Optimal. Leaf size=78 \[ \frac {3 a d \sin (e+f x) (d \sec (e+f x))^{2/3} \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\cos ^2(e+f x)\right )}{2 f \sqrt {\sin ^2(e+f x)}}+\frac {3 b (d \sec (e+f x))^{5/3}}{5 f} \]
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Rubi [A] time = 0.06, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3486, 3772, 2643} \[ \frac {3 a d \sin (e+f x) (d \sec (e+f x))^{2/3} \text {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{2},\frac {2}{3},\cos ^2(e+f x)\right )}{2 f \sqrt {\sin ^2(e+f x)}}+\frac {3 b (d \sec (e+f x))^{5/3}}{5 f} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3486
Rule 3772
Rubi steps
\begin {align*} \int (d \sec (e+f x))^{5/3} (a+b \tan (e+f x)) \, dx &=\frac {3 b (d \sec (e+f x))^{5/3}}{5 f}+a \int (d \sec (e+f x))^{5/3} \, dx\\ &=\frac {3 b (d \sec (e+f x))^{5/3}}{5 f}+\left (a \left (\frac {\cos (e+f x)}{d}\right )^{2/3} (d \sec (e+f x))^{2/3}\right ) \int \frac {1}{\left (\frac {\cos (e+f x)}{d}\right )^{5/3}} \, dx\\ &=\frac {3 b (d \sec (e+f x))^{5/3}}{5 f}+\frac {3 a d \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\cos ^2(e+f x)\right ) (d \sec (e+f x))^{2/3} \sin (e+f x)}{2 f \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 126, normalized size = 1.62 \[ \frac {d (d \sec (e+f x))^{2/3} (a+b \tan (e+f x)) \left (3 \cos ^2(e+f x)^{2/3} (5 a \sin (2 (e+f x))+4 b)-10 a \sin (e+f x) \cos ^3(e+f x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right )\right )}{20 f \cos ^2(e+f x)^{2/3} (a \cos (e+f x)+b \sin (e+f x))} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b d \sec \left (f x + e\right ) \tan \left (f x + e\right ) + a d \sec \left (f x + e\right )\right )} \left (d \sec \left (f x + e\right )\right )^{\frac {2}{3}}, x\right ) \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{\frac {5}{3}} \left (a +b \tan \left (f x +e \right )\right )\, dx \]
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 \[ \int \left (d \sec \left (f x + e\right )\right )^{\frac {5}{3}} {\left (b \tan \left (f x + e\right ) + a\right )}\,{d x} \]
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 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/3}\,\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \left (d \sec {\left (e + f x \right )}\right )^{\frac {5}{3}} \left (a + b \tan {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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