Optimal. Leaf size=119 \[ \frac {3 A \sin (c+d x) (b \sec (c+d x))^{5/3} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};\cos ^2(c+d x)\right )}{5 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 B \sin (c+d x) (b \sec (c+d x))^{8/3} \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};\cos ^2(c+d x)\right )}{8 b^2 d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {16, 3787, 3772, 2643} \[ \frac {3 A \sin (c+d x) (b \sec (c+d x))^{5/3} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};\cos ^2(c+d x)\right )}{5 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 B \sin (c+d x) (b \sec (c+d x))^{8/3} \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};\cos ^2(c+d x)\right )}{8 b^2 d \sqrt {\sin ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3772
Rule 3787
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx &=\frac {\int (b \sec (c+d x))^{8/3} (A+B \sec (c+d x)) \, dx}{b^2}\\ &=\frac {A \int (b \sec (c+d x))^{8/3} \, dx}{b^2}+\frac {B \int (b \sec (c+d x))^{11/3} \, dx}{b^3}\\ &=\frac {\left (A \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac {1}{\left (\frac {\cos (c+d x)}{b}\right )^{8/3}} \, dx}{b^2}+\frac {\left (B \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac {1}{\left (\frac {\cos (c+d x)}{b}\right )^{11/3}} \, dx}{b^3}\\ &=\frac {3 A \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {1}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/3} \sin (c+d x)}{5 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 B \, _2F_1\left (-\frac {4}{3},\frac {1}{2};-\frac {1}{3};\cos ^2(c+d x)\right ) \sec (c+d x) (b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 90, normalized size = 0.76 \[ -\frac {3 \left (-\tan ^2(c+d x)\right )^{3/2} \csc ^3(c+d x) (b \sec (c+d x))^{2/3} \left (11 A \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {4}{3};\frac {7}{3};\sec ^2(c+d x)\right )+8 B \, _2F_1\left (\frac {1}{2},\frac {11}{6};\frac {17}{6};\sec ^2(c+d x)\right )\right )}{88 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sec \left (d x + c\right )^{3} + A \sec \left (d x + c\right )^{2}\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}, x\right ) \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.76, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{2}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \sec \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}} \left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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