Optimal. Leaf size=182 \[ \frac{3 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^3 x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.248143, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2872, 3770, 3767, 8, 3768, 2638, 2635} \[ \frac{3 a^3 \cos (c+d x)}{d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac{43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{a^3 x}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2872
Rule 3770
Rule 3767
Rule 8
Rule 3768
Rule 2638
Rule 2635
Rubi steps
\begin{align*} \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac{\int \left (8 a^9 \csc (c+d x)+6 a^9 \csc ^2(c+d x)-6 a^9 \csc ^3(c+d x)-8 a^9 \csc ^4(c+d x)+3 a^9 \csc ^6(c+d x)+a^9 \csc ^7(c+d x)-3 a^9 \sin (c+d x)-a^9 \sin ^2(c+d x)\right ) \, dx}{a^6}\\ &=a^3 \int \csc ^7(c+d x) \, dx-a^3 \int \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \csc ^6(c+d x) \, dx-\left (3 a^3\right ) \int \sin (c+d x) \, dx+\left (6 a^3\right ) \int \csc ^2(c+d x) \, dx-\left (6 a^3\right ) \int \csc ^3(c+d x) \, dx+\left (8 a^3\right ) \int \csc (c+d x) \, dx-\left (8 a^3\right ) \int \csc ^4(c+d x) \, dx\\ &=-\frac{8 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^3 \cos (c+d x)}{d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} a^3 \int 1 \, dx+\frac{1}{6} \left (5 a^3\right ) \int \csc ^5(c+d x) \, dx-\left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (6 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}+\frac{\left (8 a^3\right ) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{a^3 x}{2}-\frac{5 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{8} \left (5 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac{a^3 x}{2}-\frac{5 a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{16} \left (5 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac{a^3 x}{2}-\frac{85 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac{3 a^3 \cos (c+d x)}{d}-\frac{a^3 \cot (c+d x)}{d}+\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a^3 \cot ^5(c+d x)}{5 d}+\frac{43 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{5 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 1.81905, size = 289, normalized size = 1.59 \[ \frac{a^3 (\sin (c+d x)+1)^3 \left (-960 (c+d x)+480 \sin (2 (c+d x))+5760 \cos (c+d x)+2176 \tan \left (\frac{1}{2} (c+d x)\right )-2176 \cot \left (\frac{1}{2} (c+d x)\right )-5 \csc ^6\left (\frac{1}{2} (c+d x)\right )-30 \csc ^4\left (\frac{1}{2} (c+d x)\right )+1290 \csc ^2\left (\frac{1}{2} (c+d x)\right )+5 \sec ^6\left (\frac{1}{2} (c+d x)\right )+30 \sec ^4\left (\frac{1}{2} (c+d x)\right )-1290 \sec ^2\left (\frac{1}{2} (c+d x)\right )+10200 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-10200 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-3296 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \csc ^3(c+d x)-18 \sin (c+d x) \csc ^6\left (\frac{1}{2} (c+d x)\right )+206 \sin (c+d x) \csc ^4\left (\frac{1}{2} (c+d x)\right )+36 \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right )\right )}{1920 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.102, size = 316, normalized size = 1.7 \begin{align*}{\frac{4\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16\,d}}-{\frac{3\,{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}}-3\,{\frac{{a}^{3}\cot \left ( dx+c \right ) }{d}}-{\frac{{a}^{3}c}{2\,d}}+{\frac{85\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{48\,d}}+{\frac{85\,{a}^{3}\cos \left ( dx+c \right ) }{16\,d}}+{\frac{85\,{a}^{3}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{16\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{{a}^{3}x}{2}}-{\frac{{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}+{\frac{4\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3\,d\sin \left ( dx+c \right ) }}+{\frac{5\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{5\,{a}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}-{\frac{17\,{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{24\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.63581, size = 371, normalized size = 2.04 \begin{align*} \frac{80 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a^{3} - 96 \,{\left (15 \, d x + 15 \, c + \frac{15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{3} + 5 \, a^{3}{\left (\frac{2 \,{\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \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 )} - 90 \, a^{3}{\left (\frac{2 \,{\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.24269, size = 817, normalized size = 4.49 \begin{align*} -\frac{240 \, a^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, a^{3} \cos \left (d x + c\right )^{7} - 720 \, a^{3} d x \cos \left (d x + c\right )^{4} + 5610 \, a^{3} \cos \left (d x + c\right )^{5} + 720 \, a^{3} d x \cos \left (d x + c\right )^{2} - 6800 \, a^{3} \cos \left (d x + c\right )^{3} - 240 \, a^{3} d x + 2550 \, a^{3} \cos \left (d x + c\right ) + 1275 \,{\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, 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 ) - 1275 \,{\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, 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 ) - 16 \,{\left (15 \, a^{3} \cos \left (d x + c\right )^{7} + 23 \, a^{3} \cos \left (d x + c\right )^{5} - 35 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\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 = 1.36252, size = 414, normalized size = 2.27 \begin{align*} \frac{5 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 45 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 340 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1215 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 960 \,{\left (d x + c\right )} a^{3} + 10200 \, a^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 1800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{1920 \,{\left (a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac{24990 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1800 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1215 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 340 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 45 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 36 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]