Optimal. Leaf size=141 \[ -\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \csc ^7(c+d x)}{7 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {3 a^3 \csc (c+d x)}{d}+a^3 x \]
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Rubi [A] time = 0.18, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ -\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \csc ^7(c+d x)}{7 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {3 a^3 \csc (c+d x)}{d}+a^3 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 194
Rule 270
Rule 2606
Rule 2607
Rule 3473
Rule 3886
Rubi steps
\begin {align*} \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cot ^8(c+d x)+3 a^3 \cot ^7(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^5(c+d x) \csc ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^8(c+d x) \, dx+a^3 \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cot ^7(c+d x) \csc (c+d x) \, dx+\left (3 a^3\right ) \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^3 \cot ^7(c+d x)}{7 d}-a^3 \int \cot ^6(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+a^3 \int \cot ^4(c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {4 a^3 \csc ^7(c+d x)}{7 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 {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {4 a^3 \csc ^7(c+d x)}{7 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 {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {4 a^3 \csc ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.98, size = 252, normalized size = 1.79 \[ \frac {a^3 \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \csc ^7\left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {1}{2} (c+d x)\right ) (-23282 \sin (c+d x)+23282 \sin (2 (c+d x))-9978 \sin (3 (c+d x))+1663 \sin (4 (c+d x))-13720 \sin (2 c+d x)+15512 \sin (c+2 d x)+9240 \sin (3 c+2 d x)-8088 \sin (2 c+3 d x)-2520 \sin (4 c+3 d x)+1768 \sin (3 c+4 d x)-5880 d x \cos (2 c+d x)-5880 d x \cos (c+2 d x)+5880 d x \cos (3 c+2 d x)+2520 d x \cos (2 c+3 d x)-2520 d x \cos (4 c+3 d x)-420 d x \cos (3 c+4 d x)+420 d x \cos (5 c+4 d x)+4200 \sin (c)-11032 \sin (d x)+5880 d x \cos (d x))}{215040 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 160, normalized size = 1.13 \[ \frac {221 \, a^{3} \cos \left (d x + c\right )^{4} - 348 \, a^{3} \cos \left (d x + c\right )^{3} - 25 \, a^{3} \cos \left (d x + c\right )^{2} + 303 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3} + 105 \, {\left (a^{3} d x \cos \left (d x + c\right )^{3} - 3 \, a^{3} d x \cos \left (d x + c\right )^{2} + 3 \, a^{3} d x \cos \left (d x + c\right ) - a^{3} d x\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 3 \, d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 96, normalized size = 0.68 \[ \frac {1680 \, {\left (d x + c\right )} a^{3} - 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2730 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 126 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.18, size = 293, normalized size = 2.08 \[ \frac {a^{3} \left (-\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{3} \left (-\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}\right )-\frac {3 a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}+a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 152, normalized size = 1.08 \[ \frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{3} + \frac {9 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a^{3}}{\sin \left (d x + c\right )^{7}} - \frac {{\left (35 \, \sin \left (d x + c\right )^{4} - 42 \, \sin \left (d x + c\right )^{2} + 15\right )} a^{3}}{\sin \left (d x + c\right )^{7}} - \frac {45 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.70, size = 91, normalized size = 0.65 \[ \frac {a^3\,\left (126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-560\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2730\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1680\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )-15\right )}{1680\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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