Optimal. Leaf size=198 \[ -\frac{\text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)},x\right )}{a c^2}-\frac{5 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt{a^2 c x^2+c}}+\frac{a x}{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{\sqrt{a^2 c x^2+c}}{a c^3 x \tan ^{-1}(a x)}+\frac{a x}{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
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Rubi [A] time = 1.13725, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]
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 )^{5/2} \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx\right )+\frac{\int \frac{1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac{a x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-a \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\left (2 a^3\right ) \int \frac{x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\frac{\int \frac{1}{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac{a^2 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=\frac{a x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{a x}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{c+a^2 c x^2}}{a c^3 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^2}-\frac{a \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c}-\frac{\left (a \sqrt{1+a^2 x^2}\right ) \int \frac{1}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt{c+a^2 c x^2}}+\frac{\left (2 a^3 \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{a x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{a x}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{c+a^2 c x^2}}{a c^3 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^2}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos ^3(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{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 )}{c^2 \sqrt{c+a^2 c x^2}}-\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^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{a x}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{c+a^2 c x^2}}{a c^3 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^2}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}-\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 )}{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 )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{a x}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{c+a^2 c x^2}}{a c^3 x \tan ^{-1}(a x)}-\frac{\sqrt{1+a^2 x^2} \text{Ci}\left (\tan ^{-1}(a x)\right )}{c^2 \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^2}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 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 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 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 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{a x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{a x}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}-\frac{\sqrt{c+a^2 c x^2}}{a c^3 x \tan ^{-1}(a x)}-\frac{5 \sqrt{1+a^2 x^2} \text{Ci}\left (\tan ^{-1}(a x)\right )}{4 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 c^2 \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^2}\\ \end{align*}
Mathematica [A] time = 2.20655, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.577, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( \arctan \left ( ax \right ) \right ) ^{2}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c}}{{\left (a^{6} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{5} + 3 \, a^{2} c^{3} x^{3} + c^{3} x\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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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