Optimal. Leaf size=116 \[ \frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{x}{a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.501346, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4968, 4971, 4970, 4406, 3302, 4905, 4904, 3312} \[ \frac{\sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{a^2 c x^2+c}}-\frac{x}{a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4968
Rule 4971
Rule 4970
Rule 4406
Rule 3302
Rule 4905
Rule 4904
Rule 3312
Rubi steps
\begin{align*} \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx &=-\frac{x}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{a}-(2 a) \int \frac{x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx\\ &=-\frac{x}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\sqrt{1+a^2 x^2} \int \frac{1}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{a c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 a \sqrt{1+a^2 x^2}\right ) \int \frac{x^2}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \left (\frac{3 \cos (x)}{4 x}+\frac{\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 x}-\frac{\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos (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} \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\sqrt{1+a^2 x^2} \text{Ci}\left (\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{c+a^2 c x^2}}+\frac{3 \sqrt{1+a^2 x^2} \text{Ci}\left (3 \tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.179799, size = 95, normalized size = 0.82 \[ \frac{\left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \text{CosIntegral}\left (\tan ^{-1}(a x)\right )+3 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )-4 a x}{4 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.379, size = 582, normalized size = 5. \begin{align*} \text{result too large to display} \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 )^{2}}\,{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 )^{2}}, 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}^{2}{\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 )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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