Optimal. Leaf size=146 \[ \frac {3 b C \sec ^{2+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (7+3 m)}+\frac {3 b (C (4+3 m)+A (7+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-1-3 m);\frac {1}{6} (5-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (1+3 m) (7+3 m) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 146, 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, 4131, 3857,
2722} \begin {gather*} \frac {3 b (A (3 m+7)+C (3 m+4)) \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-3 m-1);\frac {1}{6} (5-3 m);\cos ^2(c+d x)\right )}{d (3 m+1) (3 m+7) \sqrt {\sin ^2(c+d x)}}+\frac {3 b C \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \sec ^{m+2}(c+d x)}{d (3 m+7)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 3857
Rule 4131
Rubi steps
\begin {align*} \int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {\left (b \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac {4}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{\sqrt [3]{\sec (c+d x)}}\\ &=\frac {3 b C \sec ^{2+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (7+3 m)}+\frac {\left (b \left (C \left (\frac {4}{3}+m\right )+A \left (\frac {7}{3}+m\right )\right ) \sqrt [3]{b \sec (c+d x)}\right ) \int \sec ^{\frac {4}{3}+m}(c+d x) \, dx}{\left (\frac {7}{3}+m\right ) \sqrt [3]{\sec (c+d x)}}\\ &=\frac {3 b C \sec ^{2+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (7+3 m)}+\frac {\left (b \left (C \left (\frac {4}{3}+m\right )+A \left (\frac {7}{3}+m\right )\right ) \cos ^{\frac {1}{3}+m}(c+d x) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)}\right ) \int \cos ^{-\frac {4}{3}-m}(c+d x) \, dx}{\frac {7}{3}+m}\\ &=\frac {3 b C \sec ^{2+m}(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (7+3 m)}+\frac {3 b (C (4+3 m)+A (7+3 m)) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (-1-3 m);\frac {1}{6} (5-3 m);\cos ^2(c+d x)\right ) \sec ^m(c+d x) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{d (1+3 m) (7+3 m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.56, size = 333, normalized size = 2.28 \begin {gather*} -\frac {3 i 2^{\frac {7}{3}+m} e^{-\frac {1}{3} i d (4+3 m) x} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{\frac {4}{3}+m} \left (1+e^{2 i (c+d x)}\right )^{\frac {4}{3}+m} \left (\frac {2 (A+2 C) e^{\frac {1}{3} i (6 c+d (10+3 m) x)} \, _2F_1\left (\frac {5}{3}+\frac {m}{2},\frac {10}{3}+m;\frac {8}{3}+\frac {m}{2};-e^{2 i (c+d x)}\right )}{10+3 m}+\frac {A e^{4 i c+\frac {1}{3} i d (16+3 m) x} \, _2F_1\left (\frac {8}{3}+\frac {m}{2},\frac {10}{3}+m;\frac {1}{6} (22+3 m);-e^{2 i (c+d x)}\right )}{16+3 m}+\frac {A e^{\frac {1}{3} i d (4+3 m) x} \, _2F_1\left (\frac {10}{3}+m,\frac {1}{6} (4+3 m);\frac {5}{3}+\frac {m}{2};-e^{2 i (c+d x)}\right )}{4+3 m}\right ) (b \sec (c+d x))^{4/3} \left (A+C \sec ^2(c+d x)\right )}{d (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {10}{3}}(c+d x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.50, size = 0, normalized size = 0.00 \[\int \left (\sec ^{m}\left (d x +c \right )\right ) \left (b \sec \left (d x +c \right )\right )^{\frac {4}{3}} \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )\, dx\]
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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{4/3}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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