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

Optimal. Leaf size=36 \[ \frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{b \log \left (c^2 x^4+1\right )}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.01395, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5033, 260} \[ \frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{b \log \left (c^2 x^4+1\right )}{4 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcTan[c*x^2]),x]

[Out]

(x^2*(a + b*ArcTan[c*x^2]))/2 - (b*Log[1 + c^2*x^4])/(4*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 \left (a+b \tan ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-(b c) \int \frac{x^3}{1+c^2 x^4} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )-\frac{b \log \left (1+c^2 x^4\right )}{4 c}\\ \end{align*}

Mathematica [A]  time = 0.0064483, size = 41, normalized size = 1.14 \[ \frac{a x^2}{2}-\frac{b \log \left (c^2 x^4+1\right )}{4 c}+\frac{1}{2} b x^2 \tan ^{-1}\left (c x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcTan[c*x^2]),x]

[Out]

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

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Maple [A]  time = 0.019, size = 36, normalized size = 1. \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b{x}^{2}\arctan \left ( c{x}^{2} \right ) }{2}}-{\frac{b\ln \left ({c}^{2}{x}^{4}+1 \right ) }{4\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02717, size = 51, normalized size = 1.42 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.93178, size = 89, normalized size = 2.47 \begin{align*} \frac{2 \, b c x^{2} \arctan \left (c x^{2}\right ) + 2 \, a c x^{2} - b \log \left (c^{2} x^{4} + 1\right )}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 20.8985, size = 70, normalized size = 1.94 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{atan}{\left (c x^{2} \right )}}{2} - \frac{b \log{\left (x^{2} + i \sqrt{\frac{1}{c^{2}}} \right )}}{2 c} - \frac{i b \operatorname{atan}{\left (c x^{2} \right )}}{2 c^{4} \left (\frac{1}{c^{2}}\right )^{\frac{3}{2}}} & \text{for}\: c \neq 0 \\\frac{a x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15435, size = 54, normalized size = 1.5 \begin{align*} \frac{2 \, a c x^{2} +{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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