Optimal. Leaf size=216 \[ -\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2607, 14, 2611, 3768, 3770, 270} \[ -\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 270
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^6(c+d x)+a^3 \cot ^4(c+d x) \csc ^7(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx\\ &=-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {1}{10} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^7(c+d x) \, dx-\frac {1}{8} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {3 a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{80} \left (3 a^3\right ) \int \csc ^7(c+d x) \, dx+\frac {1}{16} \left (3 a^3\right ) \int \csc ^5(c+d x) \, dx+\frac {a^3 \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{32} a^3 \int \csc ^5(c+d x) \, dx+\frac {1}{64} \left (9 a^3\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {9 a^3 \cot (c+d x) \csc (c+d x)}{128 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{128} \left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\frac {1}{128} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac {9 a^3 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}+\frac {1}{256} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac {21 a^3 \tanh ^{-1}(\cos (c+d x))}{256 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{d}-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {21 a^3 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {7 a^3 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {29 a^3 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^3 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.17, size = 366, normalized size = 1.69 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (4096 \tan \left (\frac {1}{2} (c+d x)\right )-4096 \cot \left (\frac {1}{2} (c+d x)\right )-1260 \csc ^2\left (\frac {1}{2} (c+d x)\right )+6 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )+75 \sec ^8\left (\frac {1}{2} (c+d x)\right )-390 \sec ^6\left (\frac {1}{2} (c+d x)\right )-180 \sec ^4\left (\frac {1}{2} (c+d x)\right )+1260 \sec ^2\left (\frac {1}{2} (c+d x)\right )+5040 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-5040 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2 (10 \sin (c+d x)+3) \csc ^{10}\left (\frac {1}{2} (c+d x)\right )+5 (4 \sin (c+d x)-15) \csc ^8\left (\frac {1}{2} (c+d x)\right )+6 (42 \sin (c+d x)+65) \csc ^6\left (\frac {1}{2} (c+d x)\right )-4 (\sin (c+d x)-45) \csc ^4\left (\frac {1}{2} (c+d x)\right )+64 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+40 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^8\left (\frac {1}{2} (c+d x)\right )-40 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )-504 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )\right )}{61440 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 340, normalized size = 1.57 \[ \frac {630 \, a^{3} \cos \left (d x + c\right )^{9} - 2940 \, a^{3} \cos \left (d x + c\right )^{7} + 768 \, a^{3} \cos \left (d x + c\right )^{5} + 2940 \, a^{3} \cos \left (d x + c\right )^{3} - 630 \, a^{3} \cos \left (d x + c\right ) - 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (a^{3} \cos \left (d x + c\right )^{10} - 5 \, a^{3} \cos \left (d x + c\right )^{8} + 10 \, a^{3} \cos \left (d x + c\right )^{6} - 10 \, a^{3} \cos \left (d x + c\right )^{4} + 5 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 512 \, {\left (2 \, a^{3} \cos \left (d x + c\right )^{9} - 9 \, a^{3} \cos \left (d x + c\right )^{7} + 12 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{7680 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 357, normalized size = 1.65 \[ \frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5040 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {14762 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 3600 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 384 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 30 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 40 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{61440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.47, size = 248, normalized size = 1.15 \[ -\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{7}}-\frac {2 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{15 d \sin \left (d x +c \right )^{5}}-\frac {7 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{8}}-\frac {7 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{32 d \sin \left (d x +c \right )^{6}}-\frac {7 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{4}}+\frac {7 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{256 d \sin \left (d x +c \right )^{2}}+\frac {7 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{256 d}+\frac {21 a^{3} \cos \left (d x +c \right )}{256 d}+\frac {21 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256 d}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{9}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{10 d \sin \left (d x +c \right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 308, normalized size = 1.43 \[ \frac {21 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 630 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1536 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}} - \frac {512 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{3}}{\tan \left (d x + c\right )^{9}}}{53760 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.55, size = 395, normalized size = 1.83 \[ \frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}-\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{64\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{512\,d}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{1536\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {21\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}-\frac {15\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d}+\frac {15\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________