Integrand size = 29, antiderivative size = 186 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {9 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}+\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d} \]
[Out]
Time = 0.44 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2960, 2830, 2726, 2725, 3123, 3054, 3060, 2852, 212} \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {9 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{4 d}+\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
[In]
[Out]
Rule 212
Rule 2725
Rule 2726
Rule 2830
Rule 2852
Rule 2960
Rule 3054
Rule 3060
Rule 3123
Rubi steps \begin{align*} \text {integral}& = \int \sin (c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^3(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {3}{5} \int (a+a \sin (c+d x))^{3/2} \, dx+\frac {\int \csc ^2(c+d x) \left (\frac {3 a}{2}-\frac {9}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{2 a} \\ & = -\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {9 a^2}{4}-\frac {21}{4} a^2 \sin (c+d x)\right ) \, dx}{2 a}+\frac {1}{5} (4 a) \int \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}-\frac {1}{8} (9 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {\left (9 a^2\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d} \\ & = \frac {9 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}+\frac {73 a^2 \cos (c+d x)}{20 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d} \\ \end{align*}
Time = 6.14 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.73 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \csc ^7\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-118 \cos \left (\frac {1}{2} (c+d x)\right )+130 \cos \left (\frac {3}{2} (c+d x)\right )+36 \cos \left (\frac {5}{2} (c+d x)\right )-10 \cos \left (\frac {7}{2} (c+d x)\right )+2 \cos \left (\frac {9}{2} (c+d x)\right )-45 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+45 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-45 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+118 \sin \left (\frac {1}{2} (c+d x)\right )+130 \sin \left (\frac {3}{2} (c+d x)\right )-36 \sin \left (\frac {5}{2} (c+d x)\right )-10 \sin \left (\frac {7}{2} (c+d x)\right )-2 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{20 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^2} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} \left (\sin ^{2}\left (d x +c \right )\right ) \sqrt {a}-40 a^{\frac {3}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \left (\sin ^{2}\left (d x +c \right )\right )-45 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{3} \left (\sin ^{2}\left (d x +c \right )\right )-35 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}+45 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {5}{2}}\right )}{20 a^{\frac {3}{2}} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(178\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (158) = 316\).
Time = 0.30 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.17 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {45 \, {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (8 \, a \cos \left (d x + c\right )^{5} - 16 \, a \cos \left (d x + c\right )^{4} + 16 \, a \cos \left (d x + c\right )^{3} + 99 \, a \cos \left (d x + c\right )^{2} - 14 \, a \cos \left (d x + c\right ) - {\left (8 \, a \cos \left (d x + c\right )^{4} + 24 \, a \cos \left (d x + c\right )^{3} + 40 \, a \cos \left (d x + c\right )^{2} - 59 \, a \cos \left (d x + c\right ) - 73 \, a\right )} \sin \left (d x + c\right ) - 73 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{80 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{3} \,d x } \]
[In]
[Out]
none
Time = 0.77 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.17 \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\sqrt {2} {\left (128 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 320 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {20 \, {\left (14 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}\right )} \sqrt {a}}{80 \, d} \]
[In]
[Out]
Timed out. \[ \int \cos (c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^3} \,d x \]
[In]
[Out]