Optimal. Leaf size=92 \[ \frac {3 C \tan (c+d x)}{2 b d \sqrt [3]{b \sec (c+d x)}}-\frac {3 (2 A-C) \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{8 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
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Rubi [A] time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {16, 4046, 3772, 2643} \[ \frac {3 C \tan (c+d x)}{2 b d \sqrt [3]{b \sec (c+d x)}}-\frac {3 (2 A-C) \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{8 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{4/3}} \, dx &=\frac {\int \frac {A+C \sec ^2(c+d x)}{\sqrt [3]{b \sec (c+d x)}} \, dx}{b}\\ &=\frac {3 C \tan (c+d x)}{2 b d \sqrt [3]{b \sec (c+d x)}}+\frac {(2 A-C) \int \frac {1}{\sqrt [3]{b \sec (c+d x)}} \, dx}{2 b}\\ &=\frac {3 C \tan (c+d x)}{2 b d \sqrt [3]{b \sec (c+d x)}}+\frac {\left ((2 A-C) \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \sqrt [3]{\frac {\cos (c+d x)}{b}} \, dx}{2 b}\\ &=-\frac {3 (2 A-C) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{8 b^2 d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \tan (c+d x)}{2 b d \sqrt [3]{b \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.65, size = 130, normalized size = 1.41 \[ -\frac {3 i \left ((2 A-C) e^{2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {11}{6};-e^{2 i (c+d x)}\right )-5 \left (A e^{2 i (c+d x)}+A-C e^{2 i (c+d x)}\right )\right )}{5 b d \left (1+e^{2 i (c+d x)}\right ) \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{b^{2} \sec \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.87, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x +c \right ) \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\cos \left (c+d\,x\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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