Optimal. Leaf size=130 \[ \frac {a x}{c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}-\frac {\sqrt {c+a^2 c x^2}}{a c^2 x \text {ArcTan}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {CosIntegral}(\text {ArcTan}(a x))}{c \sqrt {c+a^2 c x^2}}-\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)},x\right )}{a c} \]
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Rubi [A]
time = 0.33, 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 {1}{x \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 {1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx\right )+\frac {\int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac {a x}{c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {c+a^2 c x^2}}{a c^2 x \tan ^{-1}(a x)}-a \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{a c}\\ &=\frac {a x}{c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {c+a^2 c x^2}}{a c^2 x \tan ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{a c}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {a x}{c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {c+a^2 c x^2}}{a c^2 x \tan ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{a c}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {a x}{c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {c+a^2 c x^2}}{a c^2 x \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{a c}\\ \end {align*}
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Mathematica [A]
time = 1.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \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 = 0.19, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )^{2}}\, dx\]
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 {1}{x \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {1}{x\,{\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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