Optimal. Leaf size=74 \[ \frac{2 (d \tan (a+b x))^{n+3}}{b d^3 (n+3)}+\frac{(d \tan (a+b x))^{n+5}}{b d^5 (n+5)}+\frac{(d \tan (a+b x))^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.0660517, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 270} \[ \frac{2 (d \tan (a+b x))^{n+3}}{b d^3 (n+3)}+\frac{(d \tan (a+b x))^{n+5}}{b d^5 (n+5)}+\frac{(d \tan (a+b x))^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 270
Rubi steps
\begin{align*} \int \sec ^6(a+b x) (d \tan (a+b x))^n \, dx &=\frac{\operatorname{Subst}\left (\int (d x)^n \left (1+x^2\right )^2 \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \left ((d x)^n+\frac{2 (d x)^{2+n}}{d^2}+\frac{(d x)^{4+n}}{d^4}\right ) \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{(d \tan (a+b x))^{1+n}}{b d (1+n)}+\frac{2 (d \tan (a+b x))^{3+n}}{b d^3 (3+n)}+\frac{(d \tan (a+b x))^{5+n}}{b d^5 (5+n)}\\ \end{align*}
Mathematica [A] time = 2.09949, size = 101, normalized size = 1.36 \[ \frac{d (d \tan (a+b x))^{n-1} \left (\tan ^2(a+b x) \sec ^4(a+b x) \left (2 (n+3) \cos (2 (a+b x))+\cos (4 (a+b x))+n^2+6 n+8\right )+8 \left (-\tan ^2(a+b x)\right )^{\frac{1-n}{2}}\right )}{b (n+1) (n+3) (n+5)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.196, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( bx+a \right ) \right ) ^{6} \left ( d\tan \left ( bx+a \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 [A] time = 1.66651, size = 215, normalized size = 2.91 \begin{align*} \frac{{\left (8 \, \cos \left (b x + a\right )^{4} + 4 \,{\left (n + 1\right )} \cos \left (b x + a\right )^{2} + n^{2} + 4 \, n + 3\right )} \left (\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}\right )^{n} \sin \left (b x + a\right )}{{\left (b n^{3} + 9 \, b n^{2} + 23 \, b n + 15 \, b\right )} \cos \left (b x + a\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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