Optimal. Leaf size=149 \[ \frac {3 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 B \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}}+\frac {3 (4 A+7 C) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right ) \sin (c+d x)}{7 b d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.13, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {16, 3100, 2827,
2722} \begin {gather*} \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 b d \sqrt {\sin ^2(c+d x)} \sqrt [3]{b \cos (c+d x)}}+\frac {3 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 B \sin (c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )}{4 d \sqrt {\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2722
Rule 2827
Rule 3100
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx &=b^2 \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{10/3}} \, dx\\ &=\frac {3 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 \int \frac {\frac {7 b^2 B}{3}+\frac {1}{3} b^2 (4 A+7 C) \cos (c+d x)}{(b \cos (c+d x))^{7/3}} \, dx}{7 b}\\ &=\frac {3 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+(b B) \int \frac {1}{(b \cos (c+d x))^{7/3}} \, dx+\frac {1}{7} (4 A+7 C) \int \frac {1}{(b \cos (c+d x))^{4/3}} \, dx\\ &=\frac {3 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac {3 B \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right ) \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3} \sqrt {\sin ^2(c+d x)}}+\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 b d \sqrt [3]{b \cos (c+d x)} \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 118, normalized size = 0.79 \begin {gather*} \frac {3 b^2 \cot (c+d x) \left (4 A \, _2F_1\left (-\frac {7}{6},\frac {1}{2};-\frac {1}{6};\cos ^2(c+d x)\right )+7 \cos (c+d x) \left (B \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};\cos ^2(c+d x)\right )+4 C \cos (c+d x) \, _2F_1\left (-\frac {1}{6},\frac {1}{2};\frac {5}{6};\cos ^2(c+d x)\right )\right )\right ) \sqrt {\sin ^2(c+d x)}}{28 d (b \cos (c+d x))^{10/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.35, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right ) \left (\sec ^{2}\left (d x +c \right )\right )}{\left (b \cos \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx\]
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+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^2\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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