3.6.80 \(\int \frac {x^2}{(c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^2} \, dx\) [580]

Optimal. Leaf size=98 \[ \frac {1}{a^3 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}+\frac {\sqrt {1+a^2 x^2} \text {Si}(\text {ArcTan}(a x))}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{\sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2},x\right )}{a^2 c} \]

[Out]

1/a^3/c/arctan(a*x)/(a^2*c*x^2+c)^(1/2)+Si(arctan(a*x))*(a^2*x^2+1)^(1/2)/a^3/c/(a^2*c*x^2+c)^(1/2)+Unintegrab
le(1/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2),x)/a^2/c

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Rubi [A]
time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

1/(a^3*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]) + (Sqrt[1 + a^2*x^2]*SinIntegral[ArcTan[a*x]])/(a^3*c*Sqrt[c + a^2*c
*x^2]) + Defer[Int][1/(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2), x]/(a^2*c)

Rubi steps

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

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Mathematica [A]
time = 8.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.07, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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