Optimal. Leaf size=250 \[ -\frac {a \left (a^2 (m+2)+3 b^2 (m+1)\right ) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt {\sin ^2(c+d x)}}-\frac {b \left (3 a^2 (m+3)+b^2 (m+2)\right ) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt {\sin ^2(c+d x)}}+\frac {a b^2 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac {b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))}{d (m+3)} \]
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Rubi [A] time = 0.32, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2793, 3023, 2748, 2643} \[ -\frac {a \left (a^2 (m+2)+3 b^2 (m+1)\right ) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{d (m+1) (m+2) \sqrt {\sin ^2(c+d x)}}-\frac {b \left (3 a^2 (m+3)+b^2 (m+2)\right ) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{d (m+2) (m+3) \sqrt {\sin ^2(c+d x)}}+\frac {a b^2 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac {b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))}{d (m+3)} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 2748
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int \cos ^m(c+d x) (a+b \cos (c+d x))^3 \, dx &=\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}+\frac {\int \cos ^m(c+d x) \left (a \left (b^2 (1+m)+a^2 (3+m)\right )+b \left (b^2 (2+m)+3 a^2 (3+m)\right ) \cos (c+d x)+a b^2 (7+2 m) \cos ^2(c+d x)\right ) \, dx}{3+m}\\ &=\frac {a b^2 (7+2 m) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (3+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}+\frac {\int \cos ^m(c+d x) \left (a (3+m) \left (3 b^2 (1+m)+a^2 (2+m)\right )+b (2+m) \left (b^2 (2+m)+3 a^2 (3+m)\right ) \cos (c+d x)\right ) \, dx}{6+5 m+m^2}\\ &=\frac {a b^2 (7+2 m) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (3+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}+\left (a \left (a^2+\frac {3 b^2 (1+m)}{2+m}\right )\right ) \int \cos ^m(c+d x) \, dx+\left (b \left (3 a^2+\frac {b^2 (2+m)}{3+m}\right )\right ) \int \cos ^{1+m}(c+d x) \, dx\\ &=\frac {a b^2 (7+2 m) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (3+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{d (3+m)}-\frac {a \left (a^2+\frac {3 b^2 (1+m)}{2+m}\right ) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) \sqrt {\sin ^2(c+d x)}}-\frac {b \left (3 a^2+\frac {b^2 (2+m)}{3+m}\right ) \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) \sqrt {\sin ^2(c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.91, size = 197, normalized size = 0.79 \[ \frac {\sqrt {\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left (b \cos (c+d x) \left (b \cos (c+d x) \left (-\frac {3 a \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};\cos ^2(c+d x)\right )}{m+3}-\frac {b \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+4}{2};\frac {m+6}{2};\cos ^2(c+d x)\right )}{m+4}\right )-\frac {3 a^2 \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\cos ^2(c+d x)\right )}{m+2}\right )-\frac {a^3 \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\cos ^2(c+d x)\right )}{m+1}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \cos \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.30, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{m}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{3}\, dx \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{m}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^m\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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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