Optimal. Leaf size=191 \[ -\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{45 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 a^{5/2} d}-\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.947277, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {2717, 2779, 2984, 2985, 2649, 206, 2773, 3044} \[ -\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{45 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{8 a^{5/2} d}-\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2717
Rule 2779
Rule 2984
Rule 2985
Rule 2649
Rule 206
Rule 2773
Rule 3044
Rubi steps
\begin{align*} \int \frac{\cot ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\int \frac{\csc ^4(c+d x) \left (1+\sin ^2(c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}-\frac{2 \int \frac{\csc ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{\cot (c+d x) \csc (c+d x)}{a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^3(c+d x) \left (-\frac{a}{2}+\frac{11}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{3 a^3}+\frac{\int \frac{\csc ^2(c+d x) (a-3 a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{2 a^3}\\ &=-\frac{\cot (c+d x)}{2 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^2(c+d x) \left (\frac{45 a^2}{4}-\frac{3}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{6 a^4}+\frac{\int \frac{\csc (c+d x) \left (-\frac{7 a^2}{2}+\frac{1}{2} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{2 a^4}\\ &=-\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc (c+d x) \left (-\frac{51 a^3}{8}+\frac{45}{8} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{6 a^5}-\frac{7 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{4 a^3}+\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=-\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{17 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{16 a^3}+\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{2 a^2 d}-\frac{4 \operatorname{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 )}{a^2 d}\\ &=\frac{7 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{2 a^{5/2} d}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{a^{5/2} d}-\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{17 \operatorname{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 a^2 d}-\frac{4 \operatorname{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 )}{a^2 d}\\ &=\frac{45 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{8 a^{5/2} d}-\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{a^{5/2} d}-\frac{19 \cot (c+d x)}{8 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{13 \cot (c+d x) \csc (c+d x)}{12 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 2.3391, size = 332, normalized size = 1.74 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5 \left (-\frac{8 \csc ^9\left (\frac{1}{2} (c+d x)\right ) \left (-396 \sin \left (\frac{1}{2} (c+d x)\right )-218 \sin \left (\frac{3}{2} (c+d x)\right )+114 \sin \left (\frac{5}{2} (c+d x)\right )+396 \cos \left (\frac{1}{2} (c+d x)\right )-218 \cos \left (\frac{3}{2} (c+d x)\right )-114 \cos \left (\frac{5}{2} (c+d x)\right )-405 \sin (c+d x) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+405 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+135 \sin (3 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-135 \sin (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{\left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^3}+(1536+1536 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{192 d (a (\sin (c+d x)+1))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.19, size = 182, normalized size = 1. \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{24\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( -135\,{a}^{5}{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}+57\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{5/2}+96\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{3}-88\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{7/2}+39\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{9/2} \right ){a}^{-{\frac{15}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.29704, size = 1504, normalized size = 7.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.63869, size = 938, normalized size = 4.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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