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 = 0.24, antiderivative size = 115, normalized size of antiderivative = 1.00, 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 3299
Rule 4406
Rule 4902
Rule 4970
Rule 4971
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*}
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Mathematica [A] time = 0.12, size = 84, normalized size = 0.73 \[ -\frac {\sqrt {c \left (a^2 x^2+1\right )} \left (\frac {4}{\left (a^2 x^2+1\right )^{3/2}}+3 \tan ^{-1}(a x) \text {Si}\left (\tan ^{-1}(a x)\right )+3 \tan ^{-1}(a x) \text {Si}\left (3 \tan ^{-1}(a x)\right )\right )}{4 a c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.77, size = 586, normalized size = 5.10 \[ \frac {i \left (3 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right ) x^{4} a^{4}-\sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+6 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right ) x^{2} a^{2}-3 i \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+3 \sqrt {a^{2} x^{2}+1}\, x a +3 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+i \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \sqrt {a^{2} x^{2}+1}\, \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right ) a \,c^{3}}-\frac {i \left (3 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right ) x^{4} a^{4}+6 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right ) x^{2} a^{2}-\sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+3 i \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+3 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+3 \sqrt {a^{2} x^{2}+1}\, x a -i \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \sqrt {a^{2} x^{2}+1}\, \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right ) a \,c^{3}}+\frac {3 i \left (\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right ) x^{2} a^{2}+\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sqrt {a^{2} x^{2}+1}\, x a +i \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right ) a \,c^{3}}-\frac {3 i \left (\arctan \left (a x \right ) \Ei \left (1, -i \arctan \left (a x \right )\right ) x^{2} a^{2}+\Ei \left (1, -i \arctan \left (a x \right )\right ) \arctan \left (a x \right )+\sqrt {a^{2} x^{2}+1}\, x a -i \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right ) a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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