Optimal. Leaf size=161 \[ \frac{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{n+1}}{b^5 d (n+1)}-\frac{4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^{n+2}}{b^5 d (n+2)}+\frac{2 \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^{n+3}}{b^5 d (n+3)}-\frac{4 a (a+b \tan (c+d x))^{n+4}}{b^5 d (n+4)}+\frac{(a+b \tan (c+d x))^{n+5}}{b^5 d (n+5)} \]
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Rubi [A] time = 0.123486, antiderivative size = 161, normalized size of antiderivative = 1., 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 = {3506, 697} \[ \frac{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^{n+1}}{b^5 d (n+1)}-\frac{4 a \left (a^2+b^2\right ) (a+b \tan (c+d x))^{n+2}}{b^5 d (n+2)}+\frac{2 \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^{n+3}}{b^5 d (n+3)}-\frac{4 a (a+b \tan (c+d x))^{n+4}}{b^5 d (n+4)}+\frac{(a+b \tan (c+d x))^{n+5}}{b^5 d (n+5)} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 697
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac{\operatorname{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{\operatorname{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*}
Mathematica [A] time = 2.96177, size = 161, normalized size = 1. \[ \frac{(a+b \tan (c+d x))^{n+1} \left (4 \left (a^2+b^2\right ) \left (\frac{a^2+b^2}{n+1}+\frac{(a+b \tan (c+d x))^2}{n+3}-\frac{2 a (a+b \tan (c+d x))}{n+2}\right )-4 a (a+b \tan (c+d x)) \left (\frac{a^2+b^2}{n+2}+\frac{(a+b \tan (c+d x))^2}{n+4}-\frac{2 a (a+b \tan (c+d x))}{n+3}\right )+b^4 \sec ^4(c+d x)\right )}{b^5 d (n+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.245, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{6} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.44236, size = 899, normalized size = 5.58 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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