Integrand size = 23, antiderivative size = 224 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {39 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 a^{5/2} d}+\frac {219 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {63 \cot (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.50 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2845, 3057, 3063, 3064, 2728, 212, 2852} \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {39 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{4 a^{5/2} d}+\frac {219 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {63 \cot (c+d x)}{16 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}} \]
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Rule 212
Rule 2728
Rule 2845
Rule 2852
Rule 3057
Rule 3063
Rule 3064
Rubi steps \begin{align*} \text {integral}& = \frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {\int \frac {\csc ^3(c+d x) \left (6 a-\frac {7}{2} a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{3/2}} \, dx}{4 a^2} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {\int \frac {\csc ^3(c+d x) \left (31 a^2-\frac {95}{4} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{8 a^4} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc ^2(c+d x) \left (-63 a^3+\frac {93}{2} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{16 a^5} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {63 \cot (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\int \frac {\csc (c+d x) \left (78 a^4-\frac {63}{2} a^4 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{16 a^6} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {63 \cot (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {39 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{8 a^3}-\frac {219 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{32 a^2} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {63 \cot (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {39 \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 a^2 d}+\frac {219 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{16 a^2 d} \\ & = -\frac {39 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 a^{5/2} d}+\frac {219 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cot (c+d x) \csc (c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {19 \cot (c+d x) \csc (c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}+\frac {63 \cot (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {31 \cot (c+d x) \csc (c+d x)}{16 a^2 d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.96 (sec) , antiderivative size = 680, normalized size of antiderivative = 3.04 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (-16 \sin \left (\frac {1}{2} (c+d x)\right )+8 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-108 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2+54 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3-40 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4-(438+438 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} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+20 \cot \left (\frac {1}{4} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4-\csc ^2\left (\frac {1}{4} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4-156 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+156 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+\sec ^2\left (\frac {1}{4} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+\frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}{\left (\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}-\frac {40 \sin \left (\frac {1}{4} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}{\cos \left (\frac {1}{4} (c+d x)\right )-\sin \left (\frac {1}{4} (c+d x)\right )}-\frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}{\left (\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )\right )^2}+\frac {40 \sin \left (\frac {1}{4} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}{\cos \left (\frac {1}{4} (c+d x)\right )+\sin \left (\frac {1}{4} (c+d x)\right )}+20 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \tan \left (\frac {1}{4} (c+d x)\right )\right )}{32 d (a (1+\sin (c+d x)))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(403\) vs. \(2(189)=378\).
Time = 0.99 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.80
method | result | size |
default | \(\frac {\left (219 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (\sin ^{4}\left (d x +c \right )\right ) a^{2}+438 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (\sin ^{3}\left (d x +c \right )\right ) a^{2}-312 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{4}\left (d x +c \right )\right ) a^{2}+219 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}+172 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}} \left (\sin ^{2}\left (d x +c \right )\right )-126 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}\, \left (\sin ^{2}\left (d x +c \right )\right )-624 \,\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 ) a^{2}+112 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \sin \left (d x +c \right ) a^{\frac {3}{2}}-144 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}\, \sin \left (d x +c \right )-312 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}+56 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}}-72 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{32 a^{\frac {9}{2}} \sin \left (d x +c \right )^{2} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(404\) |
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Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (189) = 378\).
Time = 0.31 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.19 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {219 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 4\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 156 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} - 3 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 4\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 (63 \, \cos \left (d x + c\right )^{4} + 95 \, \cos \left (d x + c\right )^{3} - 51 \, \cos \left (d x + c\right )^{2} + {\left (63 \, \cos \left (d x + c\right )^{3} - 32 \, \cos \left (d x + c\right )^{2} - 83 \, \cos \left (d x + c\right ) + 4\right )} \sin \left (d x + c\right ) - 87 \, \cos \left (d x + c\right ) - 4\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{64 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} - 3 \, a^{3} d \cos \left (d x + c\right )^{3} - 7 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 5 \, a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.60 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.33 \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\frac {219 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {219 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {312 \, \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {312 \, \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (252 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 568 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 399 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 85 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{64 \, d} \]
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Timed out. \[ \int \frac {\csc ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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