Optimal. Leaf size=285 \[ \frac {\sqrt {2} \sin (e+f x) \left (a^2 C+b^2 (A (m+2)+C (m+1))\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt {\cos (e+f x)+1}}-\frac {\sqrt {2} a C (a+b) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m-1;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt {\cos (e+f x)+1}}+\frac {C \sin (e+f x) (a+b \cos (e+f x))^{m+1}}{b f (m+2)} \]
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Rubi [A] time = 0.34, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3024, 2756, 2665, 139, 138} \[ \frac {\sqrt {2} \sin (e+f x) \left (a^2 C+b^2 (A (m+2)+C (m+1))\right ) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt {\cos (e+f x)+1}}-\frac {\sqrt {2} a C (a+b) \sin (e+f x) (a+b \cos (e+f x))^m \left (\frac {a+b \cos (e+f x)}{a+b}\right )^{-m} F_1\left (\frac {1}{2};\frac {1}{2},-m-1;\frac {3}{2};\frac {1}{2} (1-\cos (e+f x)),\frac {b (1-\cos (e+f x))}{a+b}\right )}{b^2 f (m+2) \sqrt {\cos (e+f x)+1}}+\frac {C \sin (e+f x) (a+b \cos (e+f x))^{m+1}}{b f (m+2)} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 2665
Rule 2756
Rule 3024
Rubi steps
\begin {align*} \int (a+b \cos (e+f x))^m \left (A+C \cos ^2(e+f x)\right ) \, dx &=\frac {C (a+b \cos (e+f x))^{1+m} \sin (e+f x)}{b f (2+m)}+\frac {\int (a+b \cos (e+f x))^m (b (C (1+m)+A (2+m))-a C \cos (e+f x)) \, dx}{b (2+m)}\\ &=\frac {C (a+b \cos (e+f x))^{1+m} \sin (e+f x)}{b f (2+m)}-\frac {(a C) \int (a+b \cos (e+f x))^{1+m} \, dx}{b^2 (2+m)}+\frac {\left (a^2 C+b^2 (C (1+m)+A (2+m))\right ) \int (a+b \cos (e+f x))^m \, dx}{b^2 (2+m)}\\ &=\frac {C (a+b \cos (e+f x))^{1+m} \sin (e+f x)}{b f (2+m)}+\frac {(a C \sin (e+f x)) \operatorname {Subst}\left (\int \frac {(a+b x)^{1+m}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}-\frac {\left (\left (a^2 C+b^2 (C (1+m)+A (2+m))\right ) \sin (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}\\ &=\frac {C (a+b \cos (e+f x))^{1+m} \sin (e+f x)}{b f (2+m)}-\frac {\left (a (-a-b) 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 ) \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{1+m}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}-\frac {\left (\left (a^2 C+b^2 (C (1+m)+A (2+m))\right ) (a+b \cos (e+f x))^m \left (-\frac {a+b \cos (e+f x)}{-a-b}\right )^{-m} \sin (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^m}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\cos (e+f x)\right )}{b^2 f (2+m) \sqrt {1-\cos (e+f x)} \sqrt {1+\cos (e+f x)}}\\ &=\frac {C (a+b \cos (e+f x))^{1+m} \sin (e+f x)}{b f (2+m)}-\frac {\sqrt {2} a (a+b) C F_1\left (\frac {1}{2};\frac {1}{2},-1-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)}{b^2 f (2+m) \sqrt {1+\cos (e+f x)}}+\frac {\sqrt {2} \left (a^2 C+b^2 (C (1+m)+A (2+m))\right ) 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)}{b^2 f (2+m) \sqrt {1+\cos (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 26.49, size = 10805, normalized size = 37.91 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \cos \left (f x + e\right )^{2} + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (C \cos \left (f x + e\right )^{2} + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.32, size = 0, normalized size = 0.00 \[ \int \left (a +b \cos \left (f x +e \right )\right )^{m} \left (A +C \left (\cos ^{2}\left (f x +e \right )\right )\right )\, dx \]
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 \[ \int {\left (C \cos \left (f x + e\right )^{2} + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int \left (C\,{\cos \left (e+f\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^m \,d x \]
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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