Optimal. Leaf size=215 \[ \frac {4 \sqrt {2} C F_1\left (\frac {1}{2};-\frac {3}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}+\frac {2 \sqrt {2} (A-C) F_1\left (\frac {1}{2};-\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}} \]
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Rubi [A]
time = 0.17, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3096, 2834,
144, 143, 2863} \begin {gather*} \frac {2 \sqrt {2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};-\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}}+\frac {4 \sqrt {2} C \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};-\frac {3}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{f \sqrt {\cos (e+f x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 2834
Rule 2863
Rule 3096
Rubi steps
\begin {align*} \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx &=(A-C) \int (1+\cos (e+f x)) (a+b \cos (e+f x))^m \, dx+C \int (1+\cos (e+f x))^2 (a+b \cos (e+f x))^m \, dx\\ &=-\frac {(C \sin (e+f x)) \text {Subst}\left (\int \frac {(1+x)^{3/2} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}+\frac {((-A+C) \sin (e+f x)) \text {Subst}\left (\int \frac {\sqrt {1+x} (a+b x)^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}\\ &=-\frac {\left (C (a+b \cos (e+f x))^m \left (-\frac {a+b \cos (e+f x)}{-a-b}\right )^{-m} \sin (e+f x)\right ) \text {Subst}\left (\int \frac {(1+x)^{3/2} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}+\frac {\left ((-A+C) (a+b \cos (e+f x))^m \left (-\frac {a+b \cos (e+f x)}{-a-b}\right )^{-m} \sin (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x}} \, dx,x,\cos (e+f x)\right )}{f \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}\\ &=\frac {4 \sqrt {2} C F_1\left (\frac {1}{2};-\frac {3}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}+\frac {2 \sqrt {2} (A-C) F_1\left (\frac {1}{2};-\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} \sin (e+f x)}{f \sqrt {1+\cos (e+f x)}}\\ \end {align*}
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Mathematica [F]
time = 3.67, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \cos (e+f x))^m \left (A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.47, size = 0, normalized size = 0.00 \[\int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (A +\left (A +C \right ) \cos \left (f x +e \right )+C \left (\cos ^{2}\left (f x +e \right )\right )\right )\, dx\]
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 \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 = -0.00 \begin {gather*} \int {\left (a+b\,\cos \left (e+f\,x\right )\right )}^m\,\left (C\,{\cos \left (e+f\,x\right )}^2+\left (A+C\right )\,\cos \left (e+f\,x\right )+A\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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