Optimal. Leaf size=66 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^2(e+f x)}{2 a^2 f} \]
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Rubi [A]
time = 0.09, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3255, 3284, 16,
44, 65, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\csc ^2(e+f x) \sqrt {a \cos ^2(e+f x)}}{2 a^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 44
Rule 65
Rule 212
Rule 3255
Rule 3284
Rubi steps
\begin {align*} \int \frac {\cot ^3(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\cot ^3(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=-\frac {\text {Subst}\left (\int \frac {x}{(1-x)^2 (a x)^{3/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{(1-x)^2 \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{2 a f}\\ &=-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^2(e+f x)}{2 a^2 f}-\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(e+f x)\right )}{4 a f}\\ &=-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^2(e+f x)}{2 a^2 f}-\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(e+f x)}\right )}{2 a^2 f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^2(e+f x)}{2 a^2 f}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 82, normalized size = 1.24 \begin {gather*} -\frac {\cos ^3(e+f x) \left (\csc ^2\left (\frac {1}{2} (e+f x)\right )+4 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )-4 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )-\sec ^2\left (\frac {1}{2} (e+f x)\right )\right )}{8 f \left (a \cos ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 8.81, size = 67, normalized size = 1.02
method | result | size |
default | \(\frac {-\frac {\sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}}{2 a^{2} \sin \left (f x +e \right )^{2}}-\frac {\ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}+2 a}{\sin \left (f x +e \right )}\right )}{2 a^{\frac {3}{2}}}}{f}\) | \(67\) |
risch | \(\frac {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}{a \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}+\frac {\ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) \cos \left (f x +e \right )}{f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 106, normalized size = 1.61 \begin {gather*} -\frac {\frac {\log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} - \frac {1}{\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 )^{2}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 83, normalized size = 1.26 \begin {gather*} -\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left ({\left (\cos \left (f x + e\right )^{2} - 1\right )} \log \left (-\frac {\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1}\right ) - 2 \, \cos \left (f x + e\right )\right )}}{4 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\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 {\cot ^{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 \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. 129 vs.
\(2 (57) = 114\).
time = 0.73, size = 129, normalized size = 1.95 \begin {gather*} -\frac {\frac {\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} + \frac {2 \, \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} - \frac {2 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \sqrt {a}}{a^{2} \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}}}{8 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^3}{{\left (a-a\,{\sin \left (e+f\,x\right )}^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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