3.99 \(\int x^2 (a+b \tan ^{-1}(c x^3)) \, dx\)

Optimal. Leaf size=36 \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \log \left (c^2 x^6+1\right )}{6 c} \]

[Out]

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

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Rubi [A]  time = 0.0206472, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5033, 260} \[ \frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \log \left (c^2 x^6+1\right )}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTan[c*x^3]),x]

[Out]

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

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 260

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]

Rubi steps

\begin{align*} \int x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-(b c) \int \frac{x^5}{1+c^2 x^6} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{b \log \left (1+c^2 x^6\right )}{6 c}\\ \end{align*}

Mathematica [A]  time = 0.007222, size = 41, normalized size = 1.14 \[ \frac{a x^3}{3}-\frac{b \log \left (c^2 x^6+1\right )}{6 c}+\frac{1}{3} b x^3 \tan ^{-1}\left (c x^3\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.021, size = 36, normalized size = 1. \begin{align*}{\frac{{x}^{3}a}{3}}+{\frac{b{x}^{3}\arctan \left ( c{x}^{3} \right ) }{3}}-{\frac{b\ln \left ({c}^{2}{x}^{6}+1 \right ) }{6\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctan(c*x^3)),x)

[Out]

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

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Maxima [A]  time = 1.00259, size = 51, normalized size = 1.42 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.66065, size = 89, normalized size = 2.47 \begin{align*} \frac{2 \, b c x^{3} \arctan \left (c x^{3}\right ) + 2 \, a c x^{3} - b \log \left (c^{2} x^{6} + 1\right )}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 83.1445, size = 808, normalized size = 22.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*(a - oo*I*b)/3, Eq(c, -I/x**3)), (x**3*(a + oo*I*b)/3, Eq(c, I/x**3)), (a*x**3/3, Eq(c, 0)), (
-I*a*c**17*x**9*(c**(-2))**(15/2)/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) - I*a*c**1
5*x**3*(c**(-2))**(15/2)/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) - I*b*c**17*x**9*(c
**(-2))**(15/2)*atan(c*x**3)/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) + I*b*c**16*x**
6*(c**(-2))**(15/2)*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(
-2))**(15/2)) + I*b*c**16*x**6*(c**(-2))**(15/2)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)
*(c**(-2))**(1/3))/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) - 2*I*b*c**16*x**6*(c**(-
2))**(15/2)*log(2)/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) - I*b*c**15*x**3*(c**(-2)
)**(15/2)*atan(c*x**3)/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) + I*b*c**14*(c**(-2))
**(15/2)*log(x - (-1)**(1/6)*(c**(-2))**(1/6))/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2
)) + I*b*c**14*(c**(-2))**(15/2)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3
))/(-3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) - 2*I*b*c**14*(c**(-2))**(15/2)*log(2)/(-
3*I*c**17*x**6*(c**(-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) + b*c*x**6*atan(c*x**3)/(-3*I*c**17*x**6*(c**(
-2))**(15/2) - 3*I*c**15*(c**(-2))**(15/2)) + b*atan(c*x**3)/(-3*I*c**18*x**6*(c**(-2))**(15/2) - 3*I*c**16*(c
**(-2))**(15/2)), True))

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Giac [A]  time = 1.161, size = 54, normalized size = 1.5 \begin{align*} \frac{2 \, a c x^{3} +{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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