Integrand size = 23, antiderivative size = 182 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {163 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {163 a^3 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a+a \sin (c+d x)}}-\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d} \]
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Time = 0.24 (sec) , antiderivative size = 182, 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, 3059, 2851, 2852, 212} \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {163 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{64 d}-\frac {163 a^3 \cot (c+d x)}{64 d \sqrt {a \sin (c+d x)+a}}-\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a \sin (c+d x)+a}}-\frac {163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a \sin (c+d x)+a}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{4 d} \]
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Rule 212
Rule 2841
Rule 2851
Rule 2852
Rule 3059
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {1}{4} a \int \csc ^4(c+d x) \left (-\frac {17 a}{2}-\frac {13}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {1}{48} \left (163 a^2\right ) \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a+a \sin (c+d x)}}-\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {1}{64} \left (163 a^2\right ) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {163 a^3 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a+a \sin (c+d x)}}-\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {1}{128} \left (163 a^2\right ) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {163 a^3 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a+a \sin (c+d x)}}-\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {\left (163 a^3\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 )}{64 d} \\ & = -\frac {163 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{64 d}-\frac {163 a^3 \cot (c+d x)}{64 d \sqrt {a+a \sin (c+d x)}}-\frac {163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt {a+a \sin (c+d x)}}-\frac {17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(370\) vs. \(2(182)=364\).
Time = 6.09 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.03 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {a^2 \csc ^{13}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-1030 \cos \left (\frac {1}{2} (c+d x)\right )+3102 \cos \left (\frac {3}{2} (c+d x)\right )-326 \cos \left (\frac {5}{2} (c+d x)\right )-978 \cos \left (\frac {7}{2} (c+d x)\right )+1467 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-1956 \cos (2 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+489 \cos (4 (c+d x)) \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-1467 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+1956 \cos (2 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-489 \cos (4 (c+d x)) \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+1030 \sin \left (\frac {1}{2} (c+d x)\right )+3102 \sin \left (\frac {3}{2} (c+d x)\right )+326 \sin \left (\frac {5}{2} (c+d x)\right )-978 \sin \left (\frac {7}{2} (c+d x)\right )\right )}{192 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^4} \]
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Time = 75.63 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (1047 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {11}{2}}-2303 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {9}{2}}+1793 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {7}{2}}-489 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {5}{2}}+489 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{6} \left (\sin ^{4}\left (d x +c \right )\right )\right )}{192 a^{\frac {7}{2}} \sin \left (d x +c \right )^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(162\) |
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Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (158) = 316\).
Time = 0.31 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.60 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {489 \, {\left (a^{2} \cos \left (d x + c\right )^{5} + a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) + a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (489 \, a^{2} \cos \left (d x + c\right )^{4} + 326 \, a^{2} \cos \left (d x + c\right )^{3} - 836 \, a^{2} \cos \left (d x + c\right )^{2} - 374 \, a^{2} \cos \left (d x + c\right ) + 299 \, a^{2} + {\left (489 \, a^{2} \cos \left (d x + c\right )^{3} + 163 \, a^{2} \cos \left (d x + c\right )^{2} - 673 \, a^{2} \cos \left (d x + c\right ) - 299 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{768 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \csc \left (d x + c\right )^{5} \,d x } \]
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Time = 0.62 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.25 \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {\sqrt {2} {\left (489 \, \sqrt {2} a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (3912 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7172 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4606 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1047 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}\right )} \sqrt {a}}{768 \, d} \]
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Timed out. \[ \int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\sin \left (c+d\,x\right )}^5} \,d x \]
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