Integrand size = 35, antiderivative size = 151 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=-\frac {(3 A c+5 B c+5 A d+19 B d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(A-B) (c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 A c+5 B c+5 A d-13 B d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3047, 3098, 2829, 2728, 212} \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=-\frac {(3 A c+5 A d+5 B c+19 B d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(3 A c+5 A d+5 B c-13 B d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac {(A-B) (c-d) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}} \]
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Rule 212
Rule 2728
Rule 2829
Rule 3047
Rule 3098
Rubi steps \begin{align*} \text {integral}& = \int \frac {A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx \\ & = -\frac {(A-B) (c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {\int \frac {-\frac {1}{2} a (3 A c+5 B c+5 A d-5 B d)-4 a B d \sin (e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx}{4 a^2} \\ & = -\frac {(A-B) (c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 A c+5 B c+5 A d-13 B d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {(3 A c+5 B c+5 A d+19 B d) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2} \\ & = -\frac {(A-B) (c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 A c+5 B c+5 A d-13 B d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(3 A c+5 B c+5 A d+19 B d) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{16 a^2 f} \\ & = -\frac {(3 A c+5 B c+5 A d+19 B d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(A-B) (c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 A c+5 B c+5 A d-13 B d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.58 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.77 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (8 (A-B) (c-d) \sin \left (\frac {1}{2} (e+f x)\right )-4 (A-B) (c-d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 (3 A c+5 B c+5 A d-13 B d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-(3 A c+5 B c+5 A d-13 B d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+(1+i) (-1)^{3/4} (3 A c+5 B c+5 A d+19 B d) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4\right )}{16 f (a (1+\sin (e+f x)))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(132)=264\).
Time = 2.59 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.97
method | result | size |
default | \(-\frac {\left (-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \left (3 A c +5 d A +5 B c +19 d B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sin \left (f x +e \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \left (3 A c +5 d A +5 B c +19 d B \right )+6 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c +10 A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d +20 A \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c +12 A \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} d -6 A \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\, c -10 A \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\, d +10 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c +38 B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d +12 B \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c -44 B \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} d -10 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\, c +26 B \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\, d \right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{32 a^{\frac {9}{2}} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(449\) |
parts | \(-\frac {A c \left (-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \left (\cos ^{2}\left (f x +e \right )\right )+6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )+6 \sqrt {a -a \sin \left (f x +e \right )}\, \sin \left (f x +e \right ) a^{\frac {3}{2}}+6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+14 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{32 a^{\frac {9}{2}} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {\left (d A +B c \right ) \left (5 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (\cos ^{2}\left (f x +e \right )\right )+10 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-12 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {5}{2}}-10 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (f x +e \right ) a^{3}-10 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{32 a^{\frac {11}{2}} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}-\frac {d B \left (-19 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \left (\cos ^{2}\left (f x +e \right )\right )+38 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )+38 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}-44 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}+26 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{32 a^{\frac {9}{2}} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(595\) |
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Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (132) = 264\).
Time = 0.27 (sec) , antiderivative size = 536, normalized size of antiderivative = 3.55 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} {\left ({\left ({\left (3 \, A + 5 \, B\right )} c + {\left (5 \, A + 19 \, B\right )} d\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (3 \, A + 5 \, B\right )} c + {\left (5 \, A + 19 \, B\right )} d\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (3 \, A + 5 \, B\right )} c - 4 \, {\left (5 \, A + 19 \, B\right )} d - 2 \, {\left ({\left (3 \, A + 5 \, B\right )} c + {\left (5 \, A + 19 \, B\right )} d\right )} \cos \left (f x + e\right ) + {\left ({\left ({\left (3 \, A + 5 \, B\right )} c + {\left (5 \, A + 19 \, B\right )} d\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left (3 \, A + 5 \, B\right )} c - 4 \, {\left (5 \, A + 19 \, B\right )} d - 2 \, {\left ({\left (3 \, A + 5 \, B\right )} c + {\left (5 \, A + 19 \, B\right )} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\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 ({\left ({\left (3 \, A + 5 \, B\right )} c + {\left (5 \, A - 13 \, B\right )} d\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (A - B\right )} c - 4 \, {\left (A - B\right )} d + {\left ({\left (7 \, A + B\right )} c + {\left (A - 9 \, B\right )} d\right )} \cos \left (f x + e\right ) - {\left (4 \, {\left (A - B\right )} c - 4 \, {\left (A - B\right )} d - {\left ({\left (3 \, A + 5 \, B\right )} c + {\left (5 \, A - 13 \, B\right )} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{64 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (d \sin \left (f x + e\right ) + c\right )}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (132) = 264\).
Time = 0.35 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.41 \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {\frac {\sqrt {2} {\left (3 \, A \sqrt {a} c + 5 \, B \sqrt {a} c + 5 \, A \sqrt {a} d + 19 \, B \sqrt {a} d\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (3 \, A \sqrt {a} c + 5 \, B \sqrt {a} c + 5 \, A \sqrt {a} d + 19 \, B \sqrt {a} d\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {2 \, {\left (3 \, \sqrt {2} A \sqrt {a} c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, \sqrt {2} B \sqrt {a} c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, \sqrt {2} A \sqrt {a} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 13 \, \sqrt {2} B \sqrt {a} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, \sqrt {2} A \sqrt {a} c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, \sqrt {2} B \sqrt {a} c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, \sqrt {2} A \sqrt {a} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11 \, \sqrt {2} B \sqrt {a} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{64 \, f} \]
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Timed out. \[ \int \frac {(A+B \sin (e+f x)) (c+d \sin (e+f x))}{(a+a \sin (e+f x))^{5/2}} \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (c+d\,\sin \left (e+f\,x\right )\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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