Optimal. Leaf size=167 \[ -\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{128 \sqrt {2} a^{5/2} f}+\frac {11 \sec (e+f x)}{96 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {11 \cos (e+f x)}{128 a f (a \sin (e+f x)+a)^{3/2}}+\frac {17 \sec (e+f x)}{48 a f (a \sin (e+f x)+a)^{3/2}}-\frac {\sec (e+f x)}{6 f (a \sin (e+f x)+a)^{5/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2712, 2859, 2687, 2650, 2649, 206} \[ \frac {11 \sec (e+f x)}{96 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{128 \sqrt {2} a^{5/2} f}-\frac {11 \cos (e+f x)}{128 a f (a \sin (e+f x)+a)^{3/2}}+\frac {17 \sec (e+f x)}{48 a f (a \sin (e+f x)+a)^{3/2}}-\frac {\sec (e+f x)}{6 f (a \sin (e+f x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2687
Rule 2712
Rule 2859
Rubi steps
\begin {align*} \int \frac {\tan ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}+\frac {\int \frac {\sec ^2(e+f x) \left (-\frac {5 a}{2}+6 a \sin (e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx}{6 a^2}\\ &=-\frac {\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}+\frac {17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac {11 \int \frac {\sec ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{96 a^2}\\ &=-\frac {\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}+\frac {17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac {11 \sec (e+f x)}{96 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {11 \int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx}{64 a}\\ &=-\frac {\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}-\frac {11 \cos (e+f x)}{128 a f (a+a \sin (e+f x))^{3/2}}+\frac {17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac {11 \sec (e+f x)}{96 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {11 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{256 a^2}\\ &=-\frac {\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}-\frac {11 \cos (e+f x)}{128 a f (a+a \sin (e+f x))^{3/2}}+\frac {17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac {11 \sec (e+f x)}{96 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {11 \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^2 f}\\ &=-\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{128 \sqrt {2} a^{5/2} f}-\frac {\sec (e+f x)}{6 f (a+a \sin (e+f x))^{5/2}}-\frac {11 \cos (e+f x)}{128 a f (a+a \sin (e+f x))^{3/2}}+\frac {17 \sec (e+f x)}{48 a f (a+a \sin (e+f x))^{3/2}}+\frac {11 \sec (e+f x)}{96 a^2 f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 284, normalized size = 1.70 \[ \frac {\frac {48 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}+15 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4-30 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3+52 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2-104 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )}+(33+33 i) (-1)^{3/4} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (e+f x)\right )-1\right )\right )-32}{384 f (a (\sin (e+f x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 279, normalized size = 1.67 \[ \frac {33 \, \sqrt {2} {\left (3 \, \cos \left (f x + e\right )^{3} + {\left (\cos \left (f x + e\right )^{3} - 4 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 4 \, \cos \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (77 \, \cos \left (f x + e\right )^{2} + {\left (33 \, \cos \left (f x + e\right )^{2} - 224\right )} \sin \left (f x + e\right ) - 160\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{1536 \, {\left (3 \, a^{3} f \cos \left (f x + e\right )^{3} - 4 \, a^{3} f \cos \left (f x + e\right ) + {\left (a^{3} f \cos \left (f x + e\right )^{3} - 4 \, a^{3} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
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 = 0.96, size = 266, normalized size = 1.59 \[ -\frac {\left (66 a^{\frac {7}{2}}-33 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (-448 a^{\frac {7}{2}}+132 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (f x +e \right )+\left (154 a^{\frac {7}{2}}-99 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right )-320 a^{\frac {7}{2}}+132 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}}{768 a^{\frac {11}{2}} \left (1+\sin \left (f x +e \right )\right )^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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