Optimal. Leaf size=44 \[ \frac {a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}-\frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3176, 3205, 16, 43} \[ \frac {a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}-\frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 43
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \frac {\tan ^3(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\tan ^3(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1-x}{x^2 (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \frac {1-x}{(a x)^{7/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {1}{(a x)^{7/2}}-\frac {1}{a (a x)^{5/2}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}-\frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 34, normalized size = 0.77 \[ \frac {a \left (3-5 \cos ^2(e+f x)\right )}{15 f \left (a \cos ^2(e+f x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 40, normalized size = 0.91 \[ -\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (5 \, \cos \left (f x + e\right )^{2} - 3\right )}}{15 \, a^{2} f \cos \left (f x + e\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 108, normalized size = 2.45 \[ \frac {\frac {5 \, {\left ({\left (a \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {a \tan \left (f x + e\right )^{2} + a} a\right )}}{a} + \frac {3 \, {\left (a \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} - 10 \, {\left (a \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {a \tan \left (f x + e\right )^{2} + a} a^{2}}{a^{2}}}{15 \, a^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.17, size = 41, normalized size = 0.93 \[ -\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (5 \left (\cos ^{2}\left (f x +e \right )\right )-3\right )}{15 a^{2} \cos \left (f x +e \right )^{6} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 48, normalized size = 1.09 \[ \frac {5 \, {\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{2} + 3 \, a^{3}}{15 \, {\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} a^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 20.21, size = 389, normalized size = 8.84 \[ -\frac {16\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {272\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {128\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {64\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \]
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 \[ \int \frac {\tan ^{3}{\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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