3.341 \(\int \frac {x \tan ^{-1}(a x)^2}{(c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac {2}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {2 x \tan ^{-1}(a x)}{a c \sqrt {a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4930, 4894} \[ \frac {2}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {2 x \tan ^{-1}(a x)}{a c \sqrt {a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[(x*ArcTan[a*x]^2)/(c + a^2*c*x^2)^(3/2),x]

[Out]

2/(a^2*c*Sqrt[c + a^2*c*x^2]) + (2*x*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) - ArcTan[a*x]^2/(a^2*c*Sqrt[c + a^
2*c*x^2])

Rule 4894

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[(x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[((d + e*x^2)^
(q + 1)*(a + b*ArcTan[c*x])^p)/(2*e*(q + 1)), x] - Dist[(b*p)/(2*c*(q + 1)), Int[(d + e*x^2)^q*(a + b*ArcTan[c
*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 50, normalized size = 0.64 \[ \frac {\sqrt {a^2 c x^2+c} \left (-\tan ^{-1}(a x)^2+2 a x \tan ^{-1}(a x)+2\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*(2 + 2*a*x*ArcTan[a*x] - ArcTan[a*x]^2))/(a^2*c^2*(1 + a^2*x^2))

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fricas [A]  time = 0.87, size = 51, normalized size = 0.65 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (2 \, a x \arctan \left (a x\right ) - \arctan \left (a x\right )^{2} + 2\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a^2*c*x^2 + c)*(2*a*x*arctan(a*x) - arctan(a*x)^2 + 2)/(a^4*c^2*x^2 + a^2*c^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [C]  time = 0.88, size = 116, normalized size = 1.49 \[ -\frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(arctan(a*x)^2-2+2*I*arctan(a*x))*(1+I*a*x)*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)/c^2/a^2+1/2*(c*(a*x-I)*
(I+a*x))^(1/2)*(-1+I*a*x)*(arctan(a*x)^2-2-2*I*arctan(a*x))/(a^2*x^2+1)/c^2/a^2

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maxima [A]  time = 0.72, size = 73, normalized size = 0.94 \[ \sqrt {c} {\left (\frac {2 \, x \arctan \left (a x\right )}{\sqrt {a^{2} x^{2} + 1} a c^{2}} - \frac {\arctan \left (a x\right )^{2}}{\sqrt {a^{2} x^{2} + 1} a^{2} c^{2}} + \frac {2}{\sqrt {a^{2} x^{2} + 1} a^{2} c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c)*(2*x*arctan(a*x)/(sqrt(a^2*x^2 + 1)*a*c^2) - arctan(a*x)^2/(sqrt(a^2*x^2 + 1)*a^2*c^2) + 2/(sqrt(a^2*x
^2 + 1)*a^2*c^2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*atan(a*x)^2)/(c + a^2*c*x^2)^(3/2),x)

[Out]

int((x*atan(a*x)^2)/(c + a^2*c*x^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*atan(a*x)**2/(c*(a**2*x**2 + 1))**(3/2), x)

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