Optimal. Leaf size=147 \[ -\frac {3 (4 A+C) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{8 d (b \sec (c+d x))^{2/3} \sqrt {\sin ^2(c+d x)}}+\frac {3 B \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{b d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sqrt [3]{b \sec (c+d x)} \tan (c+d x)}{4 b d} \]
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Rubi [A]
time = 0.10, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {16, 4132, 3857,
2722, 4131} \begin {gather*} -\frac {3 (4 A+C) \sin (c+d x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right )}{8 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac {3 B \sin (c+d x) \sqrt [3]{b \sec (c+d x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )}{b d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \tan (c+d x) \sqrt [3]{b \sec (c+d x)}}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3857
Rule 4131
Rule 4132
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(b \sec (c+d x))^{2/3}} \, dx &=\frac {\int \sqrt [3]{b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx}{b}\\ &=\frac {\int \sqrt [3]{b \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx}{b}+\frac {B \int (b \sec (c+d x))^{4/3} \, dx}{b^2}\\ &=\frac {3 C \sqrt [3]{b \sec (c+d x)} \tan (c+d x)}{4 b d}+\frac {(4 A+C) \int \sqrt [3]{b \sec (c+d x)} \, dx}{4 b}+\frac {\left (B \sqrt [3]{\frac {\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)}\right ) \int \frac {1}{\left (\frac {\cos (c+d x)}{b}\right )^{4/3}} \, dx}{b^2}\\ &=\frac {3 B \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{b d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sqrt [3]{b \sec (c+d x)} \tan (c+d x)}{4 b d}+\frac {\left ((4 A+C) \sqrt [3]{\frac {\cos (c+d x)}{b}} \sqrt [3]{b \sec (c+d x)}\right ) \int \frac {1}{\sqrt [3]{\frac {\cos (c+d x)}{b}}} \, dx}{4 b}\\ &=\frac {3 B \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{b d \sqrt {\sin ^2(c+d x)}}-\frac {3 (4 A+C) \cos (c+d x) \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\cos ^2(c+d x)\right ) \sqrt [3]{b \sec (c+d x)} \sin (c+d x)}{8 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \sqrt [3]{b \sec (c+d x)} \tan (c+d x)}{4 b d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.30, size = 305, normalized size = 2.07 \begin {gather*} \frac {3 \sqrt [3]{b \sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {i \sqrt [3]{2} e^{-i (c+d x)} \sqrt [3]{\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (4 B \left (1+e^{2 i (c+d x)}\right )+4 B \left (-1+e^{2 i c}\right ) \sqrt [3]{1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-e^{2 i (c+d x)}\right )+(4 A+C) e^{i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt [3]{1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-e^{2 i (c+d x)}\right )\right )}{-1+e^{2 i c}}+\sqrt [3]{\sec (c+d x)} (4 B \cos (d x) \csc (c)+C \tan (c+d x))\right )}{2 b d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {7}{3}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\sec \left (d x +c \right ) \left (A +B \sec \left (d x +c \right )+C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]
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 + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\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 {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\cos \left (c+d\,x\right )\,{\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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