Optimal. Leaf size=242 \[ \frac {a^5 \tan (c+d x)}{d}+\frac {5 a^4 b \tan ^2(c+d x)}{2 d}+\frac {10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac {b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac {b \left (5 a^4+20 a^2 b^2+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac {a \left (a^4+20 a^2 b^2+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {5 a b^4 \tan ^9(c+d x)}{9 d}+\frac {b^5 \tan ^{10}(c+d x)}{10 d} \]
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Rubi [A] time = 0.22, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3088, 948} \[ \frac {b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac {10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {b \left (20 a^2 b^2+5 a^4+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac {a \left (20 a^2 b^2+a^4+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac {2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {5 a^4 b \tan ^2(c+d x)}{2 d}+\frac {a^5 \tan (c+d x)}{d}+\frac {5 a b^4 \tan ^9(c+d x)}{9 d}+\frac {b^5 \tan ^{10}(c+d x)}{10 d} \]
Antiderivative was successfully verified.
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Rule 948
Rule 3088
Rubi steps
\begin {align*} \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^5 \left (1+x^2\right )^2}{x^{11}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {b^5}{x^{11}}+\frac {5 a b^4}{x^{10}}+\frac {2 \left (5 a^2 b^3+b^5\right )}{x^9}+\frac {10 a b^2 \left (a^2+b^2\right )}{x^8}+\frac {5 a^4 b+20 a^2 b^3+b^5}{x^7}+\frac {a^5+20 a^3 b^2+5 a b^4}{x^6}+\frac {10 a^2 b \left (a^2+b^2\right )}{x^5}+\frac {2 \left (a^5+5 a^3 b^2\right )}{x^4}+\frac {5 a^4 b}{x^3}+\frac {a^5}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^5 \tan (c+d x)}{d}+\frac {5 a^4 b \tan ^2(c+d x)}{2 d}+\frac {2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac {5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac {a \left (a^4+20 a^2 b^2+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac {b \left (5 a^4+20 a^2 b^2+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac {10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac {b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac {5 a b^4 \tan ^9(c+d x)}{9 d}+\frac {b^5 \tan ^{10}(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 115, normalized size = 0.48 \[ \frac {\frac {1}{4} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^8-\frac {4}{7} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^7+\frac {1}{6} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^6+\frac {1}{10} (a+b \tan (c+d x))^{10}-\frac {4}{9} a (a+b \tan (c+d x))^9}{b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 207, normalized size = 0.86 \[ \frac {126 \, b^{5} + 210 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 315 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (8 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 4 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 175 \, a b^{4} \cos \left (d x + c\right ) + 3 \, {\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 50 \, {\left (9 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1260 \, d \cos \left (d x + c\right )^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.18, size = 262, normalized size = 1.08 \[ \frac {126 \, b^{5} \tan \left (d x + c\right )^{10} + 700 \, a b^{4} \tan \left (d x + c\right )^{9} + 1575 \, a^{2} b^{3} \tan \left (d x + c\right )^{8} + 315 \, b^{5} \tan \left (d x + c\right )^{8} + 1800 \, a^{3} b^{2} \tan \left (d x + c\right )^{7} + 1800 \, a b^{4} \tan \left (d x + c\right )^{7} + 1050 \, a^{4} b \tan \left (d x + c\right )^{6} + 4200 \, a^{2} b^{3} \tan \left (d x + c\right )^{6} + 210 \, b^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 5040 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 1260 \, a b^{4} \tan \left (d x + c\right )^{5} + 3150 \, a^{4} b \tan \left (d x + c\right )^{4} + 3150 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 840 \, a^{5} \tan \left (d x + c\right )^{3} + 4200 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 3150 \, a^{4} b \tan \left (d x + c\right )^{2} + 1260 \, a^{5} \tan \left (d x + c\right )}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 299, normalized size = 1.24 \[ \frac {-a^{5} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {5 a^{4} b}{6 \cos \left (d x +c \right )^{6}}+10 a^{3} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+10 a^{2} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )+5 a \,b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )+b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.78, size = 275, normalized size = 1.14 \[ \frac {84 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{5} + 120 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{3} b^{2} + 20 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a b^{4} + \frac {525 \, {\left (4 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - \frac {21 \, {\left (10 \, \sin \left (d x + c\right )^{4} - 5 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - \frac {1050 \, a^{4} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.44, size = 548, normalized size = 2.26 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {616\,a^5}{15}-\frac {176\,a^3\,b^2}{3}+32\,a\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (40\,a^4\,b-40\,a^2\,b^3\right )-2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {616\,a^5}{15}-\frac {176\,a^3\,b^2}{3}+32\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-88\,a^5+\frac {720\,a^3\,b^2}{7}+\frac {160\,a\,b^4}{7}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (-88\,a^5+\frac {720\,a^3\,b^2}{7}+\frac {160\,a\,b^4}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {388\,a^5}{3}-\frac {4240\,a^3\,b^2}{21}+\frac {3520\,a\,b^4}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {388\,a^5}{3}-\frac {4240\,a^3\,b^2}{21}+\frac {3520\,a\,b^4}{63}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {280\,a^4\,b}{3}-\frac {80\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {280\,a^4\,b}{3}-\frac {80\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (220\,a^4\,b-160\,a^2\,b^3+\frac {192\,b^5}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (-\frac {520\,a^4\,b}{3}+\frac {200\,a^2\,b^3}{3}+\frac {64\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (-\frac {520\,a^4\,b}{3}+\frac {200\,a^2\,b^3}{3}+\frac {64\,b^5}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {38\,a^5}{3}-\frac {80\,a^3\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {38\,a^5}{3}-\frac {80\,a^3\,b^2}{3}\right )+2\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^{10}} \]
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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