Optimal. Leaf size=207 \[ -\frac{2 a^2 \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{c^2}+\frac{\text{Unintegrable}\left (\frac{1}{x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{c^2}-\frac{11 a^3 \sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 a^3 \sqrt{a^2 x^2+1} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 a^3}{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}-\frac{a^3}{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
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Rubi [A] time = 1.60702, 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 = {} \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx\right )+\frac{\int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+\frac{\int \frac{1}{x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \frac{a^2 \int \frac{1}{x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac{a^3}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\left (3 a^5\right ) \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\frac{\int \frac{1}{x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac{a^2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac{a^4 \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\right )\\ &=-\frac{a^3}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\int \frac{1}{x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac{a^3}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{a^2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}+\frac{a^5 \int \frac{x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c}\right )-\frac{\left (3 a^5 \sqrt{1+a^2 x^2}\right ) \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{a^3}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\int \frac{1}{x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac{\left (3 a^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}-2 \left (\frac{a^3}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{a^2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}+\frac{\left (a^5 \sqrt{1+a^2 x^2}\right ) \int \frac{x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{a^3}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\int \frac{1}{x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac{a^3}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{a^2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}+\frac{\left (a^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\right )-\frac{\left (3 a^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 x}+\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{a^3}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\int \frac{1}{x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac{a^3}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{a^3 \sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{a^2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}\right )-\frac{\left (3 a^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 a^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{a^3}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{3 a^3 \sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt{c+a^2 c x^2}}-\frac{3 a^3 \sqrt{1+a^2 x^2} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\int \frac{1}{x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac{a^3}{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}+\frac{a^3 \sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{a^2 \int \frac{1}{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}\right )\\ \end{align*}
Mathematica [A] time = 6.34402, size = 0, normalized size = 0. \[ \int \frac{1}{x^4 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.217, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \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^{4} \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^{10} + 3 \, a^{4} c^{3} x^{8} + 3 \, a^{2} c^{3} x^{6} + c^{3} x^{4}\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{4} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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