Optimal. Leaf size=266 \[ -\frac {a^3 (3 A-17 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.39, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3046, 2938,
2759, 2729, 2728, 212} \begin {gather*} -\frac {a^3 (3 A-17 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 c (3 A-17 B) \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rule 2759
Rule 2938
Rule 3046
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {1}{20} \left (a^3 (3 A-17 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{15/2}} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {1}{32} \left (a^3 (3 A-17 B)\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{11/2}} \, dx\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {\left (a^3 (3 A-17 B)\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx}{64 c^2}\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {\left (a^3 (3 A-17 B)\right ) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{256 c^4}\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (a^3 (3 A-17 B)\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{1024 c^5}\\ &=\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (a^3 (3 A-17 B)\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{512 c^5 f}\\ &=-\frac {a^3 (3 A-17 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{10 f (c-c \sin (e+f x))^{17/2}}+\frac {a^3 (3 A-17 B) c \cos ^5(e+f x)}{80 f (c-c \sin (e+f x))^{13/2}}-\frac {a^3 (3 A-17 B) \cos ^3(e+f x)}{96 c f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (3 A-17 B) \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 (3 A-17 B) \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.67, size = 409, normalized size = 1.54 \begin {gather*} \frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (56370 A \cos \left (\frac {1}{2} (e+f x)\right )+38970 B \cos \left (\frac {1}{2} (e+f x)\right )-31140 A \cos \left (\frac {3}{2} (e+f x)\right )-38580 B \cos \left (\frac {3}{2} (e+f x)\right )-10404 A \cos \left (\frac {5}{2} (e+f x)\right )-12724 B \cos \left (\frac {5}{2} (e+f x)\right )+435 A \cos \left (\frac {7}{2} (e+f x)\right )+7775 B \cos \left (\frac {7}{2} (e+f x)\right )-45 A \cos \left (\frac {9}{2} (e+f x)\right )+255 B \cos \left (\frac {9}{2} (e+f x)\right )+(240+240 i) \sqrt [4]{-1} (3 A-17 B) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}+56370 A \sin \left (\frac {1}{2} (e+f x)\right )+38970 B \sin \left (\frac {1}{2} (e+f x)\right )+31140 A \sin \left (\frac {3}{2} (e+f x)\right )+38580 B \sin \left (\frac {3}{2} (e+f x)\right )-10404 A \sin \left (\frac {5}{2} (e+f x)\right )-12724 B \sin \left (\frac {5}{2} (e+f x)\right )-435 A \sin \left (\frac {7}{2} (e+f x)\right )-7775 B \sin \left (\frac {7}{2} (e+f x)\right )-45 A \sin \left (\frac {9}{2} (e+f x)\right )+255 B \sin \left (\frac {9}{2} (e+f x)\right )\right )}{122880 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(525\) vs.
\(2(235)=470\).
time = 10.56, size = 526, normalized size = 1.98
method | result | size |
default | \(\frac {a^{3} \left (15 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{6} \left (3 A -17 B \right ) \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-180 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{6} \left (3 A -17 B \right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+240 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{6} \left (3 A -17 B \right ) \sin \left (f x +e \right )-75 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{6} \left (3 A -17 B \right ) \left (\cos ^{4}\left (f x +e \right )\right )+300 \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, c^{6} \left (3 A -17 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )-90 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {9}{2}} c^{\frac {3}{2}}+840 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {5}{2}}+3072 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {7}{2}}-3360 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {9}{2}}+1440 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {11}{2}}+510 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {9}{2}} c^{\frac {3}{2}}+5480 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {7}{2}} c^{\frac {5}{2}}-17408 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {7}{2}}+19040 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {9}{2}}-8160 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {11}{2}}-720 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}+4080 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{15360 c^{\frac {23}{2}} \left (\sin \left (f x +e \right )-1\right )^{4} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(526\) |
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. 806 vs.
\(2 (247) = 494\).
time = 0.39, size = 806, normalized size = 3.03 \begin {gather*} -\frac {15 \, \sqrt {2} {\left ({\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{6} - 5 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 18 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 20 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 48 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 16 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right ) - 32 \, {\left (3 \, A - 17 \, B\right )} a^{3} + {\left ({\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} + 6 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} - 12 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - 32 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 16 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right ) + 32 \, {\left (3 \, A - 17 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (15 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (39 \, A + 803 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 4 \, {\left (609 \, A + 389 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (449 \, A + 869 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 24 \, {\left (143 \, A + 43 \, B\right )} a^{3} \cos \left (f x + e\right ) - 6144 \, {\left (A + B\right )} a^{3} + {\left (15 \, {\left (3 \, A - 17 \, B\right )} a^{3} \cos \left (f x + e\right )^{4} + 80 \, {\left (3 \, A + 47 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} + 12 \, {\left (223 \, A + 443 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 24 \, {\left (113 \, A + 213 \, B\right )} a^{3} \cos \left (f x + e\right ) - 6144 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{30720 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 1037 vs.
\(2 (247) = 494\).
time = 0.76, size = 1037, normalized size = 3.90 \begin {gather*} \text {Too large to display} \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 = -0.00 \begin {gather*} \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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