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]

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

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.42, size = 39, normalized size = 1.08 \[ \frac {2 \, b c x^{2} \arctan \left (c x^{2}\right ) + 2 \, a c x^{2} - b \log \left (c^{2} x^{4} + 1\right )}{4 \, c} \]

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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giac [A]  time = 0.15, size = 40, normalized size = 1.11 \[ \frac {2 \, a c x^{2} + {\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \]

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

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maple [A]  time = 0.02, size = 36, normalized size = 1.00 \[ \frac {a \,x^{2}}{2}+\frac {b \,x^{2} \arctan \left (c \,x^{2}\right )}{2}-\frac {b \ln \left (c^{2} x^{4}+1\right )}{4 c} \]

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 = 0.31, size = 38, normalized size = 1.06 \[ \frac {1}{2} \, a x^{2} + \frac {{\left (2 \, c x^{2} \arctan \left (c x^{2}\right ) - \log \left (c^{2} x^{4} + 1\right )\right )} b}{4 \, c} \]

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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mupad [B]  time = 0.31, size = 35, normalized size = 0.97 \[ \frac {a\,x^2}{2}-\frac {b\,\ln \left (c^2\,x^4+1\right )}{4\,c}+\frac {b\,x^2\,\mathrm {atan}\left (c\,x^2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*atan(c*x^2)),x)

[Out]

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

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sympy [A]  time = 13.50, size = 66, normalized size = 1.83 \[ \begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {atan}{\left (c x^{2} \right )}}{2} - \frac {i b \sqrt {\frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{2} \right )}}{2} - \frac {b \log {\left (x^{2} + i \sqrt {\frac {1}{c^{2}}} \right )}}{2 c} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \]

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

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