Integrand size = 27, antiderivative size = 178 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {(c-d)^2 (c+11 d) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)}}\right )}{6 \sqrt {6} f}+\frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{9 f \sqrt {3+3 \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{54 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (3+3 \sin (e+f x))^{3/2}} \]
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Time = 0.34 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2844, 3047, 3102, 2830, 2728, 212} \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {(c+11 d) (c-d)^2 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d^2 (3 c-7 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{6 a^2 f}+\frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a \sin (e+f x)+a}}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a \sin (e+f x)+a)^{3/2}} \]
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Rule 212
Rule 2728
Rule 2830
Rule 2844
Rule 3047
Rule 3102
Rubi steps \begin{align*} \text {integral}& = -\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\int \frac {(c+d \sin (e+f x)) \left (-\frac {1}{2} a \left (c^2+7 c d-4 d^2\right )+\frac {1}{2} a (3 c-7 d) d \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2} \\ & = -\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{2} a c \left (c^2+7 c d-4 d^2\right )+\left (\frac {1}{2} a c (3 c-7 d) d-\frac {1}{2} a d \left (c^2+7 c d-4 d^2\right )\right ) \sin (e+f x)+\frac {1}{2} a (3 c-7 d) d^2 \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2} \\ & = \frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\int \frac {-\frac {1}{4} a^2 \left (3 c^3+21 c^2 d-15 c d^2+7 d^3\right )+\frac {1}{2} a^2 d \left (3 c^2-24 c d+13 d^2\right ) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 a^3} \\ & = \frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}+\frac {\left ((c-d)^2 (c+11 d)\right ) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 a} \\ & = \frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}}-\frac {\left ((c-d)^2 (c+11 d)\right ) \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 )}{2 a f} \\ & = -\frac {(c-d)^2 (c+11 d) \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {d \left (3 c^2-24 c d+13 d^2\right ) \cos (e+f x)}{3 a f \sqrt {a+a \sin (e+f x)}}+\frac {(3 c-7 d) d^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{6 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{2 f (a+a \sin (e+f x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.66 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (6 (c-d)^3 \sin \left (\frac {1}{2} (e+f x)\right )-3 (c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+(3+3 i) (-1)^{3/4} (c-d)^2 (c+11 d) \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-18 (2 c-d) d^2 \cos \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 d^3 \cos \left (\frac {3}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+18 (2 c-d) d^2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-2 d^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {3}{2} (e+f x)\right )\right )}{18 \sqrt {3} f (1+\sin (e+f x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(489\) vs. \(2(169)=338\).
Time = 3.18 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.75
method | result | size |
default | \(\frac {\left (\sin \left (f x +e \right ) \left (8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} d^{3} \sqrt {a}-72 a^{\frac {3}{2}} c \,d^{2} \sqrt {a -a \sin \left (f x +e \right )}+24 a^{\frac {3}{2}} d^{3} \sqrt {a -a \sin \left (f x +e \right )}-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}-27 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d +63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}\right )+8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} d^{3} \sqrt {a}-6 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{3}+18 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}} c^{2} d -90 a^{\frac {3}{2}} c \,d^{2} \sqrt {a -a \sin \left (f x +e \right )}+30 a^{\frac {3}{2}} d^{3} \sqrt {a -a \sin \left (f x +e \right )}-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{3}-27 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c^{2} d +63 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} c \,d^{2}-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} d^{3}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{12 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(490\) |
parts | \(-\frac {c^{3} \left (\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 )+2 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}+\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{4 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {d^{3} \left (8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \sqrt {a}-33 \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 )+8 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a}+24 \sqrt {a -a \sin \left (f x +e \right )}\, \sin \left (f x +e \right ) a^{\frac {3}{2}}-33 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+30 \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 )}}{12 a^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}-\frac {3 c \,d^{2} \left (-7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )+8 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a}\, \sin \left (f x +e \right )-7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +10 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{4 a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {3 c^{2} d \left (-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )-3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +2 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{4 a^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(600\) |
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Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (169) = 338\).
Time = 0.30 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.78 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {3 \, \sqrt {2} {\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} - {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (2 \, c^{3} + 18 \, c^{2} d - 42 \, c d^{2} + 22 \, d^{3} + {\left (c^{3} + 9 \, c^{2} d - 21 \, c d^{2} + 11 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (4 \, d^{3} \cos \left (f x + e\right )^{3} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} - 4 \, {\left (9 \, c d^{2} - 4 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - 3 \, {\left (c^{3} - 3 \, c^{2} d + 15 \, c d^{2} - 5 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (4 \, d^{3} \cos \left (f x + e\right )^{2} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} + 12 \, {\left (3 \, c d^{2} - d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{24 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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\[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{3}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (169) = 338\).
Time = 0.39 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.03 \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{3} + 9 \, \sqrt {a} c^{2} d - 21 \, \sqrt {a} c d^{2} + 11 \, \sqrt {a} d^{3}\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \sqrt {2} {\left (\sqrt {a} c^{3} + 9 \, \sqrt {a} c^{2} d - 21 \, \sqrt {a} c d^{2} + 11 \, \sqrt {a} d^{3}\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {6 \, \sqrt {2} {\left (\sqrt {a} c^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, \sqrt {a} c^{2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, \sqrt {a} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {a} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {16 \, \sqrt {2} {\left (2 \, a^{\frac {9}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{\frac {9}{2}} c d^{2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{\frac {9}{2}} d^{3} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{24 \, f} \]
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Timed out. \[ \int \frac {(c+d \sin (e+f x))^3}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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