Optimal. Leaf size=167 \[ \frac {64 a^2 (11 A-B) c^5 \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 (11 A-B) c^4 \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \]
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Rubi [A]
time = 0.29, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3046, 2935,
2753, 2752} \begin {gather*} \frac {64 a^2 c^5 (11 A-B) \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 c^4 (11 A-B) \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 (11 A-B) \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rule 2935
Rule 3046
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{5/2} \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx\\ &=-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{11} \left (a^2 (11 A-B) c^2\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{99} \left (8 a^2 (11 A-B) c^3\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {16 a^2 (11 A-B) c^4 \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}+\frac {1}{693} \left (32 a^2 (11 A-B) c^4\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {64 a^2 (11 A-B) c^5 \cos ^5(e+f x)}{3465 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 (11 A-B) c^4 \cos ^5(e+f x)}{693 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 (11 A-B) c^3 \cos ^5(e+f x)}{99 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 B c^2 \cos ^5(e+f x) \sqrt {c-c \sin (e+f x)}}{11 f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1173\) vs. \(2(167)=334\).
time = 6.36, size = 1173, normalized size = 7.02 \begin {gather*} \frac {(6 A-B) \cos \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(4 A+B) \cos \left (\frac {3}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{24 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(8 A-3 B) \cos \left (\frac {5}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {(2 A+3 B) \cos \left (\frac {7}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A-B) \cos \left (\frac {9}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{144 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {B \cos \left (\frac {11}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{176 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(6 A-B) \sin \left (\frac {1}{2} (e+f x)\right ) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2}}{8 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(4 A+B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {3}{2} (e+f x)\right )}{24 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(8 A-3 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {5}{2} (e+f x)\right )}{80 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A+3 B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {7}{2} (e+f x)\right )}{112 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {(2 A-B) (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {9}{2} (e+f x)\right )}{144 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {B (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \sin \left (\frac {11}{2} (e+f x)\right )}{176 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.74, size = 105, normalized size = 0.63
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (-315 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-1210 A +1370 B \right ) \sin \left (f x +e \right )+\left (-385 A +980 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+1562 A -1402 B \right )}{3465 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(105\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs.
\(2 (159) = 318\).
time = 0.36, size = 328, normalized size = 1.96 \begin {gather*} -\frac {2 \, {\left (315 \, B a^{2} c^{2} \cos \left (f x + e\right )^{6} - 35 \, {\left (11 \, A - 10 \, B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{5} + 5 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{3} + 16 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{2} - 64 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right ) - 128 \, {\left (11 \, A - B\right )} a^{2} c^{2} - {\left (315 \, B a^{2} c^{2} \cos \left (f x + e\right )^{5} + 35 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{4} + 40 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{3} + 48 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right )^{2} + 64 \, {\left (11 \, A - B\right )} a^{2} c^{2} \cos \left (f x + e\right ) + 128 \, {\left (11 \, A - B\right )} a^{2} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3465 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- 2 A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int A c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\, dx + \int B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \left (- 2 B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )}\right )\, dx + \int B c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{5}{\left (e + f x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 357 vs.
\(2 (159) = 318\).
time = 0.66, size = 357, normalized size = 2.14 \begin {gather*} -\frac {\sqrt {2} {\left (315 \, B a^{2} c^{2} \cos \left (-\frac {11}{4} \, \pi + \frac {11}{2} \, f x + \frac {11}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6930 \, {\left (6 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2310 \, {\left (4 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) - 693 \, {\left (8 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) - 495 \, {\left (2 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 385 \, {\left (2 \, A a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B a^{2} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right )\right )} \sqrt {c}}{55440 \, f} \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 = -0.01 \begin {gather*} \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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