Optimal. Leaf size=150 \[ -\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{16 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 2611, 3770, 2607, 30, 3768, 14} \[ -\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{16 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cot ^4(c+d x) \csc (c+d x)+3 a^3 \cot ^4(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^3(c+d x)+a^3 \cot ^4(c+d x) \csc ^4(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \csc (c+d x) \, dx+a^3 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac {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)}{2 d}-\frac {1}{4} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx-\frac {1}{2} \left (3 a^3\right ) \int \cot ^2(c+d x) \csc ^3(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 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 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)}{2 d}+\frac {1}{8} \left (3 a^3\right ) \int \csc (c+d x) \, dx+\frac {1}{8} \left (3 a^3\right ) \int \csc ^3(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 {3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 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)}{2 d}+\frac {1}{16} \left (3 a^3\right ) \int \csc (c+d x) \, dx\\ &=-\frac {9 a^3 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 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)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.13, size = 363, normalized size = 2.42 \[ a^3 \left (\frac {23 \tan \left (\frac {1}{2} (c+d x)\right )}{70 d}-\frac {23 \cot \left (\frac {1}{2} (c+d x)\right )}{70 d}-\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{128 d}+\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {7 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{128 d}-\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {7 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {9 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^6\left (\frac {1}{2} (c+d x)\right )}{896 d}-\frac {31 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{2240 d}+\frac {297 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{2240 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right )}{896 d}+\frac {31 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )}{2240 d}-\frac {297 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{2240 d}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 247, normalized size = 1.65 \[ -\frac {736 \, a^{3} \cos \left (d x + c\right )^{7} - 896 \, a^{3} \cos \left (d x + c\right )^{5} + 315 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (7 \, a^{3} \cos \left (d x + c\right )^{5} - 24 \, a^{3} \cos \left (d x + c\right )^{3} + 9 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 261, normalized size = 1.74 \[ \frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 77 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 665 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2520 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {6534 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 665 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 77 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{4480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.38, size = 176, normalized size = 1.17 \[ -\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{4}}+\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}+\frac {3 a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{16 d}+\frac {9 a^{3} \cos \left (d x +c \right )}{16 d}+\frac {9 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {23 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{6}}-\frac {a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 206, normalized size = 1.37 \[ \frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 70 \, a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {672 \, a^{3}}{\tan \left (d x + c\right )^{5}} - \frac {32 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.32, size = 387, normalized size = 2.58 \[ \frac {a^3\,\left (5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+35\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+77\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-455\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-665\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+665\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+455\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-77\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2520\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{4480\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
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]
________________________________________________________________________________________