Optimal. Leaf size=153 \[ -\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x-\frac {2 a b \csc ^7(c+d x)}{7 d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {2 a b \csc (c+d x)}{d}-\frac {b^2 \cot ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.15, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3886, 3473, 8, 2606, 194, 2607, 30} \[ -\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x-\frac {2 a b \csc ^7(c+d x)}{7 d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {2 a b \csc (c+d x)}{d}-\frac {b^2 \cot ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 3473
Rule 3886
Rubi steps
\begin {align*} \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx &=\int \left (a^2 \cot ^8(c+d x)+2 a b \cot ^7(c+d x) \csc (c+d x)+b^2 \cot ^6(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^8(c+d x) \, dx+(2 a b) \int \cot ^7(c+d x) \csc (c+d x) \, dx+b^2 \int \cot ^6(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a^2 \cot ^7(c+d x)}{7 d}-a^2 \int \cot ^6(c+d x) \, dx-\frac {(2 a b) \operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^2 \operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+a^2 \int \cot ^4(c+d x) \, dx-\frac {(2 a b) \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^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}-a^2 \int \cot ^2(c+d x) \, dx\\ &=\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}+a^2 \int 1 \, dx\\ &=a^2 x+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 257, normalized size = 1.68 \[ -\frac {\csc ^7(c+d x) \left (-3675 a^2 c \sin (c+d x)-3675 a^2 d x \sin (c+d x)+2205 a^2 c \sin (3 (c+d x))+2205 a^2 d x \sin (3 (c+d x))-735 a^2 c \sin (5 (c+d x))-735 a^2 d x \sin (5 (c+d x))+105 a^2 c \sin (7 (c+d x))+105 a^2 d x \sin (7 (c+d x))+1176 a^2 \cos (3 (c+d x))-392 a^2 \cos (5 (c+d x))+176 a^2 \cos (7 (c+d x))+3612 a b \cos (2 (c+d x))-840 a b \cos (4 (c+d x))+420 a b \cos (6 (c+d x))-1272 a b+525 b^2 \cos (c+d x)+315 b^2 \cos (3 (c+d x))+105 b^2 \cos (5 (c+d x))+15 b^2 \cos (7 (c+d x))\right )}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 206, normalized size = 1.35 \[ \frac {210 \, a b \cos \left (d x + c\right )^{6} + {\left (176 \, a^{2} + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 406 \, a^{2} \cos \left (d x + c\right )^{5} - 420 \, a b \cos \left (d x + c\right )^{4} + 350 \, a^{2} \cos \left (d x + c\right )^{3} + 336 \, a b \cos \left (d x + c\right )^{2} - 105 \, a^{2} \cos \left (d x + c\right ) - 96 \, a b + 105 \, {\left (a^{2} d x \cos \left (d x + c\right )^{6} - 3 \, a^{2} d x \cos \left (d x + c\right )^{4} + 3 \, a^{2} d x \cos \left (d x + c\right )^{2} - a^{2} d x\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.43, size = 366, normalized size = 2.39 \[ \frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 294 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1295 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1470 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13440 \, {\left (d x + c\right )} a^{2} - 9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7350 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 525 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 7350 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 525 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1295 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1470 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 315 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 294 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} - 30 \, a b - 15 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.95, size = 187, normalized size = 1.22 \[ \frac {a^{2} \left (-\frac {\left (\cot ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\cot ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos ^{8}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{8}\left (d x +c \right )}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}\right )-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7 \sin \left (d x +c \right )^{7}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 116, normalized size = 0.76 \[ \frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{2} + \frac {6 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a b}{\sin \left (d x + c\right )^{7}} - \frac {15 \, b^{2}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 258, normalized size = 1.69 \[ a^2\,x+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\left (a-b\right )}^2}{896\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2}{32}-\frac {5\,a\,b}{48}+\frac {b^2}{48}+\frac {{\left (a-b\right )}^2}{384}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a^2}{80}-\frac {3\,a\,b}{160}+\frac {b^2}{160}+\frac {{\left (a-b\right )}^2}{640}\right )}{d}-\frac {\frac {2\,a\,b}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {37\,a^2}{3}+14\,a\,b+3\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (93\,a^2+70\,a\,b+5\,b^2\right )+\frac {a^2}{7}+\frac {b^2}{7}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {9\,a^2}{5}+\frac {14\,a\,b}{5}+b^2\right )}{128\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {23\,a^2}{32}-\frac {17\,a\,b}{32}+\frac {b^2}{32}+\frac {{\left (a-b\right )}^2}{128}\right )}{d} \]
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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