Integrand size = 31, antiderivative size = 96 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\frac {(c-d)^2 (a+a \sin (e+f x))^{1+m}}{a f (1+m)}+\frac {2 (c-d) d (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}+\frac {d^2 (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)} \]
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Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2912, 45} \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\frac {d^2 (a \sin (e+f x)+a)^{m+3}}{a^3 f (m+3)}+\frac {2 d (c-d) (a \sin (e+f x)+a)^{m+2}}{a^2 f (m+2)}+\frac {(c-d)^2 (a \sin (e+f x)+a)^{m+1}}{a f (m+1)} \]
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Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^m \left (c+\frac {d x}{a}\right )^2 \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = \frac {\text {Subst}\left (\int \left ((c-d)^2 (a+x)^m+\frac {2 (c-d) d (a+x)^{1+m}}{a}+\frac {d^2 (a+x)^{2+m}}{a^2}\right ) \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = \frac {(c-d)^2 (a+a \sin (e+f x))^{1+m}}{a f (1+m)}+\frac {2 (c-d) d (a+a \sin (e+f x))^{2+m}}{a^2 f (2+m)}+\frac {d^2 (a+a \sin (e+f x))^{3+m}}{a^3 f (3+m)} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\frac {(a (1+\sin (e+f x)))^{1+m} \left (\frac {a^2 (c-d)^2}{1+m}+\frac {2 a^2 (c-d) d (1+\sin (e+f x))}{2+m}+\frac {d^2 (a+a \sin (e+f x))^2}{3+m}\right )}{a^3 f} \]
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Time = 1.78 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.60
method | result | size |
parallelrisch | \(-\frac {\left (4 \left (1+m \right ) \left (\frac {d m}{2}+c \left (3+m \right )\right ) d \cos \left (2 f x +2 e \right )+d^{2} \left (2+m \right ) \left (1+m \right ) \sin \left (3 f x +3 e \right )+\left (\left (-3 m^{2}-m -6\right ) d^{2}-8 c m \left (3+m \right ) d -4 c^{2} \left (3+m \right ) \left (2+m \right )\right ) \sin \left (f x +e \right )+\left (-2 m^{2}-2 m -8\right ) d^{2}-4 c \left (3+m \right ) \left (-1+m \right ) d -4 c^{2} \left (3+m \right ) \left (2+m \right )\right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{m}}{4 \left (m^{3}+6 m^{2}+11 m +6\right ) f}\) | \(154\) |
derivativedivides | \(\frac {\left (c^{2} m^{2}+5 c^{2} m -2 c d m +6 c^{2}-6 c d +2 d^{2}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {d^{2} \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (3+m \right )}+\frac {\left (c^{2} m^{2}+2 c d \,m^{2}+5 c^{2} m +6 c d m -2 d^{2} m +6 c^{2}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {\left (2 c m +d m +6 c \right ) d \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+5 m +6\right )}\) | \(226\) |
default | \(\frac {\left (c^{2} m^{2}+5 c^{2} m -2 c d m +6 c^{2}-6 c d +2 d^{2}\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {d^{2} \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (3+m \right )}+\frac {\left (c^{2} m^{2}+2 c d \,m^{2}+5 c^{2} m +6 c d m -2 d^{2} m +6 c^{2}\right ) \sin \left (f x +e \right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {\left (2 c m +d m +6 c \right ) d \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{m \ln \left (a +a \sin \left (f x +e \right )\right )}}{f \left (m^{2}+5 m +6\right )}\) | \(226\) |
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Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.97 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\frac {{\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} m^{2} - {\left ({\left (2 \, c d + d^{2}\right )} m^{2} + 6 \, c d + {\left (8 \, c d + d^{2}\right )} m\right )} \cos \left (f x + e\right )^{2} + 6 \, c^{2} + 2 \, d^{2} + {\left (5 \, c^{2} + 6 \, c d + d^{2}\right )} m + {\left ({\left (c^{2} + 2 \, c d + d^{2}\right )} m^{2} - {\left (d^{2} m^{2} + 3 \, d^{2} m + 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} + 6 \, c^{2} + 2 \, d^{2} + {\left (5 \, c^{2} + 6 \, c d + d^{2}\right )} m\right )} \sin \left (f x + e\right )\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{f m^{3} + 6 \, f m^{2} + 11 \, f m + 6 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1622 vs. \(2 (80) = 160\).
Time = 2.55 (sec) , antiderivative size = 1622, normalized size of antiderivative = 16.90 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \]
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Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.78 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\frac {\frac {2 \, {\left (a^{m} {\left (m + 1\right )} \sin \left (f x + e\right )^{2} + a^{m} m \sin \left (f x + e\right ) - a^{m}\right )} c d {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{2} + 3 \, m + 2} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} a^{m} \sin \left (f x + e\right )^{3} + {\left (m^{2} + m\right )} a^{m} \sin \left (f x + e\right )^{2} - 2 \, a^{m} m \sin \left (f x + e\right ) + 2 \, a^{m}\right )} d^{2} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} c^{2}}{a {\left (m + 1\right )}}}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (96) = 192\).
Time = 0.34 (sec) , antiderivative size = 435, normalized size of antiderivative = 4.53 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\frac {\frac {{\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} m^{2} - 2 \, {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m^{2} + {\left (a \sin \left (f x + e\right ) + a\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2} m^{2} + 3 \, {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} m - 8 \, {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m + 5 \, {\left (a \sin \left (f x + e\right ) + a\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2} m + 2 \, {\left (a \sin \left (f x + e\right ) + a\right )}^{3} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} - 6 \, {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a + 6 \, {\left (a \sin \left (f x + e\right ) + a\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a^{2}\right )} d^{2}}{a^{2} m^{3} + 6 \, a^{2} m^{2} + 11 \, a^{2} m + 6 \, a^{2}} + \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{m + 1} c^{2}}{m + 1} + \frac {2 \, {\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} m - {\left (a \sin \left (f x + e\right ) + a\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a m + {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} - 2 \, {\left (a \sin \left (f x + e\right ) + a\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} a\right )} c d}{{\left (m^{2} + 3 \, m + 2\right )} a}}{a f} \]
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Time = 11.40 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.18 \[ \int \cos (e+f x) (a+a \sin (e+f x))^m (c+d \sin (e+f x))^2 \, dx=\frac {{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\left (20\,c^2\,m-12\,c\,d+2\,d^2\,m+24\,c^2\,\sin \left (e+f\,x\right )+6\,d^2\,\sin \left (e+f\,x\right )+24\,c^2+8\,d^2+4\,c^2\,m^2+2\,d^2\,m^2-2\,d^2\,\sin \left (3\,e+3\,f\,x\right )+20\,c^2\,m\,\sin \left (e+f\,x\right )+d^2\,m\,\sin \left (e+f\,x\right )-2\,d^2\,m\,\cos \left (2\,e+2\,f\,x\right )+4\,c^2\,m^2\,\sin \left (e+f\,x\right )-3\,d^2\,m\,\sin \left (3\,e+3\,f\,x\right )+3\,d^2\,m^2\,\sin \left (e+f\,x\right )+8\,c\,d\,m-2\,d^2\,m^2\,\cos \left (2\,e+2\,f\,x\right )-d^2\,m^2\,\sin \left (3\,e+3\,f\,x\right )-12\,c\,d\,\cos \left (2\,e+2\,f\,x\right )+4\,c\,d\,m^2+24\,c\,d\,m\,\sin \left (e+f\,x\right )-16\,c\,d\,m\,\cos \left (2\,e+2\,f\,x\right )+8\,c\,d\,m^2\,\sin \left (e+f\,x\right )-4\,c\,d\,m^2\,\cos \left (2\,e+2\,f\,x\right )\right )}{4\,f\,\left (m^3+6\,m^2+11\,m+6\right )} \]
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