Optimal. Leaf size=286 \[ -\frac{a^3 \cot ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.462119, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2611, 3768, 3770, 2607, 270} \[ -\frac{a^3 \cot ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}+\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}+\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2873
Rule 2611
Rule 3768
Rule 3770
Rule 2607
Rule 270
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc ^8(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^6(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^6(c+d x)+3 a^3 \cot ^6(c+d x) \csc ^7(c+d x)+a^3 \cot ^6(c+d x) \csc ^8(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+a^3 \int \cot ^6(c+d x) \csc ^8(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^7(c+d x) \, dx\\ &=-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{1}{2} a^3 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx-\frac{1}{4} \left (5 a^3\right ) \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}+\frac{1}{16} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{1}{8} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx+\frac{a^3 \operatorname{Subst}\left (\int \left (x^6+3 x^8+3 x^{10}+x^{12}\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^{13}(c+d x)}{13 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{32 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{1}{32} a^3 \int \csc ^5(c+d x) \, dx-\frac{1}{64} \left (3 a^3\right ) \int \csc ^7(c+d x) \, dx\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^{13}(c+d x)}{13 d}+\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx-\frac{1}{128} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx\\ &=-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^{13}(c+d x)}{13 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{256 d}+\frac{9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac{1}{512} \left (15 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^{13}(c+d x)}{13 d}+\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac{9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}-\frac{\left (15 a^3\right ) \int \csc (c+d x) \, dx}{1024}\\ &=\frac{27 a^3 \tanh ^{-1}(\cos (c+d x))}{1024 d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{a^3 \cot ^9(c+d x)}{d}-\frac{6 a^3 \cot ^{11}(c+d x)}{11 d}-\frac{a^3 \cot ^{13}(c+d x)}{13 d}+\frac{27 a^3 \cot (c+d x) \csc (c+d x)}{1024 d}+\frac{9 a^3 \cot (c+d x) \csc ^3(c+d x)}{512 d}-\frac{3 a^3 \cot (c+d x) \csc ^5(c+d x)}{128 d}+\frac{a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^3 \cot ^5(c+d x) \csc ^5(c+d x)}{10 d}-\frac{3 a^3 \cot (c+d x) \csc ^7(c+d x)}{64 d}+\frac{a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{8 d}-\frac{a^3 \cot ^5(c+d x) \csc ^7(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 6.48981, size = 283, normalized size = 0.99 \[ \frac{27 (a \sin (c+d x)+a)^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{1024 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}-\frac{27 (a \sin (c+d x)+a)^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{1024 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6}+\frac{\cot (c+d x) \csc ^{12}(c+d x) (a \sin (c+d x)+a)^3 (-194159966 \sin (c+d x)-182107926 \sin (3 (c+d x))-123736613 \sin (5 (c+d x))+4571567 \sin (7 (c+d x))+1846845 \sin (9 (c+d x))-135135 \sin (11 (c+d x))-243712000 \cos (2 (c+d x))-11079680 \cos (4 (c+d x))+43294720 \cos (6 (c+d x))+9420800 \cos (8 (c+d x))-1433600 \cos (10 (c+d x))+102400 \cos (12 (c+d x))-200294400)}{5248122880 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.098, size = 312, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{12}}}-{\frac{40\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{1001\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{1024\,d}}-{\frac{27\,{a}^{3}\cos \left ( dx+c \right ) }{1024\,d}}-{\frac{27\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{1024\,d}}-{\frac{27\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5120\,d}}-{\frac{45\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{143\,d \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{20\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{143\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{40\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{27\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{320\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{640\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{9\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{2560\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{27\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{5120\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{13\,d \left ( \sin \left ( dx+c \right ) \right ) ^{13}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06445, size = 497, normalized size = 1.74 \begin{align*} -\frac{15015 \, a^{3}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{11} - 85 \, \cos \left (d x + c\right )^{9} + 198 \, \cos \left (d x + c\right )^{7} + 198 \, \cos \left (d x + c\right )^{5} - 85 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{12} - 6 \, \cos \left (d x + c\right )^{10} + 15 \, \cos \left (d x + c\right )^{8} - 20 \, \cos \left (d x + c\right )^{6} + 15 \, \cos \left (d x + c\right )^{4} - 6 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12012 \, a^{3}{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{133120 \,{\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{3}}{\tan \left (d x + c\right )^{11}} + \frac{10240 \,{\left (429 \, \tan \left (d x + c\right )^{6} + 1001 \, \tan \left (d x + c\right )^{4} + 819 \, \tan \left (d x + c\right )^{2} + 231\right )} a^{3}}{\tan \left (d x + c\right )^{13}}}{30750720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.36826, size = 1137, normalized size = 3.98 \begin{align*} \frac{409600 \, a^{3} \cos \left (d x + c\right )^{13} - 2662400 \, a^{3} \cos \left (d x + c\right )^{11} + 7321600 \, a^{3} \cos \left (d x + c\right )^{9} - 5857280 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \,{\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 135135 \,{\left (a^{3} \cos \left (d x + c\right )^{12} - 6 \, a^{3} \cos \left (d x + c\right )^{10} + 15 \, a^{3} \cos \left (d x + c\right )^{8} - 20 \, a^{3} \cos \left (d x + c\right )^{6} + 15 \, a^{3} \cos \left (d x + c\right )^{4} - 6 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 2002 \,{\left (135 \, a^{3} \cos \left (d x + c\right )^{11} - 765 \, a^{3} \cos \left (d x + c\right )^{9} + 758 \, a^{3} \cos \left (d x + c\right )^{7} + 1782 \, a^{3} \cos \left (d x + c\right )^{5} - 765 \, a^{3} \cos \left (d x + c\right )^{3} + 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{10250240 \,{\left (d \cos \left (d x + c\right )^{12} - 6 \, d \cos \left (d x + c\right )^{10} + 15 \, d \cos \left (d x + c\right )^{8} - 20 \, d \cos \left (d x + c\right )^{6} + 15 \, d \cos \left (d x + c\right )^{4} - 6 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \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 [A] time = 1.41425, size = 610, normalized size = 2.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]