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

Optimal. Leaf size=72 \[ -\frac {2 x}{c \sqrt {a^2 c x^2+c}}+\frac {x \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \tan ^{-1}(a x)}{a c \sqrt {a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4898, 191} \[ -\frac {2 x}{c \sqrt {a^2 c x^2+c}}+\frac {x \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \tan ^{-1}(a x)}{a c \sqrt {a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/(c*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^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 4898

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(b*p*(a + b*ArcTan[
c*x])^(p - 1))/(c*d*Sqrt[d + e*x^2]), x] + (-Dist[b^2*p*(p - 1), Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(
3/2), x], x] + Simp[(x*(a + b*ArcTan[c*x])^p)/(d*Sqrt[d + e*x^2]), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e,
c^2*d] && GtQ[p, 1]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 49, normalized size = 0.68 \[ \frac {\sqrt {a^2 c x^2+c} \left (-2 a x+a x \tan ^{-1}(a x)^2+2 \tan ^{-1}(a x)\right )}{c^2 \left (a^3 x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*(-2*a*x + 2*ArcTan[a*x] + a*x*ArcTan[a*x]^2))/(c^2*(a + a^3*x^2))

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fricas [A]  time = 0.67, size = 51, normalized size = 0.71 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x \arctan \left (a x\right )^{2} - 2 \, a x + 2 \, \arctan \left (a x\right )\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a^2*c*x^2 + c)*(a*x*arctan(a*x)^2 - 2*a*x + 2*arctan(a*x))/(a^3*c^2*x^2 + a*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(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 = 0.45, size = 114, normalized size = 1.58 \[ \frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2} a}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{2 \left (a^{2} x^{2}+1\right ) c^{2} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.49, size = 53, normalized size = 0.74 \[ \frac {x \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c} c} - \frac {2 \, {\left (a x - \arctan \left (a x\right )\right )}}{\sqrt {a^{2} x^{2} + 1} a c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(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 \[ \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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