Optimal. Leaf size=229 \[ \frac{149 \cot (c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{363 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.3136, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2880, 2779, 2984, 2985, 2649, 206, 2773, 3044} \[ \frac{149 \cot (c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{363 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 a^{5/2} d}+\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2880
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) \csc (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\int \frac{\csc ^5(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 ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{2 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^4(c+d x) \left (-\frac{a}{2}+\frac{15}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{4 a^3}+\frac{\int \frac{\csc ^3(c+d x) (a-5 a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{3 a^3}\\ &=-\frac{\cot (c+d x) \csc (c+d x)}{6 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^3(c+d x) \left (\frac{91 a^2}{4}-\frac{5}{4} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{12 a^4}+\frac{\int \frac{\csc ^2(c+d x) \left (-\frac{21 a^2}{2}+\frac{3}{2} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{6 a^4}\\ &=\frac{7 \cot (c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc ^2(c+d x) \left (-\frac{111 a^3}{8}+\frac{273}{8} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{24 a^5}+\frac{\int \frac{\csc (c+d x) \left (\frac{27 a^3}{4}-\frac{21}{4} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{6 a^5}\\ &=\frac{149 \cot (c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\int \frac{\csc (c+d x) \left (\frac{657 a^4}{16}-\frac{111}{16} a^4 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{24 a^6}+\frac{9 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{8 a^3}-\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{149 \cot (c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{219 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx}{128 a^3}-\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}-\frac{9 \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 )}{4 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{9 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4 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{149 \cot (c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{219 \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 )}{64 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{363 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 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{149 \cot (c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{107 \cot (c+d x) \csc (c+d x)}{96 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{17 \cot (c+d x) \csc ^2(c+d x)}{24 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 5.01605, size = 414, normalized size = 1.81 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5 \left (-\frac{16 \csc ^{12}\left (\frac{1}{2} (c+d x)\right ) \left (-6250 \sin \left (\frac{1}{2} (c+d x)\right )-4626 \sin \left (\frac{3}{2} (c+d x)\right )+1750 \sin \left (\frac{5}{2} (c+d x)\right )+894 \sin \left (\frac{7}{2} (c+d x)\right )+6250 \cos \left (\frac{1}{2} (c+d x)\right )-4626 \cos \left (\frac{3}{2} (c+d x)\right )-1750 \cos \left (\frac{5}{2} (c+d x)\right )+894 \cos \left (\frac{7}{2} (c+d x)\right )-4356 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+1089 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+3267 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+4356 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-1089 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-3267 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{\left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^4}-(24576+24576 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )\right )}{3072 d (a (\sin (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 1.201, size = 200, normalized size = 0.9 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{192\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 1089\,{a}^{7}{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}+447\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{7/2}-1127\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{9/2}-768\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{4}+1049\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{11/2}-321\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{13/2} \right ){a}^{-{\frac{19}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.28645, size = 1751, normalized size = 7.65 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.65691, size = 1141, normalized size = 4.98 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]