Optimal. Leaf size=64 \[ \frac {a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac {2 a}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
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Rubi [A]
time = 0.07, antiderivative size = 64, 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 = {3255, 3284, 16,
45} \begin {gather*} \frac {a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac {2 a}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\sqrt {a \cos ^2(e+f x)}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 45
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx &=\int \sqrt {a \cos ^2(e+f x)} \tan ^5(e+f x) \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {(1-x)^2 \sqrt {a x}}{x^3} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^3 \text {Subst}\left (\int \frac {(1-x)^2}{(a x)^{5/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^3 \text {Subst}\left (\int \left (\frac {1}{(a x)^{5/2}}-\frac {2}{a (a x)^{3/2}}+\frac {1}{a^2 \sqrt {a x}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac {2 a}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\sqrt {a \cos ^2(e+f x)}}{f}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 51, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {a \cos ^2(e+f x)} \left (-1+6 \cos ^2(e+f x)+3 \cos ^4(e+f x)\right ) \sec ^4(e+f x)}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 14.03, size = 48, normalized size = 0.75
method | result | size |
default | \(-\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (3 \left (\cos ^{4}\left (f x +e \right )\right )+6 \left (\cos ^{2}\left (f x +e \right )\right )-1\right )}{3 \cos \left (f x +e \right )^{4} f}\) | \(48\) |
risch | \(-\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, {\mathrm e}^{2 i \left (f x +e \right )}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {4 \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, \left (3 \,{\mathrm e}^{6 i \left (f x +e \right )}+4 \,{\mathrm e}^{4 i \left (f x +e \right )}+3 \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}\) | \(177\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 72, normalized size = 1.12 \begin {gather*} -\frac {3 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} a^{3} - \frac {6 \, {\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{4} + a^{5}}{{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}}{3 \, a^{3} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 47, normalized size = 0.73 \begin {gather*} -\frac {{\left (3 \, \cos \left (f x + e\right )^{4} + 6 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{3 \, f \cos \left (f x + e\right )^{4}} \end {gather*}
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 \begin {gather*} \int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \tan ^{5}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (59) = 118\).
time = 1.55, size = 136, normalized size = 2.12 \begin {gather*} \frac {2 \, \sqrt {a} {\left (\frac {3 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1} - \frac {3 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 12 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}\right )}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 19.71, size = 326, normalized size = 5.09 \begin {gather*} -\frac {\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}}{f}-\frac {8\,{\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}}{f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\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 {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\,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 {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\,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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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