Optimal. Leaf size=182 \[ -\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2873, 2611, 3770, 2607, 30, 3768} \[ -\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^6(c+d x) \csc (c+d x)+2 a^2 \cot ^6(c+d x) \csc ^2(c+d x)+a^2 \cot ^6(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \csc (c+d x) \, dx+a^2 \int \cot ^6(c+d x) \csc ^3(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{8} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx-\frac {1}{6} \left (5 a^2\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}+\frac {1}{16} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {1}{8} \left (5 a^2\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx\\ &=-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{64} \left (5 a^2\right ) \int \csc ^3(c+d x) \, dx-\frac {1}{16} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {5 a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {1}{128} \left (5 a^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {45 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.11, size = 401, normalized size = 2.20 \[ a^2 \left (-\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{7 d}+\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{7 d}-\frac {\csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {17 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {83 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {\sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {17 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {83 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {45 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}+\frac {45 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^6\left (\frac {1}{2} (c+d x)\right )}{448 d}+\frac {5 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{224 d}-\frac {19 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )}{448 d}-\frac {5 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {19 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{224 d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 255, normalized size = 1.40 \[ -\frac {512 \, a^{2} \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 1162 \, a^{2} \cos \left (d x + c\right )^{7} + 3066 \, a^{2} \cos \left (d x + c\right )^{5} - 2310 \, a^{2} \cos \left (d x + c\right )^{3} + 630 \, a^{2} \cos \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{1792 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 260, normalized size = 1.43 \[ \frac {7 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 32 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 224 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1792 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {13698 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1792 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 224 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{14336 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.39, size = 192, normalized size = 1.05 \[ -\frac {3 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{6}}+\frac {3 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{64 d \sin \left (d x +c \right )^{4}}-\frac {9 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{128 d \sin \left (d x +c \right )^{2}}-\frac {9 a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{128 d}-\frac {15 a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{128 d}-\frac {45 a^{2} \cos \left (d x +c \right )}{128 d}-\frac {45 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128 d}-\frac {2 a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}-\frac {a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 221, normalized size = 1.21 \[ -\frac {7 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 56 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1536 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 11.16, size = 387, normalized size = 2.13 \[ -\frac {a^2\,\left (7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-32\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+224\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-1792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-224\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5040\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{14336\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \]
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]
________________________________________________________________________________________