Integrand size = 21, antiderivative size = 161 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{1+n}}{b^5 d (1+n)}-\frac {4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^{2+n}}{b^5 d (2+n)}+\frac {2 \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^{3+n}}{b^5 d (3+n)}-\frac {4 a (a+b \tan (c+d x))^{4+n}}{b^5 d (4+n)}+\frac {(a+b \tan (c+d x))^{5+n}}{b^5 d (5+n)} \]
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Time = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3587, 711} \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{n+1}}{b^5 d (n+1)}-\frac {4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^{n+2}}{b^5 d (n+2)}+\frac {2 \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^{n+3}}{b^5 d (n+3)}-\frac {4 a (a+b \tan (c+d x))^{n+4}}{b^5 d (n+4)}+\frac {(a+b \tan (c+d x))^{n+5}}{b^5 d (n+5)} \]
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Rule 711
Rule 3587
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^n \left (1+\frac {x^2}{b^2}\right )^2 \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\left (a^2+b^2\right )^2 (a+x)^n}{b^4}-\frac {4 a \left (a^2+b^2\right ) (a+x)^{1+n}}{b^4}+\frac {2 \left (3 a^2+b^2\right ) (a+x)^{2+n}}{b^4}-\frac {4 a (a+x)^{3+n}}{b^4}+\frac {(a+x)^{4+n}}{b^4}\right ) \, dx,x,b \tan (c+d x)\right )}{b d} \\ & = \frac {\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{1+n}}{b^5 d (1+n)}-\frac {4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^{2+n}}{b^5 d (2+n)}+\frac {2 \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^{3+n}}{b^5 d (3+n)}-\frac {4 a (a+b \tan (c+d x))^{4+n}}{b^5 d (4+n)}+\frac {(a+b \tan (c+d x))^{5+n}}{b^5 d (5+n)} \\ \end{align*}
Time = 3.41 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {(a+b \tan (c+d x))^{1+n} \left (b^4 \sec ^4(c+d x)+4 \left (a^2+b^2\right ) \left (\frac {a^2+b^2}{1+n}-\frac {2 a (a+b \tan (c+d x))}{2+n}+\frac {(a+b \tan (c+d x))^2}{3+n}\right )-4 a (a+b \tan (c+d x)) \left (\frac {a^2+b^2}{2+n}-\frac {2 a (a+b \tan (c+d x))}{3+n}+\frac {(a+b \tan (c+d x))^2}{4+n}\right )\right )}{b^5 d (5+n)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(161)=322\).
Time = 0.12 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.87
\[\frac {\left (\tan ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (a +b \tan \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {a \left (b^{4} n^{4}+14 b^{4} n^{3}+4 a^{2} b^{2} n^{2}+71 b^{4} n^{2}+36 a^{2} b^{2} n +154 b^{4} n +24 a^{4}+80 a^{2} b^{2}+120 b^{4}\right ) {\mathrm e}^{n \ln \left (a +b \tan \left (d x +c \right )\right )}}{b^{5} d \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}+\frac {a n \left (\tan ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (a +b \tan \left (d x +c \right )\right )}}{b d \left (n^{2}+9 n +20\right )}-\frac {2 \left (-b^{2} n^{2}+2 a^{2} n -9 b^{2} n -20 b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (a +b \tan \left (d x +c \right )\right )}}{b^{2} d \left (n^{3}+12 n^{2}+47 n +60\right )}-\frac {\left (-b^{4} n^{4}+4 a^{2} b^{2} n^{3}-14 b^{4} n^{3}+36 a^{2} b^{2} n^{2}-71 b^{4} n^{2}+24 a^{4} n +80 a^{2} b^{2} n -154 b^{4} n -120 b^{4}\right ) \tan \left (d x +c \right ) {\mathrm e}^{n \ln \left (a +b \tan \left (d x +c \right )\right )}}{b^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right ) d}+\frac {2 \left (b^{2} n^{2}+9 b^{2} n +6 a^{2}+20 b^{2}\right ) a n \left (\tan ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (a +b \tan \left (d x +c \right )\right )}}{b^{3} d \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}\]
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Leaf count of result is larger than twice the leaf count of optimal. 420 vs. \(2 (161) = 322\).
Time = 0.31 (sec) , antiderivative size = 420, normalized size of antiderivative = 2.61 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {{\left (8 \, {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4} - {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} n^{2} + 3 \, {\left (a^{3} b^{2} + 5 \, a b^{4}\right )} n\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (2 \, a b^{4} n^{3} + 3 \, {\left (a^{3} b^{2} + 3 \, a b^{4}\right )} n^{2} + {\left (3 \, a^{3} b^{2} + 7 \, a b^{4}\right )} n\right )} \cos \left (d x + c\right )^{3} + {\left (a b^{4} n^{4} + 6 \, a b^{4} n^{3} + 11 \, a b^{4} n^{2} + 6 \, a b^{4} n\right )} \cos \left (d x + c\right ) + {\left (b^{5} n^{4} + 10 \, b^{5} n^{3} + 35 \, b^{5} n^{2} + 50 \, b^{5} n + 24 \, b^{5} + 8 \, {\left (8 \, b^{5} - {\left (3 \, a^{2} b^{3} - b^{5}\right )} n^{2} - 3 \, {\left (a^{4} b + 3 \, a^{2} b^{3} - 2 \, b^{5}\right )} n\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (8 \, b^{5} - {\left (a^{2} b^{3} - b^{5}\right )} n^{3} - {\left (3 \, a^{2} b^{3} - 7 \, b^{5}\right )} n^{2} - 2 \, {\left (a^{2} b^{3} - 7 \, b^{5}\right )} n\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \left (\frac {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{n}}{{\left (b^{5} d n^{5} + 15 \, b^{5} d n^{4} + 85 \, b^{5} d n^{3} + 225 \, b^{5} d n^{2} + 274 \, b^{5} d n + 120 \, b^{5} d\right )} \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\text {Timed out} \]
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none
Time = 0.41 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.78 \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\frac {\frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} + \frac {2 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} \tan \left (d x + c\right )^{3} + {\left (n^{2} + n\right )} a b^{2} \tan \left (d x + c\right )^{2} - 2 \, a^{2} b n \tan \left (d x + c\right ) + 2 \, a^{3}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} \tan \left (d x + c\right )^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} \tan \left (d x + c\right )^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} \tan \left (d x + c\right )^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} \tan \left (d x + c\right )^{2} - 24 \, a^{4} b n \tan \left (d x + c\right ) + 24 \, a^{5}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}}}{d} \]
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Exception generated. \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{{\cos \left (c+d\,x\right )}^6} \,d x \]
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