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

Optimal. Leaf size=49 \[ \frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)}{a^2 c \sqrt {a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4930, 191} \[ \frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\tan ^{-1}(a x)}{a^2 c \sqrt {a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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

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Mathematica [A]  time = 0.05, size = 42, normalized size = 0.86 \[ \frac {\sqrt {a^2 c x^2+c} \left (a x-\tan ^{-1}(a x)\right )}{a^2 c^2 \left (a^2 x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 43, normalized size = 0.88 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x - \arctan \left (a x\right )\right )}}{a^{4} c^{2} x^{2} + a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x

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maple [C]  time = 0.86, size = 100, normalized size = 2.04 \[ -\frac {\left (i+\arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.50, size = 28, normalized size = 0.57 \[ \frac {a x - \arctan \left (a x\right )}{\sqrt {a^{2} x^{2} + 1} a^{2} c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x - arctan(a*x))/(sqrt(a^2*x^2 + 1)*a^2*c^(3/2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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