Optimal. Leaf size=68 \[ \frac {a^2}{7 f \left (a \cos ^2(e+f x)\right )^{7/2}}-\frac {2 a}{5 f \left (a \cos ^2(e+f x)\right )^{5/2}}+\frac {1}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 68, 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}{7 f \left (a \cos ^2(e+f x)\right )^{7/2}}-\frac {2 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 {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 45
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \frac {\tan ^5(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\tan ^5(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {(1-x)^2}{x^3 (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^3 \text {Subst}\left (\int \frac {(1-x)^2}{(a x)^{9/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {a^3 \text {Subst}\left (\int \left (\frac {1}{(a x)^{9/2}}-\frac {2}{a (a x)^{7/2}}+\frac {1}{a^2 (a x)^{5/2}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac {a^2}{7 f \left (a \cos ^2(e+f x)\right )^{7/2}}-\frac {2 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 = 0.07, size = 51, normalized size = 0.75 \begin {gather*} \frac {\left (15-42 \cos ^2(e+f x)+35 \cos ^4(e+f x)\right ) \sec ^4(e+f x)}{105 f \left (a \cos ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 13.60, size = 51, normalized size = 0.75
method | result | size |
default | \(\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (35 \left (\cos ^{4}\left (f x +e \right )\right )-42 \left (\cos ^{2}\left (f x +e \right )\right )+15\right )}{105 a^{2} \cos \left (f x +e \right )^{8} f}\) | \(51\) |
risch | \(\frac {\frac {8 \,{\mathrm e}^{10 i \left (f x +e \right )}}{3}-\frac {32 \,{\mathrm e}^{8 i \left (f x +e \right )}}{15}+\frac {304 \,{\mathrm e}^{6 i \left (f x +e \right )}}{35}-\frac {32 \,{\mathrm e}^{4 i \left (f x +e \right )}}{15}+\frac {8 \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6} a}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 72, normalized size = 1.06 \begin {gather*} \frac {35 \, {\left (a \sin \left (f x + e\right )^{2} - a\right )}^{2} a^{3} + 42 \, {\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{4} + 15 \, a^{5}}{105 \, {\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac {7}{2}} 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 = 50, normalized size = 0.74 \begin {gather*} \frac {{\left (35 \, \cos \left (f x + e\right )^{4} - 42 \, \cos \left (f x + e\right )^{2} + 15\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{105 \, a^{2} f \cos \left (f x + e\right )^{8}} \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 \frac {\tan ^{5}{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 64, normalized size = 0.94 \begin {gather*} \frac {15 \, {\left (a \tan \left (f x + e\right )^{2} + a\right )}^{\frac {7}{2}} - 42 \, {\left (a \tan \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} a + 35 \, {\left (a \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a^{2}}{105 \, a^{5} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 33.91, size = 583, normalized size = 8.57 \begin {gather*} \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 {464\,{\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 {3072\,{\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}}{35\,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 {4736\,{\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}}{35\,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 )}+\frac {768\,{\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}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^6\,\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 {256\,{\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}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^7\,\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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