Optimal. Leaf size=330 \[ \frac {b^2 \left (b^2 (3+m)+a^2 (22+5 m)\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (4+m)}+\frac {2 a b^3 (5+m) \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}-\frac {\left (b^4 \left (3+4 m+m^2\right )+6 a^2 b^2 \left (4+5 m+m^2\right )+a^4 \left (8+6 m+m^2\right )\right ) \cos ^{1+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (1+m) (2+m) (4+m) \sqrt {\sin ^2(c+d x)}}-\frac {4 a b \left (b^2 (2+m)+a^2 (3+m)\right ) \cos ^{2+m}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(c+d x)\right ) \sin (c+d x)}{d (2+m) (3+m) \sqrt {\sin ^2(c+d x)}} \]
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Rubi [A]
time = 0.45, antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2872, 3112,
3102, 2827, 2722} \begin {gather*} -\frac {4 a b \left (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 {b^2 \left (a^2 (5 m+22)+b^2 (m+3)\right ) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}-\frac {\left (a^4 \left (m^2+6 m+8\right )+6 a^2 b^2 \left (m^2+5 m+4\right )+b^4 \left (m^2+4 m+3\right )\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) (m+4) \sqrt {\sin ^2(c+d x)}}+\frac {2 a b^3 (m+5) \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}+\frac {b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 2827
Rule 2872
Rule 3102
Rule 3112
Rubi steps
\begin {align*} \int \cos ^m(c+d x) (a+b \cos (c+d x))^4 \, dx &=\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\frac {\int \cos ^m(c+d x) (a+b \cos (c+d x)) \left (a \left (b^2 (1+m)+a^2 (4+m)\right )+b \left (b^2 (3+m)+3 a^2 (4+m)\right ) \cos (c+d x)+2 a b^2 (5+m) \cos ^2(c+d x)\right ) \, dx}{4+m}\\ &=\frac {2 a b^3 (5+m) \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\frac {\int \cos ^m(c+d x) \left (a^2 (3+m) \left (b^2 (1+m)+a^2 (4+m)\right )+4 a b (4+m) \left (b^2 (2+m)+a^2 (3+m)\right ) \cos (c+d x)+b^2 (3+m) \left (b^2 (3+m)+a^2 (22+5 m)\right ) \cos ^2(c+d x)\right ) \, dx}{12+7 m+m^2}\\ &=\frac {b^2 \left (b^2 (3+m)+a^2 (22+5 m)\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (4+m)}+\frac {2 a b^3 (5+m) \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\frac {\int \cos ^m(c+d x) \left ((3+m) \left (b^4 \left (3+4 m+m^2\right )+6 a^2 b^2 \left (4+5 m+m^2\right )+a^4 \left (8+6 m+m^2\right )\right )+4 a b (2+m) (4+m) \left (b^2 (2+m)+a^2 (3+m)\right ) \cos (c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3}\\ &=\frac {b^2 \left (b^2 (3+m)+a^2 (22+5 m)\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (4+m)}+\frac {2 a b^3 (5+m) \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}+\left (4 a b \left (a^2+\frac {b^2 (2+m)}{3+m}\right )\right ) \int \cos ^{1+m}(c+d x) \, dx+\frac {\left (b^4 \left (3+4 m+m^2\right )+6 a^2 b^2 \left (4+5 m+m^2\right )+a^4 \left (8+6 m+m^2\right )\right ) \int \cos ^m(c+d x) \, dx}{8+6 m+m^2}\\ &=\frac {b^2 \left (b^2 (3+m)+a^2 (22+5 m)\right ) \cos ^{1+m}(c+d x) \sin (c+d x)}{d (2+m) (4+m)}+\frac {2 a b^3 (5+m) \cos ^{2+m}(c+d x) \sin (c+d x)}{d (3+m) (4+m)}+\frac {b^2 \cos ^{1+m}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{d (4+m)}-\frac {\left (b^4 \left (3+4 m+m^2\right )+6 a^2 b^2 \left (4+5 m+m^2\right )+a^4 \left (8+6 m+m^2\right )\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) \left (8+6 m+m^2\right ) \sqrt {\sin ^2(c+d x)}}-\frac {4 a b \left (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 = 1.85, size = 242, normalized size = 0.73 \begin {gather*} \frac {\cos ^{1+m}(c+d x) \csc (c+d x) \left (-\frac {a^4 \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\cos ^2(c+d x)\right )}{1+m}+b \cos (c+d x) \left (-\frac {4 a^3 \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};\cos ^2(c+d x)\right )}{2+m}+b \cos (c+d x) \left (-\frac {6 a^2 \, _2F_1\left (\frac {1}{2},\frac {3+m}{2};\frac {5+m}{2};\cos ^2(c+d x)\right )}{3+m}+b \cos (c+d x) \left (-\frac {4 a \, _2F_1\left (\frac {1}{2},\frac {4+m}{2};\frac {6+m}{2};\cos ^2(c+d x)\right )}{4+m}-\frac {b \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5+m}{2};\frac {7+m}{2};\cos ^2(c+d x)\right )}{5+m}\right )\right )\right )\right ) \sqrt {\sin ^2(c+d x)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.32, size = 0, normalized size = 0.00 \[\int \left (\cos ^{m}\left (d x +c \right )\right ) \left (a +b \cos \left (d x +c \right )\right )^{4}\, dx\]
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 {\cos \left (c+d\,x\right )}^m\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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