Optimal. Leaf size=94 \[ -\frac {2 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{4-2 m}}{d e \left (m^2-5 m+6\right )}-\frac {a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{4-2 m}}{d e (3-m)} \]
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Rubi [A] time = 0.14, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2674, 2673} \[ -\frac {2 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{4-2 m}}{d e \left (m^2-5 m+6\right )}-\frac {a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{4-2 m}}{d e (3-m)} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3-2 m} (a+a \sin (c+d x))^m \, dx &=-\frac {a (e \cos (c+d x))^{4-2 m} (a+a \sin (c+d x))^{-1+m}}{d e (3-m)}+\frac {(2 a) \int (e \cos (c+d x))^{3-2 m} (a+a \sin (c+d x))^{-1+m} \, dx}{3-m}\\ &=-\frac {2 a^2 (e \cos (c+d x))^{4-2 m} (a+a \sin (c+d x))^{-2+m}}{d e \left (6-5 m+m^2\right )}-\frac {a (e \cos (c+d x))^{4-2 m} (a+a \sin (c+d x))^{-1+m}}{d e (3-m)}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 72, normalized size = 0.77 \[ \frac {e^3 \cos ^4(c+d x) ((m-2) \sin (c+d x)+m-4) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m}}{d (m-3) (m-2) (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 170, normalized size = 1.81 \[ \frac {{\left ({\left (m - 2\right )} \cos \left (d x + c\right )^{2} + {\left (m - 4\right )} \cos \left (d x + c\right ) + {\left ({\left (m - 2\right )} \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - 2\right )} \left (e \cos \left (d x + c\right )\right )^{-2 \, m + 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}}{2 \, d m^{2} - {\left (d m^{2} - 5 \, d m + 6 \, d\right )} \cos \left (d x + c\right )^{2} - 10 \, d m + {\left (d m^{2} - 5 \, d m + 6 \, d\right )} \cos \left (d x + c\right ) + {\left (2 \, d m^{2} - 10 \, d m + {\left (d m^{2} - 5 \, d m + 6 \, d\right )} \cos \left (d x + c\right ) + 12 \, d\right )} \sin \left (d x + c\right ) + 12 \, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{-2 \, m + 3} {\left (a \sin \left (d x + c\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.57, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x +c \right )\right )^{3-2 m} \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 351, normalized size = 3.73 \[ \frac {{\left (a^{m} e^{3} {\left (m - 4\right )} - \frac {2 \, a^{m} e^{3} {\left (m - 6\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {a^{m} e^{3} {\left (m + 12\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{m} e^{3} {\left (m + 2\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {a^{m} e^{3} {\left (m + 12\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2 \, a^{m} e^{3} {\left (m - 6\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a^{m} e^{3} {\left (m - 4\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} e^{\left (-2 \, m \log \left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + m \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )\right )}}{{\left ({\left (m^{2} - 5 \, m + 6\right )} e^{2 \, m} + \frac {3 \, {\left (m^{2} - 5 \, m + 6\right )} e^{2 \, m} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, {\left (m^{2} - 5 \, m + 6\right )} e^{2 \, m} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (m^{2} - 5 \, m + 6\right )} e^{2 \, m} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.77, size = 241, normalized size = 2.56 \[ \frac {e^3\,{\left (a\,\left (\sin \left (c+d\,x\right )+1\right )\right )}^m\,\left (14\,m-24\,\sin \left (c+d\,x\right )-36\,\sin \left (3\,c+3\,d\,x\right )-12\,\sin \left (5\,c+5\,d\,x\right )+24\,{\sin \left (2\,c+2\,d\,x\right )}^2-4\,{\sin \left (3\,c+3\,d\,x\right )}^2+8\,m\,\sin \left (c+d\,x\right )-17\,m\,\left (2\,{\sin \left (c+d\,x\right )}^2-1\right )+12\,m\,\sin \left (3\,c+3\,d\,x\right )+4\,m\,\sin \left (5\,c+5\,d\,x\right )-2\,m\,\left (2\,{\sin \left (2\,c+2\,d\,x\right )}^2-1\right )+m\,\left (2\,{\sin \left (3\,c+3\,d\,x\right )}^2-1\right )+132\,{\sin \left (c+d\,x\right )}^2-128\right )}{8\,d\,{\left (-e\,\left (2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\right )}^{2\,m}\,\left (m^2-5\,m+6\right )\,\left (12\,{\sin \left (c+d\,x\right )}^2+15\,\sin \left (c+d\,x\right )-\sin \left (3\,c+3\,d\,x\right )+4\right )} \]
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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