Optimal. Leaf size=227 \[ \frac {3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}-\frac {3 (C (2+3 m)+A (5+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1-3 m);\frac {1}{6} (7-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (1-3 m) (5+3 m) \sqrt {\sin ^2(c+d x)}}+\frac {3 B \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-2-3 m);\frac {1}{6} (4-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.14, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {20, 4132, 3857,
2722, 4131} \begin {gather*} -\frac {3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1-3 m);\frac {1}{6} (7-3 m);\cos ^2(c+d x)\right )}{d (1-3 m) (3 m+5) \sqrt {\sin ^2(c+d x)}}+\frac {3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-3 m-2);\frac {1}{6} (4-3 m);\cos ^2(c+d x)\right )}{d (3 m+2) \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m+1}(c+d x)}{d (3 m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 3857
Rule 4131
Rule 4132
Rubi steps
\begin {align*} \int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {(b \sec (c+d x))^{2/3} \int \sec ^{\frac {2}{3}+m}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx}{\sec ^{\frac {2}{3}}(c+d x)}\\ &=\frac {(b \sec (c+d x))^{2/3} \int \sec ^{\frac {2}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{\sec ^{\frac {2}{3}}(c+d x)}+\frac {\left (B (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac {5}{3}+m}(c+d x) \, dx}{\sec ^{\frac {2}{3}}(c+d x)}\\ &=\frac {3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}+\frac {\left (\left (C \left (\frac {2}{3}+m\right )+A \left (\frac {5}{3}+m\right )\right ) (b \sec (c+d x))^{2/3}\right ) \int \sec ^{\frac {2}{3}+m}(c+d x) \, dx}{\left (\frac {5}{3}+m\right ) \sec ^{\frac {2}{3}}(c+d x)}+\left (B \cos ^{\frac {2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac {5}{3}-m}(c+d x) \, dx\\ &=\frac {3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}+\frac {3 B \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-2-3 m);\frac {1}{6} (4-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt {\sin ^2(c+d x)}}+\frac {\left (\left (C \left (\frac {2}{3}+m\right )+A \left (\frac {5}{3}+m\right )\right ) \cos ^{\frac {2}{3}+m}(c+d x) \sec ^m(c+d x) (b \sec (c+d x))^{2/3}\right ) \int \cos ^{-\frac {2}{3}-m}(c+d x) \, dx}{\frac {5}{3}+m}\\ &=\frac {3 C \sec ^{1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (5+3 m)}-\frac {3 (C (2+3 m)+A (5+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (1-3 m);\frac {1}{6} (7-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (1-3 m) (5+3 m) \sqrt {\sin ^2(c+d x)}}+\frac {3 B \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-2-3 m);\frac {1}{6} (4-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \sin (c+d x)}{d (2+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 8.53, size = 547, normalized size = 2.41 \begin {gather*} -\frac {3 i 2^{\frac {5}{3}+m} e^{-\frac {1}{3} i d (2+3 m) x} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{\frac {2}{3}+m} \left (1+e^{2 i (c+d x)}\right )^{\frac {2}{3}+m} \left (\frac {A e^{4 i c+\frac {1}{3} i d (14+3 m) x} \, _2F_1\left (\frac {7}{3}+\frac {m}{2},\frac {8}{3}+m;\frac {1}{6} (20+3 m);-e^{2 i (c+d x)}\right )}{14+3 m}+\frac {A e^{\frac {1}{3} i d (2+3 m) x} \, _2F_1\left (\frac {8}{3}+m,\frac {1}{6} (2+3 m);\frac {1}{6} (8+3 m);-e^{2 i (c+d x)}\right )}{2+3 m}+\frac {2 B e^{\frac {1}{3} i (3 c+d (5+3 m) x)} \, _2F_1\left (\frac {8}{3}+m,\frac {1}{6} (5+3 m);\frac {1}{6} (11+3 m);-e^{2 i (c+d x)}\right )}{5+3 m}+\frac {2 A e^{\frac {1}{3} i (6 c+d (8+3 m) x)} \, _2F_1\left (\frac {8}{3}+m,\frac {1}{6} (8+3 m);\frac {7}{3}+\frac {m}{2};-e^{2 i (c+d x)}\right )}{8+3 m}+\frac {4 C e^{\frac {1}{3} i (6 c+d (8+3 m) x)} \, _2F_1\left (\frac {8}{3}+m,\frac {1}{6} (8+3 m);\frac {7}{3}+\frac {m}{2};-e^{2 i (c+d x)}\right )}{8+3 m}+\frac {2 B e^{\frac {1}{3} i (9 c+d (11+3 m) x)} \, _2F_1\left (\frac {8}{3}+m,\frac {1}{6} (11+3 m);\frac {1}{6} (17+3 m);-e^{2 i (c+d x)}\right )}{11+3 m}\right ) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {8}{3}}(c+d x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.40, size = 0, normalized size = 0.00 \[\int \left (\sec ^{m}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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