Optimal. Leaf size=429 \[ \frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3465 d^2 f}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 d f}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f} \]
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Rubi [A] time = 1.07, antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {2976, 2981, 2761, 2751, 2646} \[ \frac {2 a^3 \left (11 A d (3 c-19 d)-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt {a \sin (e+f x)+a}}-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3465 d^2 f}+\frac {2 a^2 (-11 A d+5 B c-14 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (-40 c^2 d+5 c^3+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{1155 d f}-\frac {2 a B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c+d \sin (e+f x))^3}{11 d f} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2761
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx &=-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {2 \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \left (\frac {1}{2} a (11 A d+3 B (c+2 d))-\frac {1}{2} a (5 B c-11 A d-14 B d) \sin (e+f x)\right ) \, dx}{11 d}\\ &=\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {4 \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \left (\frac {1}{4} a^2 \left (11 A d (c+15 d)-B \left (5 c^2-11 c d-138 d^2\right )\right )-\frac {1}{4} a^2 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{99 d^2}\\ &=\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {\left (a^2 \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{231 d^3}\\ &=-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {\left (2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{1155 d^3}\\ &=-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}+\frac {\left (a^2 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{3465 d^3}\\ &=-\frac {2 a^3 \left (15 c^2+10 c d+7 d^2\right ) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x)}{3465 d^3 f \sqrt {a+a \sin (e+f x)}}-\frac {4 a^2 (5 c-d) \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3465 d^2 f}-\frac {2 a \left (11 A d \left (c^2-10 c d+73 d^2\right )-B \left (5 c^3-40 c^2 d+169 c d^2-710 d^3\right )\right ) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{1155 d f}+\frac {2 a^3 \left (11 A (3 c-19 d) d-B \left (15 c^2-65 c d+194 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^3}{693 d^3 f \sqrt {a+a \sin (e+f x)}}+\frac {2 a^2 (5 B c-11 A d-14 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3}{99 d^2 f}-\frac {2 a B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3}{11 d f}\\ \end {align*}
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Mathematica [B] time = 6.61, size = 891, normalized size = 2.08 \[ \frac {(a (\sin (e+f x)+1))^{5/2} \left (-277200 A \cos \left (\frac {1}{2} (e+f x)\right ) c^2-207900 B \cos \left (\frac {1}{2} (e+f x)\right ) c^2-46200 A \cos \left (\frac {3}{2} (e+f x)\right ) c^2-50820 B \cos \left (\frac {3}{2} (e+f x)\right ) c^2+5544 A \cos \left (\frac {5}{2} (e+f x)\right ) c^2+13860 B \cos \left (\frac {5}{2} (e+f x)\right ) c^2+1980 B \cos \left (\frac {7}{2} (e+f x)\right ) c^2+277200 A \sin \left (\frac {1}{2} (e+f x)\right ) c^2+207900 B \sin \left (\frac {1}{2} (e+f x)\right ) c^2-46200 A \sin \left (\frac {3}{2} (e+f x)\right ) c^2-50820 B \sin \left (\frac {3}{2} (e+f x)\right ) c^2-5544 A \sin \left (\frac {5}{2} (e+f x)\right ) c^2-13860 B \sin \left (\frac {5}{2} (e+f x)\right ) c^2+1980 B \sin \left (\frac {7}{2} (e+f x)\right ) c^2-415800 A d \cos \left (\frac {1}{2} (e+f x)\right ) c-360360 B d \cos \left (\frac {1}{2} (e+f x)\right ) c-101640 A d \cos \left (\frac {3}{2} (e+f x)\right ) c-92400 B d \cos \left (\frac {3}{2} (e+f x)\right ) c+27720 A d \cos \left (\frac {5}{2} (e+f x)\right ) c+33264 B d \cos \left (\frac {5}{2} (e+f x)\right ) c+3960 A d \cos \left (\frac {7}{2} (e+f x)\right ) c+9900 B d \cos \left (\frac {7}{2} (e+f x)\right ) c-1540 B d \cos \left (\frac {9}{2} (e+f x)\right ) c+415800 A d \sin \left (\frac {1}{2} (e+f x)\right ) c+360360 B d \sin \left (\frac {1}{2} (e+f x)\right ) c-101640 A d \sin \left (\frac {3}{2} (e+f x)\right ) c-92400 B d \sin \left (\frac {3}{2} (e+f x)\right ) c-27720 A d \sin \left (\frac {5}{2} (e+f x)\right ) c-33264 B d \sin \left (\frac {5}{2} (e+f x)\right ) c+3960 A d \sin \left (\frac {7}{2} (e+f x)\right ) c+9900 B d \sin \left (\frac {7}{2} (e+f x)\right ) c+1540 B d \sin \left (\frac {9}{2} (e+f x)\right ) c-180180 A d^2 \cos \left (\frac {1}{2} (e+f x)\right )-159390 B d^2 \cos \left (\frac {1}{2} (e+f x)\right )-46200 A d^2 \cos \left (\frac {3}{2} (e+f x)\right )-43890 B d^2 \cos \left (\frac {3}{2} (e+f x)\right )+16632 A d^2 \cos \left (\frac {5}{2} (e+f x)\right )+17325 B d^2 \cos \left (\frac {5}{2} (e+f x)\right )+4950 A d^2 \cos \left (\frac {7}{2} (e+f x)\right )+6435 B d^2 \cos \left (\frac {7}{2} (e+f x)\right )-770 A d^2 \cos \left (\frac {9}{2} (e+f x)\right )-1925 B d^2 \cos \left (\frac {9}{2} (e+f x)\right )-315 B d^2 \cos \left (\frac {11}{2} (e+f x)\right )+180180 A d^2 \sin \left (\frac {1}{2} (e+f x)\right )+159390 B d^2 \sin \left (\frac {1}{2} (e+f x)\right )-46200 A d^2 \sin \left (\frac {3}{2} (e+f x)\right )-43890 B d^2 \sin \left (\frac {3}{2} (e+f x)\right )-16632 A d^2 \sin \left (\frac {5}{2} (e+f x)\right )-17325 B d^2 \sin \left (\frac {5}{2} (e+f x)\right )+4950 A d^2 \sin \left (\frac {7}{2} (e+f x)\right )+6435 B d^2 \sin \left (\frac {7}{2} (e+f x)\right )+770 A d^2 \sin \left (\frac {9}{2} (e+f x)\right )+1925 B d^2 \sin \left (\frac {9}{2} (e+f x)\right )-315 B d^2 \sin \left (\frac {11}{2} (e+f x)\right )\right )}{55440 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 593, normalized size = 1.38 \[ -\frac {2 \, {\left (315 \, B a^{2} d^{2} \cos \left (f x + e\right )^{6} + 35 \, {\left (22 \, B a^{2} c d + {\left (11 \, A + 32 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{5} + 1056 \, {\left (7 \, A + 5 \, B\right )} a^{2} c^{2} + 704 \, {\left (15 \, A + 13 \, B\right )} a^{2} c d + 32 \, {\left (143 \, A + 125 \, B\right )} a^{2} d^{2} - 5 \, {\left (99 \, B a^{2} c^{2} + 22 \, {\left (9 \, A + 19 \, B\right )} a^{2} c d + {\left (209 \, A + 320 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (99 \, {\left (7 \, A + 20 \, B\right )} a^{2} c^{2} + 22 \, {\left (180 \, A + 289 \, B\right )} a^{2} c d + {\left (3179 \, A + 4370 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (33 \, {\left (77 \, A + 85 \, B\right )} a^{2} c^{2} + 22 \, {\left (255 \, A + 263 \, B\right )} a^{2} c d + {\left (2893 \, A + 2965 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (33 \, {\left (161 \, A + 145 \, B\right )} a^{2} c^{2} + 22 \, {\left (435 \, A + 419 \, B\right )} a^{2} c d + {\left (4609 \, A + 4465 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right ) + {\left (315 \, B a^{2} d^{2} \cos \left (f x + e\right )^{5} - 1056 \, {\left (7 \, A + 5 \, B\right )} a^{2} c^{2} - 704 \, {\left (15 \, A + 13 \, B\right )} a^{2} c d - 32 \, {\left (143 \, A + 125 \, B\right )} a^{2} d^{2} - 35 \, {\left (22 \, B a^{2} c d + {\left (11 \, A + 23 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} - 5 \, {\left (99 \, B a^{2} c^{2} + 22 \, {\left (9 \, A + 26 \, B\right )} a^{2} c d + 13 \, {\left (22 \, A + 37 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (33 \, {\left (7 \, A + 15 \, B\right )} a^{2} c^{2} + 22 \, {\left (45 \, A + 53 \, B\right )} a^{2} c d + {\left (583 \, A + 655 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (33 \, {\left (49 \, A + 65 \, B\right )} a^{2} c^{2} + 22 \, {\left (195 \, A + 211 \, B\right )} a^{2} c d + {\left (2321 \, A + 2465 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3465 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.43, size = 257, normalized size = 0.60 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (315 B \,d^{2} \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-990 A c d -1430 A \,d^{2}-495 B \,c^{2}-2860 B c d -2405 B \,d^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (3234 A \,c^{2}+8580 A c d +4642 A \,d^{2}+4290 B \,c^{2}+9284 B c d +4930 B \,d^{2}\right ) \sin \left (f x +e \right )+\left (385 A \,d^{2}+770 B c d +1120 B \,d^{2}\right ) \left (\cos ^{4}\left (f x +e \right )\right )+\left (-693 A \,c^{2}-3960 A c d -3179 A \,d^{2}-1980 B \,c^{2}-6358 B c d -4370 B \,d^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+10626 A \,c^{2}+19140 A c d +9218 A \,d^{2}+9570 B \,c^{2}+18436 B c d +8930 B \,d^{2}\right )}{3465 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{2}\,{d x} \]
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 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]
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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