Optimal. Leaf size=87 \[ \frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.198889, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4971, 4970, 4406, 3299} \[ \frac{\sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4971
Rule 4970
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{x}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 x}+\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.124767, size = 51, normalized size = 0.59 \[ \frac{\sqrt{c \left (a^2 x^2+1\right )} \left (\text{Si}\left (\tan ^{-1}(a x)\right )+\text{Si}\left (3 \tan ^{-1}(a x)\right )\right )}{4 a^2 c^3 \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.283, size = 125, normalized size = 1.4 \begin{align*} -{\frac{{\it csgn} \left ( \arctan \left ( ax \right ) \right ) \pi }{4\,{c}^{3}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{\it Si} \left ( 3\,\arctan \left ( ax \right ) \right ) }{4\,{c}^{3}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{{\it Si} \left ( \arctan \left ( ax \right ) \right ) }{4\,{c}^{3}{a}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} x}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}} \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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