Optimal. Leaf size=93 \[ \frac {3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac {3 (2 A+5 C) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b d \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {16, 3091, 2722}
\begin {gather*} \frac {3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac {3 (2 A+5 C) \sin (c+d x) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right )}{5 b d \sqrt {\sin ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3091
Rubi steps
\begin {align*} \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx &=b^2 \int \frac {A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{8/3}} \, dx\\ &=\frac {3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}+\frac {1}{5} (2 A+5 C) \int \frac {1}{(b \cos (c+d x))^{2/3}} \, dx\\ &=\frac {3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac {3 (2 A+5 C) \sqrt [3]{b \cos (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {1}{2};\frac {7}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{5 b d \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 103, normalized size = 1.11 \begin {gather*} \frac {-\left ((2 A+5 C) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {3}{2};\cos ^2(d x-\text {ArcTan}(\cot (c)))\right ) \sin (2 d x-2 \text {ArcTan}(\cot (c)))\right )+6 A \sqrt [6]{\sin ^2(d x-\text {ArcTan}(\cot (c)))} \tan (c+d x)}{10 d (b \cos (c+d x))^{2/3} \sqrt [6]{\sin ^2(d x-\text {ArcTan}(\cot (c)))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{2}\left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {2}{3}}}\, dx\]
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 \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (b \cos {\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 {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^2\,{\left (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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