Optimal. Leaf size=118 \[ -\frac {4 \cos ^2(x) \cot (x)}{3 a^2 \sqrt {a \sin ^4(x)}}-\frac {6 \cos ^2(x) \cot ^3(x)}{5 a^2 \sqrt {a \sin ^4(x)}}-\frac {4 \cos ^2(x) \cot ^5(x)}{7 a^2 \sqrt {a \sin ^4(x)}}-\frac {\cos ^2(x) \cot ^7(x)}{9 a^2 \sqrt {a \sin ^4(x)}}-\frac {\cos (x) \sin (x)}{a^2 \sqrt {a \sin ^4(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3286, 3852}
\begin {gather*} -\frac {\sin (x) \cos (x)}{a^2 \sqrt {a \sin ^4(x)}}-\frac {\cos ^2(x) \cot ^7(x)}{9 a^2 \sqrt {a \sin ^4(x)}}-\frac {4 \cos ^2(x) \cot ^5(x)}{7 a^2 \sqrt {a \sin ^4(x)}}-\frac {6 \cos ^2(x) \cot ^3(x)}{5 a^2 \sqrt {a \sin ^4(x)}}-\frac {4 \cos ^2(x) \cot (x)}{3 a^2 \sqrt {a \sin ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3286
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sin ^4(x)\right )^{5/2}} \, dx &=\frac {\sin ^2(x) \int \csc ^{10}(x) \, dx}{a^2 \sqrt {a \sin ^4(x)}}\\ &=-\frac {\sin ^2(x) \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,\cot (x)\right )}{a^2 \sqrt {a \sin ^4(x)}}\\ &=-\frac {4 \cos ^2(x) \cot (x)}{3 a^2 \sqrt {a \sin ^4(x)}}-\frac {6 \cos ^2(x) \cot ^3(x)}{5 a^2 \sqrt {a \sin ^4(x)}}-\frac {4 \cos ^2(x) \cot ^5(x)}{7 a^2 \sqrt {a \sin ^4(x)}}-\frac {\cos ^2(x) \cot ^7(x)}{9 a^2 \sqrt {a \sin ^4(x)}}-\frac {\cos (x) \sin (x)}{a^2 \sqrt {a \sin ^4(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 47, normalized size = 0.40 \begin {gather*} -\frac {\cos (x) \left (128+64 \csc ^2(x)+48 \csc ^4(x)+40 \csc ^6(x)+35 \csc ^8(x)\right ) \sin (x)}{315 a^2 \sqrt {a \sin ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 41, normalized size = 0.35
method | result | size |
default | \(-\frac {\left (128 \left (\cos ^{8}\left (x \right )\right )-576 \left (\cos ^{6}\left (x \right )\right )+1008 \left (\cos ^{4}\left (x \right )\right )-840 \left (\cos ^{2}\left (x \right )\right )+315\right ) \cos \left (x \right ) \sin \left (x \right )}{315 \left (a \left (\sin ^{4}\left (x \right )\right )\right )^{\frac {5}{2}}}\) | \(41\) |
risch | \(\frac {256 i \left (126 \,{\mathrm e}^{6 i x}-84 \,{\mathrm e}^{4 i x}-9+37 \cos \left (2 x \right )+35 i \sin \left (2 x \right )\right )}{315 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{7} \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 35, normalized size = 0.30 \begin {gather*} -\frac {315 \, \tan \left (x\right )^{8} + 420 \, \tan \left (x\right )^{6} + 378 \, \tan \left (x\right )^{4} + 180 \, \tan \left (x\right )^{2} + 35}{315 \, a^{\frac {5}{2}} \tan \left (x\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 104, normalized size = 0.88 \begin {gather*} \frac {{\left (128 \, \cos \left (x\right )^{9} - 576 \, \cos \left (x\right )^{7} + 1008 \, \cos \left (x\right )^{5} - 840 \, \cos \left (x\right )^{3} + 315 \, \cos \left (x\right )\right )} \sqrt {a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a}}{315 \, {\left (a^{3} \cos \left (x\right )^{10} - 5 \, a^{3} \cos \left (x\right )^{8} + 10 \, a^{3} \cos \left (x\right )^{6} - 10 \, a^{3} \cos \left (x\right )^{4} + 5 \, a^{3} \cos \left (x\right )^{2} - a^{3}\right )} \sin \left (x\right )} \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 {1}{\left (a \sin ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.56, size = 117, normalized size = 0.99 \begin {gather*} \frac {256\,\left ({\mathrm {e}}^{x\,46{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{x\,48{}\mathrm {i}}\,9{}\mathrm {i}+{\mathrm {e}}^{x\,50{}\mathrm {i}}\,36{}\mathrm {i}-{\mathrm {e}}^{x\,52{}\mathrm {i}}\,84{}\mathrm {i}+{\mathrm {e}}^{x\,54{}\mathrm {i}}\,126{}\mathrm {i}\right )}{315\,a^{5/2}\,\left ({\mathrm {e}}^{x\,46{}\mathrm {i}}-9\,{\mathrm {e}}^{x\,48{}\mathrm {i}}+36\,{\mathrm {e}}^{x\,50{}\mathrm {i}}-84\,{\mathrm {e}}^{x\,52{}\mathrm {i}}+126\,{\mathrm {e}}^{x\,54{}\mathrm {i}}-126\,{\mathrm {e}}^{x\,56{}\mathrm {i}}+84\,{\mathrm {e}}^{x\,58{}\mathrm {i}}-36\,{\mathrm {e}}^{x\,60{}\mathrm {i}}+9\,{\mathrm {e}}^{x\,62{}\mathrm {i}}-{\mathrm {e}}^{x\,64{}\mathrm {i}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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