Optimal. Leaf size=90 \[ \frac {3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac {3 (A+4 C) \sin (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right )}{4 b d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4045, 3772, 2643} \[ \frac {3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac {3 (A+4 C) \sin (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right )}{4 b d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3772
Rule 4045
Rubi steps
\begin {align*} \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx &=\frac {3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}+\frac {(A+4 C) \int (b \sec (c+d x))^{2/3} \, dx}{4 b^2}\\ &=\frac {3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}+\frac {\left ((A+4 C) \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 )^{2/3}} \, dx}{4 b^2}\\ &=-\frac {3 (A+4 C) \cos (c+d x) \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{4 b^2 d \sqrt {\sin ^2(c+d x)}}+\frac {3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.53, size = 124, normalized size = 1.38 \[ -\frac {3 i e^{-i (c+d x)} \left (2 (A+4 C) e^{2 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};-e^{2 i (c+d x)}\right )+A \left (-1+e^{4 i (c+d x)}\right )\right )}{8 b d \left (1+e^{2 i (c+d x)}\right ) \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{b^{2} \sec \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.05, size = 0, normalized size = 0.00 \[ \int \frac {A +C \left (\sec ^{2}\left (d x +c \right )\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {C \sec \left (d x + c\right )^{2} + A}{\left (b \sec \left (d x + c\right )\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \]
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 \[ \int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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