Optimal. Leaf size=147 \[ \frac {3 (C (1-3 m)-A (3 m+2)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4-3 m);\frac {1}{6} (10-3 m);\cos ^2(c+d x)\right )}{d (4-3 m) (3 m+2) \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}}+\frac {3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+2) \sqrt [3]{b \sec (c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 147, 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 = {20, 4046, 3772, 2643} \[ \frac {3 (C (1-3 m)-A (3 m+2)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4-3 m);\frac {1}{6} (10-3 m);\cos ^2(c+d x)\right )}{d (4-3 m) (3 m+2) \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \sec (c+d x)}}+\frac {3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+2) \sqrt [3]{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 20
Rule 2643
Rule 3772
Rule 4046
Rubi steps
\begin {align*} \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx &=\frac {\sqrt [3]{\sec (c+d x)} \int \sec ^{-\frac {1}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{\sqrt [3]{b \sec (c+d x)}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (2+3 m) \sqrt [3]{b \sec (c+d x)}}+\frac {\left (\left (C \left (-\frac {1}{3}+m\right )+A \left (\frac {2}{3}+m\right )\right ) \sqrt [3]{\sec (c+d x)}\right ) \int \sec ^{-\frac {1}{3}+m}(c+d x) \, dx}{\left (\frac {2}{3}+m\right ) \sqrt [3]{b \sec (c+d x)}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (2+3 m) \sqrt [3]{b \sec (c+d x)}}+\frac {\left (\left (C \left (-\frac {1}{3}+m\right )+A \left (\frac {2}{3}+m\right )\right ) \cos ^{\frac {2}{3}+m}(c+d x) \sec ^{1+m}(c+d x)\right ) \int \cos ^{\frac {1}{3}-m}(c+d x) \, dx}{\left (\frac {2}{3}+m\right ) \sqrt [3]{b \sec (c+d x)}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (2+3 m) \sqrt [3]{b \sec (c+d x)}}+\frac {3 (C (1-3 m)-A (2+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (4-3 m);\frac {1}{6} (10-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (4-3 m) (2+3 m) \sqrt [3]{b \sec (c+d x)} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] time = 8.54, size = 311, normalized size = 2.12 \[ -\frac {3 i 2^{m+\frac {2}{3}} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{m-\frac {1}{3}} \left (1+e^{2 i (c+d x)}\right )^{m-\frac {1}{3}} \left (A+C \sec ^2(c+d x)\right ) \left ((3 m-1) e^{2 i (c+d x)} \left (2 (3 m+11) (A+2 C) \, _2F_1\left (m+\frac {5}{3},\frac {1}{6} (3 m+5);\frac {1}{6} (3 m+11);-e^{2 i (c+d x)}\right )+A (3 m+5) e^{2 i (c+d x)} \, _2F_1\left (m+\frac {5}{3},\frac {1}{6} (3 m+11);\frac {1}{6} (3 m+17);-e^{2 i (c+d x)}\right )\right )+A \left (9 m^2+48 m+55\right ) \, _2F_1\left (m+\frac {5}{3},\frac {1}{6} (3 m-1);\frac {1}{6} (3 m+5);-e^{2 i (c+d x)}\right )\right )}{d (3 m-1) (3 m+5) (3 m+11) \sec ^{\frac {5}{3}}(c+d x) \sqrt [3]{b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 C)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{m}}{b \sec \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (\sec ^{m}\left (d x +c \right )\right ) \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{m}}{\left (b \sec \left (d x + c\right )\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}}{\sqrt [3]{b \sec {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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