Optimal. Leaf size=168 \[ -\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+a^3 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30, 3768} \[ -\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+a^3 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3473
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc (c+d x)+3 a^3 \cot ^4(c+d x) \csc ^2(c+d x)+a^3 \cot ^4(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {1}{2} a^3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx-a^3 \int \cot ^2(c+d x) \, dx-\frac {1}{4} \left (9 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {9 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{8} a^3 \int \csc ^3(c+d x) \, dx+a^3 \int 1 \, dx+\frac {1}{8} \left (9 a^3\right ) \int \csc (c+d x) \, dx\\ &=a^3 x-\frac {9 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{16} a^3 \int \csc (c+d x) \, dx\\ &=a^3 x-\frac {19 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.78, size = 217, normalized size = 1.29 \[ \frac {a^3 \left (-704 \tan \left (\frac {1}{2} (c+d x)\right )+704 \cot \left (\frac {1}{2} (c+d x)\right )+870 \csc ^2\left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right )+60 \sec ^4\left (\frac {1}{2} (c+d x)\right )-870 \sec ^2\left (\frac {1}{2} (c+d x)\right )+2280 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2280 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\left ((18 \sin (c+d x)+5) \csc ^6\left (\frac {1}{2} (c+d x)\right )\right )+(86 \sin (c+d x)-60) \csc ^4\left (\frac {1}{2} (c+d x)\right )-1376 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+36 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )+1920 c+1920 d x\right )}{1920 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 290, normalized size = 1.73 \[ \frac {480 \, a^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, a^{3} d x \cos \left (d x + c\right )^{4} - 870 \, a^{3} \cos \left (d x + c\right )^{5} + 1440 \, a^{3} d x \cos \left (d x + c\right )^{2} + 1520 \, a^{3} \cos \left (d x + c\right )^{3} - 480 \, a^{3} d x - 570 \, a^{3} \cos \left (d x + c\right ) - 285 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 285 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (11 \, a^{3} \cos \left (d x + c\right )^{5} - 35 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 239, normalized size = 1.42 \[ \frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 75 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 100 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1920 \, {\left (d x + c\right )} a^{3} + 2280 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {5586 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 100 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.36, size = 194, normalized size = 1.15 \[ -\frac {a^{3} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{3} \cot \left (d x +c \right )}{d}+a^{3} x +\frac {a^{3} c}{d}-\frac {19 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}+\frac {19 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{2}}+\frac {19 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}+\frac {19 a^{3} \cos \left (d x +c \right )}{16 d}+\frac {19 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 215, normalized size = 1.28 \[ \frac {160 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} + 5 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 90 \, a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {288 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 9.38, size = 313, normalized size = 1.86 \[ \frac {49\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {49\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+19\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{19\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {19\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{16\,d}+\frac {7\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]
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]
________________________________________________________________________________________