Optimal. Leaf size=179 \[ -a^3 x-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3971, 3554, 8,
2686, 200, 2687, 30, 276} \begin {gather*} -\frac {4 a^3 \cot ^9(c+d x)}{9 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {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 {4 a^3 \csc ^9(c+d x)}{9 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc (c+d x)}{d}-a^3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 200
Rule 276
Rule 2686
Rule 2687
Rule 3554
Rule 3971
Rubi steps
\begin {align*} \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^{10}(c+d x)+3 a^3 \cot ^9(c+d x) \csc (c+d x)+3 a^3 \cot ^8(c+d x) \csc ^2(c+d x)+a^3 \cot ^7(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^{10}(c+d x) \, dx+a^3 \int \cot ^7(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^9(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^8(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^3 \cot ^9(c+d x)}{9 d}-a^3 \int \cot ^8(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^8 \, dx,x,-\cot (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}+a^3 \int \cot ^6(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d}-a^3 \int \cot ^4(c+d x) \, dx\\ &=\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d}+a^3 \int \cot ^2(c+d x) \, dx\\ &=-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d}-a^3 \int 1 \, dx\\ &=-a^3 x-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(370\) vs. \(2(179)=358\).
time = 1.36, size = 370, normalized size = 2.07 \begin {gather*} \frac {a^3 \csc \left (\frac {c}{2}\right ) \csc ^9\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) (-181440 d x \cos (d x)+181440 d x \cos (2 c+d x)+136080 d x \cos (c+2 d x)-136080 d x \cos (3 c+2 d x)+10080 d x \cos (2 c+3 d x)-10080 d x \cos (4 c+3 d x)-60480 d x \cos (3 c+4 d x)+60480 d x \cos (5 c+4 d x)+30240 d x \cos (4 c+5 d x)-30240 d x \cos (6 c+5 d x)-5040 d x \cos (5 c+6 d x)+5040 d x \cos (7 c+6 d x)-169344 \sin (c)+338112 \sin (d x)+675036 \sin (c+d x)-506277 \sin (2 (c+d x))-37502 \sin (3 (c+d x))+225012 \sin (4 (c+d x))-112506 \sin (5 (c+d x))+18751 \sin (6 (c+d x))+431424 \sin (2 c+d x)-375552 \sin (c+2 d x)-201600 \sin (3 c+2 d x)+41248 \sin (2 c+3 d x)-84000 \sin (4 c+3 d x)+155712 \sin (3 c+4 d x)+100800 \sin (5 c+4 d x)-98016 \sin (4 c+5 d x)-30240 \sin (6 c+5 d x)+21376 \sin (5 c+6 d x))}{41287680 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(363\) vs.
\(2(163)=326\).
time = 0.13, size = 364, normalized size = 2.03
method | result | size |
risch | \(-a^{3} x -\frac {2 i a^{3} \left (945 \,{\mathrm e}^{11 i \left (d x +c \right )}-3150 \,{\mathrm e}^{10 i \left (d x +c \right )}+2625 \,{\mathrm e}^{9 i \left (d x +c \right )}+6300 \,{\mathrm e}^{8 i \left (d x +c \right )}-13482 \,{\mathrm e}^{7 i \left (d x +c \right )}+5292 \,{\mathrm e}^{6 i \left (d x +c \right )}+10566 \,{\mathrm e}^{5 i \left (d x +c \right )}-11736 \,{\mathrm e}^{4 i \left (d x +c \right )}+1289 \,{\mathrm e}^{3 i \left (d x +c \right )}+4866 \,{\mathrm e}^{2 i \left (d x +c \right )}-3063 \,{\mathrm e}^{i \left (d x +c \right )}+668\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(166\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{8}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {\cos ^{8}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{63 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {a^{3} \left (\cos ^{9}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{9}}+3 a^{3} \left (-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{10}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a^{3} \left (-\frac {\left (\cot ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(364\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{8}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {\cos ^{8}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{63 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {a^{3} \left (\cos ^{9}\left (d x +c \right )\right )}{3 \sin \left (d x +c \right )^{9}}+3 a^{3} \left (-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{10}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{10}\left (d x +c \right )}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{10}\left (d x +c \right )}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a^{3} \left (-\frac {\left (\cot ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(364\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 182, normalized size = 1.02 \begin {gather*} -\frac {{\left (315 \, d x + 315 \, c + \frac {315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a^{3} + \frac {3 \, {\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a^{3}}{\sin \left (d x + c\right )^{9}} - \frac {{\left (105 \, \sin \left (d x + c\right )^{6} - 189 \, \sin \left (d x + c\right )^{4} + 135 \, \sin \left (d x + c\right )^{2} - 35\right )} a^{3}}{\sin \left (d x + c\right )^{9}} + \frac {105 \, a^{3}}{\tan \left (d x + c\right )^{9}}}{315 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.98, size = 235, normalized size = 1.31 \begin {gather*} -\frac {668 \, a^{3} \cos \left (d x + c\right )^{6} - 1059 \, a^{3} \cos \left (d x + c\right )^{5} - 573 \, a^{3} \cos \left (d x + c\right )^{4} + 1813 \, a^{3} \cos \left (d x + c\right )^{3} - 393 \, a^{3} \cos \left (d x + c\right )^{2} - 789 \, a^{3} \cos \left (d x + c\right ) + 368 \, a^{3} + 315 \, {\left (a^{3} d x \cos \left (d x + c\right )^{5} - 3 \, a^{3} d x \cos \left (d x + c\right )^{4} + 2 \, a^{3} d x \cos \left (d x + c\right )^{3} + 2 \, a^{3} d x \cos \left (d x + c\right )^{2} - 3 \, a^{3} d x \cos \left (d x + c\right ) + a^{3} d x\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - 3 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 128, normalized size = 0.72 \begin {gather*} -\frac {105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20160 \, {\left (d x + c\right )} a^{3} - 2520 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {31185 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 6720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1827 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 360 \, 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 )^{9}}}{20160 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.94, size = 206, normalized size = 1.15 \begin {gather*} -\frac {a^3\,\left (35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+31185\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1827\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+20160\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )\right )}{20160\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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