Integrand size = 31, antiderivative size = 101 \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\frac {a^2 (c-d)^2 (c+d \sin (e+f x))^{1+n}}{d^3 f (1+n)}-\frac {2 a^2 (c-d) (c+d \sin (e+f x))^{2+n}}{d^3 f (2+n)}+\frac {a^2 (c+d \sin (e+f x))^{3+n}}{d^3 f (3+n)} \]
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Time = 0.10 (sec) , antiderivative size = 101, 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} \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\frac {a^2 (c-d)^2 (c+d \sin (e+f x))^{n+1}}{d^3 f (n+1)}-\frac {2 a^2 (c-d) (c+d \sin (e+f x))^{n+2}}{d^3 f (n+2)}+\frac {a^2 (c+d \sin (e+f x))^{n+3}}{d^3 f (n+3)} \]
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Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^2 \left (c+\frac {d x}{a}\right )^n \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2 (c-d)^2 \left (c+\frac {d x}{a}\right )^n}{d^2}-\frac {2 a^2 (c-d) \left (c+\frac {d x}{a}\right )^{1+n}}{d^2}+\frac {a^2 \left (c+\frac {d x}{a}\right )^{2+n}}{d^2}\right ) \, dx,x,a \sin (e+f x)\right )}{a f} \\ & = \frac {a^2 (c-d)^2 (c+d \sin (e+f x))^{1+n}}{d^3 f (1+n)}-\frac {2 a^2 (c-d) (c+d \sin (e+f x))^{2+n}}{d^3 f (2+n)}+\frac {a^2 (c+d \sin (e+f x))^{3+n}}{d^3 f (3+n)} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.77 \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\frac {a^2 (c+d \sin (e+f x))^{1+n} \left (\frac {(c-d)^2}{1+n}-\frac {2 (c-d) (c+d \sin (e+f x))}{2+n}+\frac {(c+d \sin (e+f x))^2}{3+n}\right )}{d^3 f} \]
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Time = 2.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.64
method | result | size |
parallelrisch | \(-\frac {\left (c +d \sin \left (f x +e \right )\right )^{n} \left (2 \left (1+n \right ) \left (\left (2 n +6\right ) d +c n \right ) d^{2} \cos \left (2 f x +2 e \right )+d^{3} \left (2+n \right ) \left (1+n \right ) \sin \left (3 f x +3 e \right )+8 \left (\left (-\frac {7}{8} n^{2}-\frac {29}{8} n -\frac {15}{4}\right ) d^{2}-c n \left (3+n \right ) d +c^{2} n \right ) d \sin \left (f x +e \right )+\left (-4 n^{2}-16 n -12\right ) d^{3}-6 c \left (n^{2}+\frac {11}{3} n +4\right ) d^{2}+8 c^{2} \left (3+n \right ) d -8 c^{3}\right ) a^{2}}{4 f \,d^{3} \left (1+n \right ) \left (3+n \right ) \left (2+n \right )}\) | \(166\) |
derivativedivides | \(\frac {a^{2} \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{f \left (3+n \right )}+\frac {a^{2} c \left (d^{2} n^{2}-2 c d n +5 d^{2} n +2 c^{2}-6 c d +6 d^{2}\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{3} f \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (c n +2 n d +6 d \right ) a^{2} \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{f d \left (n^{2}+5 n +6\right )}-\frac {a^{2} \left (-2 c d \,n^{2}-d^{2} n^{2}+2 c^{2} n -6 c d n -5 d^{2} n -6 d^{2}\right ) \sin \left (f x +e \right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{2} \left (n^{3}+6 n^{2}+11 n +6\right ) f}\) | \(246\) |
default | \(\frac {a^{2} \left (\sin ^{3}\left (f x +e \right )\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{f \left (3+n \right )}+\frac {a^{2} c \left (d^{2} n^{2}-2 c d n +5 d^{2} n +2 c^{2}-6 c d +6 d^{2}\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{3} f \left (n^{3}+6 n^{2}+11 n +6\right )}+\frac {\left (c n +2 n d +6 d \right ) a^{2} \left (\sin ^{2}\left (f x +e \right )\right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{f d \left (n^{2}+5 n +6\right )}-\frac {a^{2} \left (-2 c d \,n^{2}-d^{2} n^{2}+2 c^{2} n -6 c d n -5 d^{2} n -6 d^{2}\right ) \sin \left (f x +e \right ) {\mathrm e}^{n \ln \left (c +d \sin \left (f x +e \right )\right )}}{d^{2} \left (n^{3}+6 n^{2}+11 n +6\right ) f}\) | \(246\) |
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (101) = 202\).
Time = 0.30 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.91 \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\frac {{\left (2 \, a^{2} c^{3} - 6 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} + 6 \, a^{2} d^{3} + 2 \, {\left (a^{2} c d^{2} + a^{2} d^{3}\right )} n^{2} - {\left (6 \, a^{2} d^{3} + {\left (a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} n^{2} + {\left (a^{2} c d^{2} + 8 \, a^{2} d^{3}\right )} n\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 4 \, a^{2} d^{3}\right )} n + {\left (8 \, a^{2} d^{3} + 2 \, {\left (a^{2} c d^{2} + a^{2} d^{3}\right )} n^{2} - {\left (a^{2} d^{3} n^{2} + 3 \, a^{2} d^{3} n + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 4 \, a^{2} d^{3}\right )} n\right )} \sin \left (f x + e\right )\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{d^{3} f n^{3} + 6 \, d^{3} f n^{2} + 11 \, d^{3} f n + 6 \, d^{3} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2159 vs. \(2 (85) = 170\).
Time = 3.32 (sec) , antiderivative size = 2159, normalized size of antiderivative = 21.38 \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\text {Too large to display} \]
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Time = 0.22 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.81 \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\frac {\frac {2 \, {\left (d^{2} {\left (n + 1\right )} \sin \left (f x + e\right )^{2} + c d n \sin \left (f x + e\right ) - c^{2}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{2}}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a^{2}}{d {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} \sin \left (f x + e\right )^{3} + {\left (n^{2} + n\right )} c d^{2} \sin \left (f x + e\right )^{2} - 2 \, c^{2} d n \sin \left (f x + e\right ) + 2 \, c^{3}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}}}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (101) = 202\).
Time = 0.34 (sec) , antiderivative size = 436, normalized size of antiderivative = 4.32 \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\frac {\frac {{\left ({\left (d \sin \left (f x + e\right ) + c\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} n^{2} - 2 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n^{2} + {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2} n^{2} + 3 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} n - 8 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n + 5 \, {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2} n + 2 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} - 6 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c + 6 \, {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2}\right )} a^{2}}{d^{2} n^{3} + 6 \, d^{2} n^{2} + 11 \, d^{2} n + 6 \, d^{2}} + \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a^{2}}{n + 1} + \frac {2 \, {\left ({\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} n - {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n + {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} - 2 \, {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c\right )} a^{2}}{{\left (n^{2} + 3 \, n + 2\right )} d}}{d f} \]
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Time = 12.41 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.99 \[ \int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx=\frac {a^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (24\,c\,d^2-24\,c^2\,d+16\,d^3\,n+30\,d^3\,\sin \left (e+f\,x\right )+8\,c^3+12\,d^3-12\,d^3\,\cos \left (2\,e+2\,f\,x\right )+4\,d^3\,n^2-2\,d^3\,\sin \left (3\,e+3\,f\,x\right )+29\,d^3\,n\,\sin \left (e+f\,x\right )+6\,c\,d^2\,n^2-16\,d^3\,n\,\cos \left (2\,e+2\,f\,x\right )-3\,d^3\,n\,\sin \left (3\,e+3\,f\,x\right )+7\,d^3\,n^2\,\sin \left (e+f\,x\right )-4\,d^3\,n^2\,\cos \left (2\,e+2\,f\,x\right )-d^3\,n^2\,\sin \left (3\,e+3\,f\,x\right )+22\,c\,d^2\,n-8\,c^2\,d\,n-2\,c\,d^2\,n^2\,\cos \left (2\,e+2\,f\,x\right )+24\,c\,d^2\,n\,\sin \left (e+f\,x\right )-8\,c^2\,d\,n\,\sin \left (e+f\,x\right )-2\,c\,d^2\,n\,\cos \left (2\,e+2\,f\,x\right )+8\,c\,d^2\,n^2\,\sin \left (e+f\,x\right )\right )}{4\,d^3\,f\,\left (n^3+6\,n^2+11\,n+6\right )} \]
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