Integrand size = 38, antiderivative size = 218 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {2} a^3 (5 A+9 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.39 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {3046, 2938, 2758, 2728, 212} \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {2} a^3 (5 A+9 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac {a^3 c^3 (A+B) \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 c (5 A+9 B) \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \]
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Rule 212
Rule 2728
Rule 2758
Rule 2938
Rule 3046
Rubi steps \begin{align*} \text {integral}& = \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))^{9/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}-\frac {1}{4} \left (a^3 (5 A+9 B) c^2\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}-\frac {1}{2} \left (a^3 (5 A+9 B) c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\left (a^3 (5 A+9 B)\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}-\frac {\left (2 a^3 (5 A+9 B)\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{c} \\ & = \frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}+\frac {\left (4 a^3 (5 A+9 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 )}{c f} \\ & = -\frac {2 \sqrt {2} a^3 (5 A+9 B) \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac {a^3 (A+B) c^3 \cos ^7(e+f x)}{2 f (c-c \sin (e+f x))^{9/2}}+\frac {a^3 (5 A+9 B) c \cos ^5(e+f x)}{10 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 (5 A+9 B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^3 (5 A+9 B) \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 11.75 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.04 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \left (120 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+(120+120 i) \sqrt [4]{-1} (5 A+9 B) \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+30 (9 A+20 B) \cos \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-5 (2 A+9 B) \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-3 B \cos \left (\frac {5}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+240 (A+B) \sin \left (\frac {1}{2} (e+f x)\right )+30 (9 A+20 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+5 (2 A+9 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {3}{2} (e+f x)\right )-3 B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {5}{2} (e+f x)\right )\right )}{30 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{3/2}} \]
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Time = 3.46 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.62
method | result | size |
default | \(\frac {2 a^{3} \left (\sin \left (f x +e \right ) \left (-60 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {5}{2}}-5 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {3}{2}}-120 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {5}{2}}-15 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {3}{2}}-3 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c}+75 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3}+135 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3}\right )+90 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {5}{2}}+5 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {3}{2}}+150 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {5}{2}}+15 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {3}{2}}+3 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c}-75 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3}-135 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{15 c^{\frac {9}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(354\) |
parts | \(\text {Expression too large to display}\) | \(787\) |
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Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (195) = 390\).
Time = 0.29 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.97 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {\frac {15 \, \sqrt {2} {\left ({\left (5 \, A + 9 \, B\right )} a^{3} c \cos \left (f x + e\right )^{2} - {\left (5 \, A + 9 \, B\right )} a^{3} c \cos \left (f x + e\right ) - 2 \, {\left (5 \, A + 9 \, B\right )} a^{3} c + {\left ({\left (5 \, A + 9 \, B\right )} a^{3} c \cos \left (f x + e\right ) + 2 \, {\left (5 \, A + 9 \, B\right )} a^{3} c\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} + 2 \, {\left (3 \, B a^{3} \cos \left (f x + e\right )^{4} - {\left (5 \, A + 18 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - {\left (65 \, A + 141 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 30 \, {\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right ) - 30 \, {\left (A + B\right )} a^{3} - {\left (3 \, B a^{3} \cos \left (f x + e\right )^{3} + {\left (5 \, A + 21 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 60 \, {\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right ) + 30 \, {\left (A + B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{15 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f + {\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (195) = 390\).
Time = 0.43 (sec) , antiderivative size = 730, normalized size of antiderivative = 3.35 \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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