Integrand size = 27, antiderivative size = 277 \[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}-\frac {3 \sqrt {2} a (a+b) C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 d \sqrt {1+\cos (c+d x)} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\sqrt {2} \left (3 a^2 C+b^2 (7 A+4 C)\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 d \sqrt {1+\cos (c+d x)} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}} \]
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Time = 0.37 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3103, 2835, 2744, 144, 143} \[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} \left (3 a^2 C+b^2 (7 A+4 C)\right ) \sin (c+d x) \sqrt [3]{a+b \cos (c+d x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right )}{7 b^2 d \sqrt {\cos (c+d x)+1} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}-\frac {3 \sqrt {2} a C (a+b) \sin (c+d x) \sqrt [3]{a+b \cos (c+d x)} \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right )}{7 b^2 d \sqrt {\cos (c+d x)+1} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}+\frac {3 C \sin (c+d x) (a+b \cos (c+d x))^{4/3}}{7 b d} \]
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Rule 143
Rule 144
Rule 2744
Rule 2835
Rule 3103
Rubi steps \begin{align*} \text {integral}& = \frac {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}+\frac {3 \int \sqrt [3]{a+b \cos (c+d x)} \left (\frac {1}{3} b (7 A+4 C)-a C \cos (c+d x)\right ) \, dx}{7 b} \\ & = \frac {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}-\frac {(3 a C) \int (a+b \cos (c+d x))^{4/3} \, dx}{7 b^2}+\frac {1}{7} \left (7 A+\left (4+\frac {3 a^2}{b^2}\right ) C\right ) \int \sqrt [3]{a+b \cos (c+d x)} \, dx \\ & = \frac {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}+\frac {(3 a C \sin (c+d x)) \text {Subst}\left (\int \frac {(a+b x)^{4/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{7 b^2 d \sqrt {1-\cos (c+d x)} \sqrt {1+\cos (c+d x)}}+\frac {\left (\left (-7 A-\left (4+\frac {3 a^2}{b^2}\right ) C\right ) \sin (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{7 d \sqrt {1-\cos (c+d x)} \sqrt {1+\cos (c+d x)}} \\ & = \frac {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}-\frac {\left (3 a (-a-b) C \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{4/3}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{7 b^2 d \sqrt {1-\cos (c+d x)} \sqrt {1+\cos (c+d x)} \sqrt [3]{-\frac {a+b \cos (c+d x)}{-a-b}}}+\frac {\left (\left (-7 A-\left (4+\frac {3 a^2}{b^2}\right ) C\right ) \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt [3]{-\frac {a}{-a-b}-\frac {b x}{-a-b}}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{7 d \sqrt {1-\cos (c+d x)} \sqrt {1+\cos (c+d x)} \sqrt [3]{-\frac {a+b \cos (c+d x)}{-a-b}}} \\ & = \frac {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}-\frac {3 \sqrt {2} a (a+b) C \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 d \sqrt {1+\cos (c+d x)} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\sqrt {2} \left (7 A+\left (4+\frac {3 a^2}{b^2}\right ) C\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right ) \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)}{7 d \sqrt {1+\cos (c+d x)} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}} \\ \end{align*}
Time = 2.00 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {3 \sqrt [3]{a+b \cos (c+d x)} \csc (c+d x) \left (12 a \left (a^2-b^2\right ) C \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},\frac {a+b \cos (c+d x)}{a-b},\frac {a+b \cos (c+d x)}{a+b}\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {-\frac {b (1+\cos (c+d x))}{a-b}}+\left (28 A b^2-3 a^2 C+16 b^2 C\right ) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},\frac {a+b \cos (c+d x)}{a-b},\frac {a+b \cos (c+d x)}{a+b}\right ) \sqrt {-\frac {b (-1+\cos (c+d x))}{a+b}} \sqrt {\frac {b (1+\cos (c+d x))}{-a+b}} (a+b \cos (c+d x))-4 b^2 C (a+4 b \cos (c+d x)) \sin ^2(c+d x)\right )}{112 b^3 d} \]
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\[\int \left (a +\cos \left (d x +c \right ) b \right )^{\frac {1}{3}} \left (A +C \left (\cos ^{2}\left (d x +c \right )\right )\right )d x\]
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\[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sqrt [3]{a + b \cos {\left (c + d x \right )}}\, dx \]
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\[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
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\[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \,d x } \]
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Timed out. \[ \int \sqrt [3]{a+b \cos (c+d x)} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \]
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