Optimal. Leaf size=93 \[ -\frac {3 b (4 A+7 C) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{28 d (b \sec (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}}+\frac {3 A b^2 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}} \]
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Rubi [A]
time = 0.08, antiderivative size = 93, 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 = {16, 4130, 3857,
2722} \begin {gather*} \frac {3 A b^2 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\frac {3 b (4 A+7 C) \sin (c+d x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right )}{28 d \sqrt {\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 3857
Rule 4130
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+C \sec ^2(c+d x)\right )}{\sqrt [3]{b \sec (c+d x)}} \, dx &=b^2 \int \frac {A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{7/3}} \, dx\\ &=\frac {3 A b^2 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}+\frac {1}{7} (4 A+7 C) \int \frac {1}{\sqrt [3]{b \sec (c+d x)}} \, dx\\ &=\frac {3 A b^2 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}+\frac {1}{7} \left ((4 A+7 C) \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \sqrt [3]{\frac {\cos (c+d x)}{b}} \, dx\\ &=-\frac {3 (4 A+7 C) \cos ^2(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\cos ^2(c+d x)\right ) (b \sec (c+d x))^{2/3} \sin (c+d x)}{28 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 A b^2 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 89, normalized size = 0.96 \begin {gather*} -\frac {3 \cot (c+d x) \left (A \cos ^2(c+d x) \, _2F_1\left (-\frac {7}{6},\frac {1}{2};-\frac {1}{6};\sec ^2(c+d x)\right )+7 C \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\sec ^2(c+d x)\right )\right ) \sqrt {-\tan ^2(c+d x)}}{7 d \sqrt [3]{b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.68, size = 0, normalized size = 0.00 \[\int \frac {\left (\cos ^{2}\left (d x +c \right )\right ) \left (A +C \left (\sec ^{2}\left (d x +c \right )\right )\right )}{\left (b \sec \left (d x +c \right )\right )^{\frac {1}{3}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{2}{\left (c + d x \right )}}{\sqrt [3]{b \sec {\left (c + d x \right )}}}\, 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 {{\cos \left (c+d\,x\right )}^2\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {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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