Integrand size = 14, antiderivative size = 96 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)}}\right )}{48 \sqrt {6} f}-\frac {\cos (e+f x)}{4 f (3+3 \sin (e+f x))^{5/2}}-\frac {\cos (e+f x)}{16 f (3+3 \sin (e+f x))^{3/2}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2729, 2728, 212} \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {3 \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {3 \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac {\cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}} \]
[In]
[Out]
Rule 212
Rule 2728
Rule 2729
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {3 \int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx}{8 a} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {3 \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2} \\ & = -\frac {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {3 \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {3 \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 \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 {\cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {3 \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 = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.58 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (11 \cos \left (\frac {1}{2} (e+f x)\right )+3 \cos \left (\frac {3}{2} (e+f x)\right )-11 \sin \left (\frac {1}{2} (e+f x)\right )-(6+6 i) (-1)^{3/4} \text {arctanh}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4+3 \sin \left (\frac {3}{2} (e+f x)\right )\right )}{288 \sqrt {3} f (1+\sin (e+f x))^{5/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(88)=176\).
Time = 0.86 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.07
method | result | size |
default | \(-\frac {\left (-3 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {2}\, a^{2} \left (\cos ^{2}\left (f x +e \right )\right )+6 \sqrt {a -a \sin \left (f x +e \right )}\, \sin \left (f x +e \right ) a^{\frac {3}{2}}+6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \sin \left (f x +e \right )+14 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}+6 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{32 a^{\frac {9}{2}} \left (\sin \left (f x +e \right )+1\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(199\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (88) = 176\).
Time = 0.27 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.33 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {3 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 4\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (3 \, \cos \left (f x + e\right )^{2} + {\left (3 \, \cos \left (f x + e\right ) - 4\right )} \sin \left (f x + e\right ) + 7 \, \cos \left (f x + e\right ) + 4\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{64 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (a \sin {\left (e + f x \right )} + a\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.38 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (\frac {3 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {3 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {2 \, {\left (3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{64 \, \sqrt {a} f} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
[In]
[Out]