Optimal. Leaf size=406 \[ -\frac {\sin (e+f x) \left (b (m+1) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right )+a^2 (m+2) (a A (m+4)+b B (m+1))\right ) (c \cos (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{c f (m+1) (m+2) (m+4) \sqrt {\sin ^2(e+f x)}}+\frac {b \sin (e+f x) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right ) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4)}-\frac {\sin (e+f x) \left (a^3 B (m+3)+3 a^2 A b (m+3)+3 a b^2 B (m+2)+A b^3 (m+2)\right ) (c \cos (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )}{c^2 f (m+2) (m+3) \sqrt {\sin ^2(e+f x)}}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)} \]
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Rubi [A] time = 1.05, antiderivative size = 406, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2990, 3033, 3023, 2748, 2643} \[ -\frac {\sin (e+f x) \left (3 a^2 A b (m+3)+a^3 B (m+3)+3 a b^2 B (m+2)+A b^3 (m+2)\right ) (c \cos (e+f x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )}{c^2 f (m+2) (m+3) \sqrt {\sin ^2(e+f x)}}-\frac {\sin (e+f x) \left (b (m+1) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right )+a^2 (m+2) (a A (m+4)+b B (m+1))\right ) (c \cos (e+f x))^{m+1} \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{c f (m+1) (m+2) (m+4) \sqrt {\sin ^2(e+f x)}}+\frac {b \sin (e+f x) \left (2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right ) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4)}+\frac {b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4)}+\frac {b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2990
Rule 3023
Rule 3033
Rubi steps
\begin {align*} \int (c \cos (e+f x))^m (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) \, dx &=\frac {b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^2 \sin (e+f x)}{c f (4+m)}+\frac {\int (c \cos (e+f x))^m (a+b \cos (e+f x)) \left (a c (b B (1+m)+a A (4+m))+c \left (b^2 B (3+m)+a (2 A b+a B) (4+m)\right ) \cos (e+f x)+b c (A b (4+m)+a B (6+m)) \cos ^2(e+f x)\right ) \, dx}{c (4+m)}\\ &=\frac {b^2 (A b (4+m)+a B (6+m)) \cos (e+f x) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (3+m) (4+m)}+\frac {b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^2 \sin (e+f x)}{c f (4+m)}+\frac {\int (c \cos (e+f x))^m \left (a^2 c^2 (3+m) (b B (1+m)+a A (4+m))+c^2 (4+m) \left (A b^3 (2+m)+3 a b^2 B (2+m)+3 a^2 A b (3+m)+a^3 B (3+m)\right ) \cos (e+f x)+b c^2 (3+m) \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right ) \cos ^2(e+f x)\right ) \, dx}{c^2 (3+m) (4+m)}\\ &=\frac {b \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right ) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (4+m)}+\frac {b^2 (A b (4+m)+a B (6+m)) \cos (e+f x) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (3+m) (4+m)}+\frac {b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^2 \sin (e+f x)}{c f (4+m)}+\frac {\int (c \cos (e+f x))^m \left (c^3 (3+m) \left (a^2 (2+m) (b B (1+m)+a A (4+m))+b (1+m) \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right )\right )+c^3 (2+m) (4+m) \left (A b^3 (2+m)+3 a b^2 B (2+m)+3 a^2 A b (3+m)+a^3 B (3+m)\right ) \cos (e+f x)\right ) \, dx}{c^3 (2+m) (3+m) (4+m)}\\ &=\frac {b \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right ) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (4+m)}+\frac {b^2 (A b (4+m)+a B (6+m)) \cos (e+f x) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (3+m) (4+m)}+\frac {b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^2 \sin (e+f x)}{c f (4+m)}+\frac {\left (A b^3 (2+m)+3 a b^2 B (2+m)+3 a^2 A b (3+m)+a^3 B (3+m)\right ) \int (c \cos (e+f x))^{1+m} \, dx}{c (3+m)}+\frac {\left (a^2 (2+m) (b B (1+m)+a A (4+m))+b (1+m) \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right )\right ) \int (c \cos (e+f x))^m \, dx}{(2+m) (4+m)}\\ &=\frac {b \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right ) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (2+m) (4+m)}+\frac {b^2 (A b (4+m)+a B (6+m)) \cos (e+f x) (c \cos (e+f x))^{1+m} \sin (e+f x)}{c f (3+m) (4+m)}+\frac {b B (c \cos (e+f x))^{1+m} (a+b \cos (e+f x))^2 \sin (e+f x)}{c f (4+m)}-\frac {\left (a^2 (2+m) (b B (1+m)+a A (4+m))+b (1+m) \left (b^2 B (3+m)+3 a A b (4+m)+2 a^2 B (5+m)\right )\right ) (c \cos (e+f x))^{1+m} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{c f (1+m) (2+m) (4+m) \sqrt {\sin ^2(e+f x)}}-\frac {\left (A b^3 (2+m)+3 a b^2 B (2+m)+3 a^2 A b (3+m)+a^3 B (3+m)\right ) (c \cos (e+f x))^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{c^2 f (2+m) (3+m) \sqrt {\sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.81, size = 269, normalized size = 0.66 \[ \frac {\sin (e+f x) \cos (e+f x) (c \cos (e+f x))^m \left (\cos (e+f x) \left (b \cos (e+f x) \left (b \cos (e+f x) \left (-\frac {(3 a B+A b) \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};\cos ^2(e+f x)\right )}{m+4}-\frac {b B \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {m+5}{2};\frac {m+7}{2};\cos ^2(e+f x)\right )}{m+5}\right )-\frac {3 a (a B+A b) \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};\cos ^2(e+f x)\right )}{m+3}\right )-\frac {a^2 (a B+3 A b) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(e+f x)\right )}{m+2}\right )-\frac {a^3 A \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(e+f x)\right )}{m+1}\right )}{f \sqrt {\sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b^{3} \cos \left (f x + e\right )^{4} + A a^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (f x + e\right )\right )} \left (c \cos \left (f x + e\right )\right )^{m}, x\right ) \]
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 \[ \int {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{3} \left (c \cos \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.00, size = 0, normalized size = 0.00 \[ \int \left (c \cos \left (f x +e \right )\right )^{m} \left (a +b \cos \left (f x +e \right )\right )^{3} \left (A +B \cos \left (f x +e \right )\right )\, dx \]
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 \[ \int {\left (B \cos \left (f x + e\right ) + A\right )} {\left (b \cos \left (f x + e\right ) + a\right )}^{3} \left (c \cos \left (f x + e\right )\right )^{m}\,{d x} \]
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 \[ \int {\left (c\,\cos \left (e+f\,x\right )\right )}^m\,\left (A+B\,\cos \left (e+f\,x\right )\right )\,{\left (a+b\,\cos \left (e+f\,x\right )\right )}^3 \,d x \]
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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