Optimal. Leaf size=115 \[ -\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.244665, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4902, 4971, 4970, 4406, 3299} \[ -\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{a^2 c x^2+c}}-\frac{1}{a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4902
Rule 4971
Rule 4970
Rule 4406
Rule 3299
Rubi steps
\begin{align*} \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx &=-\frac{1}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-(3 a) \int \frac{x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx\\ &=-\frac{1}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{\left (3 a \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{1}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{\left (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 )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{\left (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 )}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (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 a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac{3 \sqrt{1+a^2 x^2} \text{Si}\left (\tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \text{Si}\left (3 \tan ^{-1}(a x)\right )}{4 a c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.186117, size = 61, normalized size = 0.53 \[ \frac{-3 \left (a^2 x^2+1\right )^{3/2} \left (\text{Si}\left (\tan ^{-1}(a x)\right )+\text{Si}\left (3 \tan ^{-1}(a x)\right )\right )-\frac{4}{\tan ^{-1}(a x)}}{4 a c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.341, size = 586, normalized size = 5.1 \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{1}{{\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}}{{\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{1}{\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{1}{{\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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