Optimal. Leaf size=30 \[ \frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}}-\frac {1}{a \sqrt {a \sec ^2(x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3657, 4124, 43} \[ \frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}}-\frac {1}{a \sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3657
Rule 4124
Rubi steps
\begin {align*} \int \frac {\tan ^3(x)}{\left (a+a \tan ^2(x)\right )^{3/2}} \, dx &=\int \frac {\tan ^3(x)}{\left (a \sec ^2(x)\right )^{3/2}} \, dx\\ &=\frac {1}{2} a \operatorname {Subst}\left (\int \frac {-1+x}{(a x)^{5/2}} \, dx,x,\sec ^2(x)\right )\\ &=\frac {1}{2} a \operatorname {Subst}\left (\int \left (-\frac {1}{(a x)^{5/2}}+\frac {1}{a (a x)^{3/2}}\right ) \, dx,x,\sec ^2(x)\right )\\ &=\frac {1}{3 \left (a \sec ^2(x)\right )^{3/2}}-\frac {1}{a \sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 23, normalized size = 0.77 \[ \frac {\cos (2 x)-5}{6 a \sqrt {a \sec ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 43, normalized size = 1.43 \[ -\frac {\sqrt {a \tan \relax (x)^{2} + a} {\left (3 \, \tan \relax (x)^{2} + 2\right )}}{3 \, {\left (a^{2} \tan \relax (x)^{4} + 2 \, a^{2} \tan \relax (x)^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 26, normalized size = 0.87 \[ -\frac {3 \, a \tan \relax (x)^{2} + 2 \, a}{3 \, {\left (a \tan \relax (x)^{2} + a\right )}^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 29, normalized size = 0.97 \[ -\frac {1}{a \sqrt {a +a \left (\tan ^{2}\relax (x )\right )}}+\frac {1}{3 \left (a +a \left (\tan ^{2}\relax (x )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 38, normalized size = 1.27 \[ \frac {{\left (\sin \relax (x)^{2} + 2\right )} {\left (\sin \relax (x) + 1\right )}^{\frac {3}{2}} {\left (-\sin \relax (x) + 1\right )}^{\frac {3}{2}}}{3 \, {\left (a^{\frac {3}{2}} \sin \relax (x)^{2} - a^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.75, size = 29, normalized size = 0.97 \[ -\frac {\left ({\mathrm {tan}\relax (x)}^2+\frac {2}{3}\right )\,\sqrt {a\,{\mathrm {tan}\relax (x)}^2+a}}{a^2\,{\left ({\mathrm {tan}\relax (x)}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.57, size = 36, normalized size = 1.20 \[ \begin {cases} \frac {\frac {a}{3 \left (a \tan ^{2}{\relax (x )} + a\right )^{\frac {3}{2}}} - \frac {1}{\sqrt {a \tan ^{2}{\relax (x )} + a}}}{a} & \text {for}\: a \neq 0 \\\tilde {\infty } \tan ^{4}{\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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