Optimal. Leaf size=162 \[ \frac {5 \tan ^3(e+f x) \sqrt {a (\sin (e+f x)+1)}}{12 f}+\frac {29 \tan (e+f x) \sqrt {a \sin (e+f x)+a}}{12 f}-\frac {\sec ^3(e+f x) \sqrt {a (\sin (e+f x)+1)}}{12 f}-\frac {27 \sec (e+f x) \sqrt {a (\sin (e+f x)+1)}}{8 f}+\frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{8 \sqrt {2} f} \]
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Rubi [A] time = 0.92, 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 = {2714, 2646, 4401, 2675, 2687, 2650, 2649, 206, 2878, 2855} \[ \frac {11 a^2 \cos (e+f x)}{8 f (a \sin (e+f x)+a)^{3/2}}-\frac {2 a \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}+\frac {4 \sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 a f}-\frac {7 \sec ^3(e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}-\frac {11 a \sec (e+f x)}{6 f \sqrt {a \sin (e+f x)+a}}+\frac {11 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{8 \sqrt {2} f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2646
Rule 2649
Rule 2650
Rule 2675
Rule 2687
Rule 2714
Rule 2855
Rule 2878
Rule 4401
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) \operatorname {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) \operatorname {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] time = 5.57, size = 394, normalized size = 2.43 \[ \frac {\sqrt {a (\sin (e+f x)+1)} \left (-48 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \cos \left (\frac {f x}{2}\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2+48 \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \sin \left (\frac {f x}{2}\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2-\frac {36 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}+\frac {4 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {3 \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}{\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )}+\frac {6 \sin \left (\frac {f x}{2}\right )}{\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )}+(33+33 i) (-1)^{3/4} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {f x}{4}\right ) \left (\cos \left (\frac {1}{4} (2 e+f x)\right )-\sin \left (\frac {1}{4} (2 e+f x)\right )\right )\right )\right )}{24 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 200, normalized size = 1.23 \[ \frac {33 \, \sqrt {2} \sqrt {a} \cos \left (f x + e\right )^{3} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {a \sin \left (f x + e\right ) + a} {\left (\sqrt {2} \cos \left (f x + e\right ) - \sqrt {2} \sin \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (81 \, \cos \left (f x + e\right )^{2} - 2 \, {\left (24 \, \cos \left (f x + e\right )^{2} + 5\right )} \sin \left (f x + e\right ) + 2\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{96 \, f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \sin \left (f x + e\right ) + a} \tan \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 172, normalized size = 1.06 \[ -\frac {96 a^{\frac {5}{2}} \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (33 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +20 a^{\frac {5}{2}}\right ) \sin \left (f x +e \right )-162 a^{\frac {5}{2}} \left (\cos ^{2}\left (f x +e \right )\right )+33 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -4 a^{\frac {5}{2}}}{48 a^{\frac {3}{2}} \left (\sin \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \sqrt {a \sin \left (f x + e\right ) + a} \tan \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^4\,\sqrt {a+a\,\sin \left (e+f\,x\right )} \,d x \]
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 \[ \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \tan ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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