Optimal. Leaf size=69 \[ \frac {\sqrt {a^2 x^2+1} \text {Ci}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.17, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4942, 4905, 4904, 3302} \[ \frac {\sqrt {a^2 x^2+1} \text {CosIntegral}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 4904
Rule 4905
Rule 4942
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx &=-\frac {x}{a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{a}\\ &=-\frac {x}{a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{a c \sqrt {c+a^2 c x^2}}\\ &=-\frac {x}{a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c \sqrt {c+a^2 c x^2}}\\ &=-\frac {x}{a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{a^2 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 55, normalized size = 0.80 \[ \frac {\sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Ci}\left (\tan ^{-1}(a x)\right )-a x}{a^2 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x}{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.12, size = 210, normalized size = 3.04 \[ -\frac {\left (\arctan \left (a x \right ) \Ei \left (1, -i \arctan \left (a x \right )\right ) x^{2} a^{2}+\Ei \left (1, -i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sqrt {a^{2} x^{2}+1}\, x a -i \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right ) c^{2} a^{2}}-\frac {\left (\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right ) x^{2} a^{2}+\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sqrt {a^{2} x^{2}+1}\, x a +i \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right ) c^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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