )-2 \sqrt {3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )\right )}{4 \sqrt [3]{c}}+b x \tan ^{-1}\left (c x^3\right ) \]

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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fricas [A]  time = 0.46, size = 234, normalized size = 2.32 \[ \left [\frac {4 \, b c x \arctan \left (c x^{3}\right ) + \sqrt {3} b c \sqrt {-\frac {1}{c^{\frac {2}{3}}}} \log \left (\frac {2 \, c^{2} x^{6} - 3 \, c^{\frac {2}{3}} x^{2} - \sqrt {3} {\left (2 \, c^{\frac {5}{3}} x^{4} + c x^{2} - c^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{c^{\frac {2}{3}}}} - 1}{c^{2} x^{6} + 1}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} - c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {1}{3}}\right )}{4 \, c}, \frac {4 \, b c x \arctan \left (c x^{3}\right ) + 2 \, \sqrt {3} b c^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} {\left (2 \, c x^{2} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 4 \, a c x - b c^{\frac {2}{3}} \log \left (c^{2} x^{4} - c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right ) + 2 \, b c^{\frac {2}{3}} \log \left (c x^{2} + c^{\frac {1}{3}}\right )}{4 \, c}\right ] \]

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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giac [A]  time = 0.16, size = 95, normalized size = 0.94 \[ -\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{2}} + \frac {{\left | c \right |}^{\frac {2}{3}} \log \left (x^{4} - \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{2}} - \frac {2 \, \log \left (x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b + a x \]

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

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maple [A]  time = 0.03, size = 98, normalized size = 0.97 \[ a x +b x \arctan \left (c \,x^{3}\right )+\frac {b \ln \left (x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{4}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}} \]

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 = 0.42, size = 92, normalized size = 0.91 \[ -\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} - c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} + \frac {\log \left (c^{\frac {4}{3}} x^{4} - c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {4}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} + 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}}\right )} - 4 \, x \arctan \left (c x^{3}\right )\right )} b + a x \]

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)*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

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mupad [B]  time = 2.30, size = 91, normalized size = 0.90 \[ a\,x+b\,x\,\mathrm {atan}\left (c\,x^3\right )+\frac {b\,\ln \left (c^{2/3}\,x^2+1\right )}{2\,c^{1/3}}-\frac {\ln \left (2-4\,c^{2/3}\,x^2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}}-\frac {\ln \left (4\,c^{2/3}\,x^2-2+\sqrt {3}\,2{}\mathrm {i}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{4\,c^{1/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 31.24, size = 898, normalized size = 8.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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*(-1)**(2/3)*c**4*x**6*
(c**(-2))**(5/3)*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(4*c*x**6 + 4/c) - 3*(-1)**(2/3)*c**4*x**6*(c**(-2))**(
5/3)*log(4*x**2 - 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(4*c*x**6 + 4/c) + (-1)**
(2/3)*c**4*x**6*(c**(-2))**(5/3)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3
))/(4*c*x**6 + 4/c) + 2*(-1)**(2/3)*sqrt(3)*c**4*x**6*(c**(-2))**(5/3)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2
))**(1/6)) - sqrt(3)/3)/(4*c*x**6 + 4/c) - 2*(-1)**(2/3)*sqrt(3)*c**4*x**6*(c**(-2))**(5/3)*atan(2*(-1)**(5/6)
*sqrt(3)*x/(3*(c**(-2))**(1/6)) + sqrt(3)/3)/(4*c*x**6 + 4/c) + 4*(-1)**(2/3)*c**4*x**6*(c**(-2))**(5/3)*log(2
)/(4*c*x**6 + 4/c) + 4*(-1)**(1/6)*c**3*x**6*(c**(-2))**(7/6)*atan(c*x**3)/(4*c*x**6 + 4/c) + 4*(-1)**(2/3)*c*
*2*(c**(-2))**(5/3)*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(4*c*x**6 + 4/c) - 3*(-1)**(2/3)*c**2*(c**(-2))**(5/
3)*log(4*x**2 - 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(4*c*x**6 + 4/c) + (-1)**(2
/3)*c**2*(c**(-2))**(5/3)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/(4*c
*x**6 + 4/c) + 2*(-1)**(2/3)*sqrt(3)*c**2*(c**(-2))**(5/3)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) -
sqrt(3)/3)/(4*c*x**6 + 4/c) - 2*(-1)**(2/3)*sqrt(3)*c**2*(c**(-2))**(5/3)*atan(2*(-1)**(5/6)*sqrt(3)*x/(3*(c*
*(-2))**(1/6)) + sqrt(3)/3)/(4*c*x**6 + 4/c) + 4*(-1)**(2/3)*c**2*(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*(-1)**(1/6)*c*(c**(-2))**(7/6)*atan(c*x**3)/(4*c*x**6 + 4/c) + 4*x*a
tan(c*x**3)/(4*c**2*x**6 + 4), True))

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