Optimal. Leaf size=92 \[ \frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{7 d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 92, 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^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \sin (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )}{7 d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (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 ^3(c+d x)}{\sqrt [3]{b \cos (c+d x)}} \, dx &=b^3 \int \frac {A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{10/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {1}{7} (b (4 A+7 C)) \int \frac {1}{(b \cos (c+d x))^{4/3}} \, dx\\ &=\frac {3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 (4 A+7 C) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{7 d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.31, size = 481, normalized size = 5.23 \begin {gather*} b \left (-\frac {i (4 A+7 C) \cos ^{\frac {10}{3}}(c+d x) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \left (C+A \sec ^2(c+d x)\right ) \left (-\frac {3 i e^{-i d x} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right ) \sqrt [3]{1+e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)}}{2^{2/3} d \sqrt [3]{e^{-i d x} \left (\left (1+e^{2 i d x}\right ) \cos (c)+i \left (-1+e^{2 i d x}\right ) \sin (c)\right )}}-\frac {3 i e^{i d x} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right ) \sqrt [3]{1+e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)}}{2\ 2^{2/3} d \sqrt [3]{e^{-i d x} \left (\left (1+e^{2 i d x}\right ) \cos (c)+i \left (-1+e^{2 i d x}\right ) \sin (c)\right )}}\right )}{7 (b \cos (c+d x))^{4/3} (2 A+C+C \cos (2 c+2 d x))}+\frac {\cos ^4(c+d x) \left (C+A \sec ^2(c+d x)\right ) \left (\frac {6 (4 A+7 C) \csc (c) \sec (c)}{7 d}+\frac {6 A \sec (c) \sec ^3(c+d x) \sin (d x)}{7 d}+\frac {6 \sec (c) \sec (c+d x) (4 A \sin (d x)+7 C \sin (d x))}{7 d}+\frac {6 A \sec ^2(c+d x) \tan (c)}{7 d}\right )}{(b \cos (c+d x))^{4/3} (2 A+C+C \cos (2 c+2 d x))}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{3}\left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^3\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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