Optimal. Leaf size=21 \[ \frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3176, 3205, 16, 32} \[ \frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 32
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \frac {\tan (e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\tan (e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {1}{(a x)^{5/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 21, normalized size = 1.00 \[ \frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 28, normalized size = 1.33 \[ \frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{3 \, a^{2} f \cos \left (f x + e\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.55, size = 64, normalized size = 3.05 \[ \frac {3 \, \sqrt {a \tan \left (f x + e\right )^{2} + a} + \frac {{\left (a \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {a \tan \left (f x + e\right )^{2} + a} a}{a}}{3 \, a^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 21, normalized size = 1.00 \[ \frac {1}{3 f \left (a -a \left (\sin ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 95, normalized size = 4.52 \[ \frac {\frac {1}{\sqrt {-a \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right ) + \sqrt {-a \sin \left (f x + e\right )^{2} + a} a} - \frac {1}{\sqrt {-a \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right ) - \sqrt {-a \sin \left (f x + e\right )^{2} + a} a}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.31, size = 72, normalized size = 3.43 \[ \frac {16\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4} \]
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 \[ \int \frac {\tan {\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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