Optimal. Leaf size=145 \[ \frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}-\frac {3 (A-C (2-3 m)+3 A m) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (5-3 m) (1+3 m) (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 145, 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 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+1) (b \sec (c+d x))^{2/3}}-\frac {3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right )}{d (5-3 m) (3 m+1) \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 2722
Rule 3857
Rule 4131
Rubi steps
\begin {align*} \int \frac {\sec ^m(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx &=\frac {\sec ^{\frac {2}{3}}(c+d x) \int \sec ^{-\frac {2}{3}+m}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx}{(b \sec (c+d x))^{2/3}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}+\frac {\left (\left (C \left (-\frac {2}{3}+m\right )+A \left (\frac {1}{3}+m\right )\right ) \sec ^{\frac {2}{3}}(c+d x)\right ) \int \sec ^{-\frac {2}{3}+m}(c+d x) \, dx}{\left (\frac {1}{3}+m\right ) (b \sec (c+d x))^{2/3}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}+\frac {\left (\left (C \left (-\frac {2}{3}+m\right )+A \left (\frac {1}{3}+m\right )\right ) \cos ^{\frac {1}{3}+m}(c+d x) \sec ^{1+m}(c+d x)\right ) \int \cos ^{\frac {2}{3}-m}(c+d x) \, dx}{\left (\frac {1}{3}+m\right ) (b \sec (c+d x))^{2/3}}\\ &=\frac {3 C \sec ^{1+m}(c+d x) \sin (c+d x)}{d (1+3 m) (b \sec (c+d x))^{2/3}}-\frac {3 (A-C (2-3 m)+3 A m) \, _2F_1\left (\frac {1}{2},\frac {1}{6} (5-3 m);\frac {1}{6} (11-3 m);\cos ^2(c+d x)\right ) \sec ^{-1+m}(c+d x) \sin (c+d x)}{d (5-3 m) (1+3 m) (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.39, size = 311, normalized size = 2.14 \begin {gather*} -\frac {3 i 2^{\frac {1}{3}+m} \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 (A e^{4 i (c+d x)} \left (-8+6 m+9 m^2\right ) \, _2F_1\left (\frac {5}{3}+\frac {m}{2},\frac {4}{3}+m;\frac {8}{3}+\frac {m}{2};-e^{2 i (c+d x)}\right )+(10+3 m) \left (A (4+3 m) \, _2F_1\left (\frac {4}{3}+m,\frac {1}{6} (-2+3 m);\frac {1}{6} (4+3 m);-e^{2 i (c+d x)}\right )+2 (A+2 C) e^{2 i (c+d x)} (-2+3 m) \, _2F_1\left (\frac {4}{3}+m,\frac {1}{6} (4+3 m);\frac {5}{3}+\frac {m}{2};-e^{2 i (c+d x)}\right )\right )\right ) \left (A+C \sec ^2(c+d x)\right )}{d (-2+3 m) (4+3 m) (10+3 m) (A+2 C+A \cos (2 c+2 d x)) \sec ^{\frac {4}{3}}(c+d x) (b \sec (c+d x))^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.39, size = 0, normalized size = 0.00 \[\int \frac {\left (\sec ^{m}\left (d x +c \right )\right ) \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]
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 \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 \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{m}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, 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 = -0.01 \begin {gather*} \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 )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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