3.8.78 \(\int \frac {\text {ArcTan}(a x)^{3/2}}{c+a^2 c x^2} \, dx\) [778]

Optimal. Leaf size=18 \[ \frac {2 \text {ArcTan}(a x)^{5/2}}{5 a c} \]

[Out]

2/5*arctan(a*x)^(5/2)/a/c

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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5004} \begin {gather*} \frac {2 \text {ArcTan}(a x)^{5/2}}{5 a c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[a*x]^(5/2))/(5*a*c)

Rule 5004

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

Rubi steps

\begin {align*} \int \frac {\tan ^{-1}(a x)^{3/2}}{c+a^2 c x^2} \, dx &=\frac {2 \tan ^{-1}(a x)^{5/2}}{5 a c}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \text {ArcTan}(a x)^{5/2}}{5 a c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[a*x]^(5/2))/(5*a*c)

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Maple [A]
time = 0.20, size = 15, normalized size = 0.83

method	result	size
default	\(\frac {2 \arctan \left (a x \right )^{\frac {5}{2}}}{5 a c}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*arctan(a*x)^(5/2)/a/c

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 3.13, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, \arctan \left (a x\right )^{\frac {5}{2}}}{5 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*arctan(a*x)^(5/2)/(a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, \arctan \left (a x\right )^{\frac {5}{2}}}{5 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*arctan(a*x)^(5/2)/(a*c)

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Mupad [B]
time = 0.39, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\mathrm {atan}\left (a\,x\right )}^{5/2}}{5\,a\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atan(a*x)^(5/2))/(5*a*c)

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