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