Optimal. Leaf size=162 \[ \frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{8 \sqrt {2} f}-\frac {27 \sec (e+f x) \sqrt {a (1+\sin (e+f x))}}{8 f}-\frac {\sec ^3(e+f x) \sqrt {a (1+\sin (e+f x))}}{12 f}+\frac {29 \sqrt {a+a \sin (e+f x)} \tan (e+f x)}{12 f}+\frac {5 \sqrt {a (1+\sin (e+f x))} \tan ^3(e+f x)}{12 f} \]
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Rubi [A]
time = 0.63, antiderivative size = 195, normalized size of antiderivative = 1.20, number of steps
used = 15, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2793, 2725,
4486, 2754, 2766, 2729, 2728, 212, 2957, 2934} \begin {gather*} \frac {11 a^2 \cos (e+f x)}{8 f (a \sin (e+f x)+a)^{3/2}}-\frac {2 a \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}+\frac {4 \sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 a f}-\frac {7 \sec ^3(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}-\frac {11 a \sec (e+f x)}{6 f \sqrt {a \sin (e+f x)+a}}+\frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{8 \sqrt {2} f} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2725
Rule 2728
Rule 2729
Rule 2754
Rule 2766
Rule 2793
Rule 2934
Rule 2957
Rule 4486
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} \tan ^4(e+f x) \, dx &=\int \sqrt {a+a \sin (e+f x)} \, dx-\int \sec ^4(e+f x) \sqrt {a+a \sin (e+f x)} \left (1-2 \sin ^2(e+f x)\right ) \, dx\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\int \left (\sec ^4(e+f x) \sqrt {a (1+\sin (e+f x))}-2 \sec ^2(e+f x) \sqrt {a (1+\sin (e+f x))} \tan ^2(e+f x)\right ) \, dx\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}+2 \int \sec ^2(e+f x) \sqrt {a (1+\sin (e+f x))} \tan ^2(e+f x) \, dx-\int \sec ^4(e+f x) \sqrt {a (1+\sin (e+f x))} \, dx\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {\sec ^3(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 a f}-\frac {4 \int \sec ^4(e+f x) \sqrt {a+a \sin (e+f x)} \left (\frac {3 a}{2}+3 a \sin (e+f x)\right ) \, dx}{3 a}-\frac {1}{6} (5 a) \int \frac {\sec ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx\\ &=-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {5 a \sec (e+f x)}{6 f \sqrt {a+a \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 a f}-a \int \frac {\sec ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx-\frac {1}{4} \left (5 a^2\right ) \int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx\\ &=\frac {5 a^2 \cos (e+f x)}{8 f (a+a \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {11 a \sec (e+f x)}{6 f \sqrt {a+a \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 a f}-\frac {1}{16} (5 a) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx-\frac {1}{2} \left (3 a^2\right ) \int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx\\ &=\frac {11 a^2 \cos (e+f x)}{8 f (a+a \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {11 a \sec (e+f x)}{6 f \sqrt {a+a \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 a f}-\frac {1}{8} (3 a) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx+\frac {(5 a) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 f}\\ &=\frac {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{8 \sqrt {2} f}+\frac {11 a^2 \cos (e+f x)}{8 f (a+a \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {11 a \sec (e+f x)}{6 f \sqrt {a+a \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 a f}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{4 f}\\ &=\frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{8 \sqrt {2} f}+\frac {11 a^2 \cos (e+f x)}{8 f (a+a \sin (e+f x))^{3/2}}-\frac {2 a \cos (e+f x)}{f \sqrt {a+a \sin (e+f x)}}-\frac {11 a \sec (e+f x)}{6 f \sqrt {a+a \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 a f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.38, size = 394, normalized size = 2.43 \begin {gather*} \frac {\left (\frac {6 \sin \left (\frac {f x}{2}\right )}{\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )}-\frac {3 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}{\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )}+(33+33 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {f x}{4}\right ) \left (\cos \left (\frac {1}{4} (2 e+f x)\right )-\sin \left (\frac {1}{4} (2 e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-48 \cos \left (\frac {f x}{2}\right ) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+48 \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+\frac {4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {36 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}\right ) \sqrt {a (1+\sin (e+f x))}}{24 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.87, size = 172, normalized size = 1.06
method | result | size |
default | \(-\frac {96 a^{\frac {5}{2}} \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (33 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +20 a^{\frac {5}{2}}\right ) \sin \left (f x +e \right )-162 a^{\frac {5}{2}} \left (\cos ^{2}\left (f x +e \right )\right )+33 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -4 a^{\frac {5}{2}}}{48 a^{\frac {3}{2}} \left (\sin \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 217, normalized size = 1.34 \begin {gather*} \frac {33 \, \sqrt {2} \sqrt {a} \cos \left (f x + e\right )^{3} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a \sin \left (f x + e\right ) + a} {\left (\sqrt {2} \cos \left (f x + e\right ) - \sqrt {2} \sin \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (81 \, \cos \left (f x + e\right )^{2} - 2 \, {\left (24 \, \cos \left (f x + e\right )^{2} + 5\right )} \sin \left (f x + e\right ) + 2\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{96 \, f \cos \left (f x + e\right )^{3}} \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 )} \tan ^{4}{\left (e + f x \right )}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^4\,\sqrt {a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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