Optimal. Leaf size=167 \[ -\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{2 \sqrt {2} f}+\frac {2 a^3 \cos ^3(e+f x)}{3 f (a \sin (e+f x)+a)^{3/2}}-\frac {4 a^2 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}+\frac {\sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}-\frac {7 a \sec (e+f x) \sqrt {a \sin (e+f x)+a}}{2 f} \]
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Rubi [A] time = 0.98, antiderivative size = 195, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2714, 2647, 2646, 4401, 2675, 2649, 206, 2878, 2855} \[ -\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{2 \sqrt {2} f}-\frac {2 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}+\frac {4 \sec ^3(e+f x) (a \sin (e+f x)+a)^{5/2}}{a f}-\frac {23 \sec ^3(e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f}+\frac {a \sec (e+f x) \sqrt {a \sin (e+f x)+a}}{2 f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2646
Rule 2647
Rule 2649
Rule 2675
Rule 2714
Rule 2855
Rule 2878
Rule 4401
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} \tan ^4(e+f x) \, dx &=\int (a+a \sin (e+f x))^{3/2} \, dx-\int \sec ^4(e+f x) (a+a \sin (e+f x))^{3/2} \left (1-2 \sin ^2(e+f x)\right ) \, dx\\ &=-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {1}{3} (4 a) \int \sqrt {a+a \sin (e+f x)} \, dx-\int \left (\sec ^4(e+f x) (a (1+\sin (e+f x)))^{3/2}-2 \sec ^2(e+f x) (a (1+\sin (e+f x)))^{3/2} \tan ^2(e+f x)\right ) \, dx\\ &=-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+2 \int \sec ^2(e+f x) (a (1+\sin (e+f x)))^{3/2} \tan ^2(e+f x) \, dx-\int \sec ^4(e+f x) (a (1+\sin (e+f x)))^{3/2} \, dx\\ &=-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {\sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{5/2}}{a f}-\frac {4 \int \sec ^4(e+f x) (a+a \sin (e+f x))^{3/2} \left (\frac {5 a}{2}+3 a \sin (e+f x)\right ) \, dx}{a}-\frac {1}{2} a \int \sec ^2(e+f x) \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {a \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{2 f}-\frac {23 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{5/2}}{a f}+a \int \sec ^2(e+f x) \sqrt {a+a \sin (e+f x)} \, dx-\frac {1}{4} a^2 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx\\ &=-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {a \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{2 f}-\frac {23 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{5/2}}{a f}+\frac {1}{2} a^2 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx+\frac {a^2 \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 )}{2 f}\\ &=\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} f}-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {a \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{2 f}-\frac {23 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{5/2}}{a f}-\frac {a^2 \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 )}{f}\\ &=-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} f}-\frac {8 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {a \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{2 f}-\frac {23 \sec ^3(e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}+\frac {4 \sec ^3(e+f x) (a+a \sin (e+f x))^{5/2}}{a f}\\ \end {align*}
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Mathematica [C] time = 5.55, size = 141, normalized size = 0.84 \[ \frac {a \sec ^3(e+f x) \sqrt {a (\sin (e+f x)+1)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (54 \sin (e+f x)+\sin (3 (e+f x))+6 \cos (2 (e+f x))+(3+3 i) (-1)^{3/4} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (e+f x)\right )-1\right )\right )-45\right )}{6 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 239, normalized size = 1.43 \[ \frac {3 \, {\left (\sqrt {2} a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - \sqrt {2} a \cos \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a \sin \left (f x + e\right ) + a} {\left (\sqrt {2} \cos \left (f x + e\right ) - \sqrt {2} \sin \left (f x + e\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (12 \, a \cos \left (f x + e\right )^{2} + {\left (4 \, a \cos \left (f x + e\right )^{2} + 53 \, a\right )} \sin \left (f x + e\right ) - 51 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{24 \, {\left (f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - f \cos \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 139, normalized size = 0.83 \[ \frac {\left (1+\sin \left (f x +e \right )\right ) \left (3 a^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}}-8 a^{3} \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-24 a^{3} \left (\cos ^{2}\left (f x +e \right )\right )-106 \sin \left (f x +e \right ) a^{3}+102 a^{3}\right )}{12 a \left (\sin \left (f x +e \right )-1\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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