Integrand size = 23, antiderivative size = 144 \[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {11 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2841, 21, 2851, 2852, 212} \[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {11 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{8 d}-\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a \sin (c+d x)+a}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a \sin (c+d x)+a}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a \sin (c+d x)+a}} \]
[In]
[Out]
Rule 21
Rule 212
Rule 2841
Rule 2851
Rule 2852
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {1}{3} a \int \frac {\csc ^3(c+d x) \left (-\frac {11 a}{2}-\frac {11}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{6} (11 a) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{8} (11 a) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{16} (11 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {\left (11 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 )}{8 d} \\ & = -\frac {11 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {11 a^2 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {11 a^2 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x)}{3 d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Time = 2.70 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.99 \[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {a \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-108 \cos \left (\frac {1}{2} (c+d x)\right )-22 \cos \left (\frac {3}{2} (c+d x)\right )+66 \cos \left (\frac {5}{2} (c+d x)\right )+108 \sin \left (\frac {1}{2} (c+d x)\right )-99 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+99 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-22 \sin \left (\frac {3}{2} (c+d x)\right )-66 \sin \left (\frac {5}{2} (c+d x)\right )+33 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-33 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))\right )}{24 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3} \]
[In]
[Out]
Time = 0.73 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (33 a^{\frac {5}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}}-88 a^{\frac {7}{2}} \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}}+33 a^{5} \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{3}\left (d x +c \right )\right )+63 a^{\frac {9}{2}} \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\right )}{24 a^{\frac {7}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(144\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (124) = 248\).
Time = 0.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.64 \[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {33 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (33 \, a \cos \left (d x + c\right )^{3} + 11 \, a \cos \left (d x + c\right )^{2} - 41 \, a \cos \left (d x + c\right ) - {\left (33 \, a \cos \left (d x + c\right )^{2} + 22 \, a \cos \left (d x + c\right ) - 19 \, a\right )} \sin \left (d x + c\right ) - 19 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]
[In]
[Out]
Timed out. \[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \csc \left (d x + c\right )^{4} \,d x } \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.31 \[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (33 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (132 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 176 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 63 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{96 \, d} \]
[In]
[Out]
Timed out. \[ \int \csc ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^4} \,d x \]
[In]
[Out]