Optimal. Leaf size=91 \[ \frac {3 b^2 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}-\frac {3 b^3 (A-2 C) \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right )}{5 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}} \]
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Rubi [A] time = 0.12, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {16, 4046, 3772, 2643} \[ \frac {3 b^2 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}-\frac {3 b^3 (A-2 C) \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right )}{5 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \, dx &=b^2 \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{2/3}} \, dx\\ &=\frac {3 b^2 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}+\left (b^2 (A-2 C)\right ) \int \frac {1}{(b \sec (c+d x))^{2/3}} \, dx\\ &=\frac {3 b^2 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}+\left (b^2 (A-2 C) \sqrt [3]{\frac {\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)}\right ) \int \left (\frac {\cos (c+d x)}{b}\right )^{2/3} \, dx\\ &=-\frac {3 b (A-2 C) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{5 d \sqrt {\sin ^2(c+d x)}}+\frac {3 b^2 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 90, normalized size = 0.99 \[ -\frac {3 \sqrt {-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{4/3} \left (2 A \cos ^2(c+d x) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};\sec ^2(c+d x)\right )-C \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\sec ^2(c+d x)\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{3} + A b \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.52, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}} \cos \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \]
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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