Optimal. Leaf size=284 \[ -\frac {\sqrt {2} \left (-3 a^2 C+7 a b B-4 b^2 C\right ) \sin (c+d x) \sqrt [3]{a+b \cos (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right )}{7 b^2 d \sqrt {\cos (c+d x)+1} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\sqrt {2} (a+b) (7 b B-3 a C) \sin (c+d x) \sqrt [3]{a+b \cos (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right )}{7 b^2 d \sqrt {\cos (c+d x)+1} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}+\frac {3 C \sin (c+d x) (a+b \cos (c+d x))^{4/3}}{7 b d} \]
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Rubi [A] time = 0.33, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3023, 2756, 2665, 139, 138} \[ -\frac {\sqrt {2} \left (-3 a^2 C+7 a b B-4 b^2 C\right ) \sin (c+d x) \sqrt [3]{a+b \cos (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {1}{3};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right )}{7 b^2 d \sqrt {\cos (c+d x)+1} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\sqrt {2} (a+b) (7 b B-3 a C) \sin (c+d x) \sqrt [3]{a+b \cos (c+d x)} F_1\left (\frac {1}{2};\frac {1}{2},-\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\cos (c+d x)),\frac {b (1-\cos (c+d x))}{a+b}\right )}{7 b^2 d \sqrt {\cos (c+d x)+1} \sqrt [3]{\frac {a+b \cos (c+d x)}{a+b}}}+\frac {3 C \sin (c+d x) (a+b \cos (c+d x))^{4/3}}{7 b d} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 2665
Rule 2756
Rule 3023
Rubi steps
\begin {align*} \int \sqrt [3]{a+b \cos (c+d x)} \left (B \cos (c+d x)+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 \int \sqrt [3]{a+b \cos (c+d x)} \left (\frac {4 b C}{3}+\frac {1}{3} (7 b B-3 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 {(7 b B-3 a C) \int (a+b \cos (c+d x))^{4/3} \, dx}{7 b^2}-\frac {\left (7 a b B-3 a^2 C-4 b^2 C\right ) \int \sqrt [3]{a+b \cos (c+d x)} \, dx}{7 b^2}\\ &=\frac {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}-\frac {((7 b B-3 a C) \sin (c+d x)) \operatorname {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 b B-3 a^2 C-4 b^2 C\right ) \sin (c+d x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{\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 {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}+\frac {\left ((-a-b) (7 b B-3 a C) \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)\right ) \operatorname {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 b B-3 a^2 C-4 b^2 C\right ) \sqrt [3]{a+b \cos (c+d x)} \sin (c+d x)\right ) \operatorname {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 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 {3 C (a+b \cos (c+d x))^{4/3} \sin (c+d x)}{7 b d}+\frac {\sqrt {2} (a+b) (7 b B-3 a C) F_1\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 b B-3 a^2 C-4 b^2 C\right ) F_1\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}}}\\ \end {align*}
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Mathematica [A] time = 3.01, size = 289, normalized size = 1.02 \[ -\frac {3 \csc (c+d x) \sqrt [3]{a+b \cos (c+d x)} \left (\left (-3 a^2 C+7 a b B+16 b^2 C\right ) \sqrt {-\frac {b (\cos (c+d x)-1)}{a+b}} \sqrt {-\frac {b (\cos (c+d x)+1)}{a-b}} (a+b \cos (c+d x)) F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};\frac {a+b \cos (c+d x)}{a-b},\frac {a+b \cos (c+d x)}{a+b}\right )+4 \left (b^2-a^2\right ) (7 b B-3 a C) \sqrt {-\frac {b (\cos (c+d x)-1)}{a+b}} \sqrt {-\frac {b (\cos (c+d x)+1)}{a-b}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};\frac {a+b \cos (c+d x)}{a-b},\frac {a+b \cos (c+d x)}{a+b}\right )-4 b^2 \sin ^2(c+d x) (a C+7 b B+4 b C \cos (c+d x))\right )}{112 b^3 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \left (a +b \cos \left (d x +c \right )\right )^{\frac {1}{3}} \left (B \cos \left (d x +c \right )+C \left (\cos ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{1/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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