Optimal. Leaf size=250 \[ -\frac {\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac {a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac {a \left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac {\left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}+\frac {\left (a^6-4 a^4 b^2+6 a^2 b^4-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac {a \sec ^6(c+d x)}{6 b^2 d}-\frac {\log (\cos (c+d x))}{a d}+\frac {\sec ^7(c+d x)}{7 b d} \]
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Rubi [A] time = 0.20, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3885, 894} \[ \frac {\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac {a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac {\left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac {a \left (-4 a^2 b^2+a^4+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac {\left (-4 a^4 b^2+6 a^2 b^4+a^6-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac {\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}-\frac {a \sec ^6(c+d x)}{6 b^2 d}-\frac {\log (\cos (c+d x))}{a d}+\frac {\sec ^7(c+d x)}{7 b d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3885
Rubi steps
\begin {align*} \int \frac {\tan ^9(c+d x)}{a+b \sec (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^4}{x (a+x)} \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^6 \left (1+\frac {-4 a^4 b^2+6 a^2 b^4-4 b^6}{a^6}\right )+\frac {b^8}{a x}-a \left (a^4-4 a^2 b^2+6 b^4\right ) x+\left (a^4-4 a^2 b^2+6 b^4\right ) x^2-a \left (a^2-4 b^2\right ) x^3+\left (a^2-4 b^2\right ) x^4-a x^5+x^6-\frac {\left (a^2-b^2\right )^4}{a (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{b^8 d}\\ &=-\frac {\log (\cos (c+d x))}{a d}-\frac {\left (a^2-b^2\right )^4 \log (a+b \sec (c+d x))}{a b^8 d}+\frac {\left (a^6-4 a^4 b^2+6 a^2 b^4-4 b^6\right ) \sec (c+d x)}{b^7 d}-\frac {a \left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^2(c+d x)}{2 b^6 d}+\frac {\left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-4 b^2\right ) \sec ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-4 b^2\right ) \sec ^5(c+d x)}{5 b^3 d}-\frac {a \sec ^6(c+d x)}{6 b^2 d}+\frac {\sec ^7(c+d x)}{7 b d}\\ \end {align*}
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Mathematica [B] time = 6.24, size = 520, normalized size = 2.08 \[ -\frac {\left (2 b^2-a^2\right ) \left (a^4-2 a^2 b^2+2 b^4\right ) \sec ^2(c+d x) (a \cos (c+d x)+b)}{b^7 d (a+b \sec (c+d x))}-\frac {a \left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^3(c+d x) (a \cos (c+d x)+b)}{2 b^6 d (a+b \sec (c+d x))}+\frac {\left (a^4-4 a^2 b^2+6 b^4\right ) \sec ^4(c+d x) (a \cos (c+d x)+b)}{3 b^5 d (a+b \sec (c+d x))}+\frac {\left (a^7-4 a^5 b^2+6 a^3 b^4-4 a b^6\right ) \sec (c+d x) \log (\cos (c+d x)) (a \cos (c+d x)+b)}{b^8 d (a+b \sec (c+d x))}+\frac {\left (-a^8+4 a^6 b^2-6 a^4 b^4+4 a^2 b^6-b^8\right ) \sec (c+d x) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{a b^8 d (a+b \sec (c+d x))}+\frac {a (2 b-a) (a+2 b) \sec ^5(c+d x) (a \cos (c+d x)+b)}{4 b^4 d (a+b \sec (c+d x))}-\frac {(2 b-a) (a+2 b) \sec ^6(c+d x) (a \cos (c+d x)+b)}{5 b^3 d (a+b \sec (c+d x))}-\frac {a \sec ^7(c+d x) (a \cos (c+d x)+b)}{6 b^2 d (a+b \sec (c+d x))}+\frac {\sec ^8(c+d x) (a \cos (c+d x)+b)}{7 b d (a+b \sec (c+d x))} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 293, normalized size = 1.17 \[ -\frac {70 \, a^{2} b^{6} \cos \left (d x + c\right ) + 420 \, {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{7} \log \left (a \cos \left (d x + c\right ) + b\right ) - 420 \, {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 60 \, a b^{7} - 420 \, {\left (a^{7} b - 4 \, a^{5} b^{3} + 6 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{6} + 210 \, {\left (a^{6} b^{2} - 4 \, a^{4} b^{4} + 6 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (a^{5} b^{3} - 4 \, a^{3} b^{5} + 6 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 105 \, {\left (a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} - 84 \, {\left (a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )^{2}}{420 \, a b^{8} d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 17.34, size = 1768, normalized size = 7.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 460, normalized size = 1.84 \[ -\frac {\ln \left (b +a \cos \left (d x +c \right )\right )}{d a}-\frac {4}{5 d b \cos \left (d x +c \right )^{5}}+\frac {2}{d b \cos \left (d x +c \right )^{3}}-\frac {4}{d b \cos \left (d x +c \right )}+\frac {1}{7 d b \cos \left (d x +c \right )^{7}}-\frac {a}{6 d \,b^{2} \cos \left (d x +c \right )^{6}}+\frac {a^{2}}{5 d \,b^{3} \cos \left (d x +c \right )^{5}}+\frac {a^{4}}{3 d \,b^{5} \cos \left (d x +c \right )^{3}}-\frac {4 a^{2}}{3 d \,b^{3} \cos \left (d x +c \right )^{3}}+\frac {a^{6}}{d \,b^{7} \cos \left (d x +c \right )}-\frac {4 a^{4}}{d \,b^{5} \cos \left (d x +c \right )}-\frac {3 a}{d \,b^{2} \cos \left (d x +c \right )^{2}}+\frac {a^{7} \ln \left (\cos \left (d x +c \right )\right )}{d \,b^{8}}-\frac {4 a^{5} \ln \left (\cos \left (d x +c \right )\right )}{d \,b^{6}}+\frac {6 a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d \,b^{4}}-\frac {4 a \ln \left (\cos \left (d x +c \right )\right )}{d \,b^{2}}-\frac {a^{7} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,b^{8}}+\frac {4 a^{5} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,b^{6}}-\frac {6 a^{3} \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,b^{4}}+\frac {4 a \ln \left (b +a \cos \left (d x +c \right )\right )}{d \,b^{2}}-\frac {a^{5}}{2 d \,b^{6} \cos \left (d x +c \right )^{2}}+\frac {2 a^{3}}{d \,b^{4} \cos \left (d x +c \right )^{2}}+\frac {6 a^{2}}{d \,b^{3} \cos \left (d x +c \right )}-\frac {a^{3}}{4 d \,b^{4} \cos \left (d x +c \right )^{4}}+\frac {a}{d \,b^{2} \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 268, normalized size = 1.07 \[ \frac {\frac {420 \, {\left (a^{7} - 4 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 4 \, a b^{6}\right )} \log \left (\cos \left (d x + c\right )\right )}{b^{8}} - \frac {420 \, {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a b^{8}} - \frac {70 \, a b^{5} \cos \left (d x + c\right ) - 420 \, {\left (a^{6} - 4 \, a^{4} b^{2} + 6 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - 60 \, b^{6} + 210 \, {\left (a^{5} b - 4 \, a^{3} b^{3} + 6 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (a^{4} b^{2} - 4 \, a^{2} b^{4} + 6 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 105 \, {\left (a^{3} b^{3} - 4 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 84 \, {\left (a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{2}}{b^{7} \cos \left (d x + c\right )^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.97, size = 631, normalized size = 2.52 \[ \frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {\frac {2\,\left (105\,a^6-385\,a^4\,b^2+511\,a^2\,b^4-279\,b^6\right )}{105\,b^7}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (a^6+a^5\,b-3\,a^4\,b^2-3\,a^3\,b^3+3\,a^2\,b^4+3\,a\,b^5-b^6\right )}{b^7}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (6\,a^6+5\,a^5\,b-20\,a^4\,b^2-17\,a^3\,b^3+22\,a^2\,b^4+19\,a\,b^5-8\,b^6\right )}{b^7}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (30\,a^6+15\,a^5\,b-112\,a^4\,b^2-54\,a^3\,b^3+154\,a^2\,b^4+71\,a\,b^5-96\,b^6\right )}{3\,b^7}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (45\,a^6+30\,a^5\,b-161\,a^4\,b^2-108\,a^3\,b^3+203\,a^2\,b^4+142\,a\,b^5-87\,b^6\right )}{3\,b^7}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (75\,a^6+25\,a^5\,b-285\,a^4\,b^2-85\,a^3\,b^3+401\,a^2\,b^4+95\,a\,b^5-239\,b^6\right )}{5\,b^7}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (90\,a^6+15\,a^5\,b-340\,a^4\,b^2-45\,a^3\,b^3+466\,a^2\,b^4+45\,a\,b^5-264\,b^6\right )}{15\,b^7}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )\,\left (-a^7+4\,a^5\,b^2-6\,a^3\,b^4+4\,a\,b^6\right )}{b^8\,d}-\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,{\left (a^2-b^2\right )}^4}{a\,b^8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{9}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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