Optimal. Leaf size=125 \[ \frac {3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} F_1\left (\frac {7}{3};-\frac {1}{2},-\frac {1}{2};\frac {10}{3};\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right )}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 0.13, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2704, 138} \[ \frac {3 \cos (e+f x) (c+d \sin (e+f x))^{7/3} F_1\left (\frac {7}{3};-\frac {1}{2},-\frac {1}{2};\frac {10}{3};\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right )}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 2704
Rubi steps
\begin {align*} \int \cos ^2(e+f x) (c+d \sin (e+f x))^{4/3} \, dx &=\frac {\cos (e+f x) \operatorname {Subst}\left (\int (c+d x)^{4/3} \sqrt {-\frac {d}{c-d}-\frac {d x}{c-d}} \sqrt {\frac {d}{c+d}-\frac {d x}{c+d}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}}\\ &=\frac {3 F_1\left (\frac {7}{3};-\frac {1}{2},-\frac {1}{2};\frac {10}{3};\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right ) \cos (e+f x) (c+d \sin (e+f x))^{7/3}}{7 d f \sqrt {1-\frac {c+d \sin (e+f x)}{c-d}} \sqrt {1-\frac {c+d \sin (e+f x)}{c+d}}}\\ \end {align*}
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Mathematica [B] time = 2.13, size = 301, normalized size = 2.41 \[ -\frac {3 \sec (e+f x) \sqrt [3]{c+d \sin (e+f x)} \left (-3 c \left (4 c^2+51 d^2\right ) \sqrt {-\frac {d (\sin (e+f x)-1)}{c+d}} \sqrt {-\frac {d (\sin (e+f x)+1)}{c-d}} (c+d \sin (e+f x)) F_1\left (\frac {4}{3};\frac {1}{2},\frac {1}{2};\frac {7}{3};\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right )+12 \left (4 c^4+3 c^2 d^2-7 d^4\right ) \sqrt {-\frac {d (\sin (e+f x)-1)}{c+d}} \sqrt {-\frac {d (\sin (e+f x)+1)}{c-d}} F_1\left (\frac {1}{3};\frac {1}{2},\frac {1}{2};\frac {4}{3};\frac {c+d \sin (e+f x)}{c-d},\frac {c+d \sin (e+f x)}{c+d}\right )+4 d^2 \cos ^2(e+f x) \left (-4 c^2-44 c d \sin (e+f x)+14 d^2 \cos (2 (e+f x))+7 d^2\right )\right )}{1120 d^3 f} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (d \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) + c \cos \left (f x + e\right )^{2}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {1}{3}}, x\right ) \]
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 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{2}\left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}\, dx \]
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 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}} \cos \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (e+f\,x\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3} \,d x \]
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 \[ \int \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {4}{3}} \cos ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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