Optimal. Leaf size=177 \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{256 \sqrt {2} a^{3/2} f}+\frac {\tan ^3(e+f x)}{3 f (a \sin (e+f x)+a)^{3/2}}+\frac {a \sin (e+f x) \tan (e+f x)}{12 f (a \sin (e+f x)+a)^{5/2}}+\frac {7 \cos (e+f x)}{256 f (a \sin (e+f x)+a)^{3/2}}-\frac {(87 \sin (e+f x)+65) \sec (e+f x)}{192 f (a \sin (e+f x)+a)^{3/2}} \]
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Rubi [A] time = 1.20, antiderivative size = 195, normalized size of antiderivative = 1.10, number of steps used = 20, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2714, 2650, 2649, 206, 4401, 2681, 2687, 2877, 2855} \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{256 \sqrt {2} a^{3/2} f}+\frac {7 \cos (e+f x)}{256 f (a \sin (e+f x)+a)^{3/2}}+\frac {\sec ^3(e+f x)}{4 a f \sqrt {a \sin (e+f x)+a}}-\frac {\sec ^3(e+f x)}{6 f (a \sin (e+f x)+a)^{3/2}}-\frac {45 \sec (e+f x)}{64 a f \sqrt {a \sin (e+f x)+a}}+\frac {9 \sec (e+f x)}{32 f (a \sin (e+f x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2681
Rule 2687
Rule 2714
Rule 2855
Rule 2877
Rule 4401
Rubi steps
\begin {align*} \int \frac {\tan ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx &=\int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx-\int \frac {\sec ^4(e+f x) \left (1-2 \sin ^2(e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx\\ &=-\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 a}-\int \left (\frac {\sec ^4(e+f x)}{(a (1+\sin (e+f x)))^{3/2}}-\frac {2 \sec ^2(e+f x) \tan ^2(e+f x)}{(a (1+\sin (e+f x)))^{3/2}}\right ) \, dx\\ &=-\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}+2 \int \frac {\sec ^2(e+f x) \tan ^2(e+f x)}{(a (1+\sin (e+f x)))^{3/2}} \, dx-\frac {\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 a f}-\int \frac {\sec ^4(e+f x)}{(a (1+\sin (e+f x)))^{3/2}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{6 f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {\sec ^4(e+f x) \left (-\frac {3 a}{2}+6 a \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 a^2}-\frac {3 \int \frac {\sec ^4(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 a}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{6 f (a+a \sin (e+f x))^{3/2}}+\frac {\sec ^3(e+f x)}{4 a f \sqrt {a+a \sin (e+f x)}}-\frac {1}{4} \int \frac {\sec ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx-\frac {7}{8} \int \frac {\sec ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {9 \sec (e+f x)}{32 f (a+a \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{6 f (a+a \sin (e+f x))^{3/2}}+\frac {\sec ^3(e+f x)}{4 a f \sqrt {a+a \sin (e+f x)}}-\frac {5 \int \frac {\sec ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a}-\frac {35 \int \frac {\sec ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{64 a}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}-\frac {\cos (e+f x)}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {9 \sec (e+f x)}{32 f (a+a \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{6 f (a+a \sin (e+f x))^{3/2}}-\frac {45 \sec (e+f x)}{64 a f \sqrt {a+a \sin (e+f x)}}+\frac {\sec ^3(e+f x)}{4 a f \sqrt {a+a \sin (e+f x)}}-\frac {15}{64} \int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx-\frac {105}{128} \int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {7 \cos (e+f x)}{256 f (a+a \sin (e+f x))^{3/2}}+\frac {9 \sec (e+f x)}{32 f (a+a \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{6 f (a+a \sin (e+f x))^{3/2}}-\frac {45 \sec (e+f x)}{64 a f \sqrt {a+a \sin (e+f x)}}+\frac {\sec ^3(e+f x)}{4 a f \sqrt {a+a \sin (e+f x)}}-\frac {15 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{256 a}-\frac {105 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{512 a}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {7 \cos (e+f x)}{256 f (a+a \sin (e+f x))^{3/2}}+\frac {9 \sec (e+f x)}{32 f (a+a \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{6 f (a+a \sin (e+f x))^{3/2}}-\frac {45 \sec (e+f x)}{64 a f \sqrt {a+a \sin (e+f x)}}+\frac {\sec ^3(e+f x)}{4 a f \sqrt {a+a \sin (e+f x)}}+\frac {15 \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 )}{128 a f}+\frac {105 \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 )}{256 a f}\\ &=\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{256 \sqrt {2} a^{3/2} f}+\frac {7 \cos (e+f x)}{256 f (a+a \sin (e+f x))^{3/2}}+\frac {9 \sec (e+f x)}{32 f (a+a \sin (e+f x))^{3/2}}-\frac {\sec ^3(e+f x)}{6 f (a+a \sin (e+f x))^{3/2}}-\frac {45 \sec (e+f x)}{64 a f \sqrt {a+a \sin (e+f x)}}+\frac {\sec ^3(e+f x)}{4 a f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 334, normalized size = 1.89 \[ \frac {-\frac {192 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}+\frac {32 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-171 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2+342 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-\frac {248 \sin \left (\frac {1}{2} (e+f x)\right )}{\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )}-\frac {32}{\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}+(-21-21 i) (-1)^{3/4} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (e+f x)\right )-1\right )\right )+124}{768 f (a (\sin (e+f x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 270, normalized size = 1.53 \[ \frac {21 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{5} - 2 \, \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right )^{3}\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (21 \, \cos \left (f x + e\right )^{4} - 324 \, \cos \left (f x + e\right )^{2} - 12 \, {\left (45 \, \cos \left (f x + e\right )^{2} - 16\right )} \sin \left (f x + e\right ) + 64\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3072 \, {\left (a^{2} f \cos \left (f x + e\right )^{5} - 2 \, a^{2} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) - 2 \, a^{2} f \cos \left (f x + e\right )^{3}\right )}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 289, normalized size = 1.63 \[ -\frac {\left (-1080 a^{\frac {9}{2}}-21 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (384 a^{\frac {9}{2}}+84 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (f x +e \right )+42 a^{\frac {9}{2}} \left (\cos ^{4}\left (f x +e \right )\right )+\left (-648 a^{\frac {9}{2}}-63 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+128 a^{\frac {9}{2}}+84 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}}{1536 a^{\frac {11}{2}} \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{4}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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