Optimal. Leaf size=53 \[ \frac{\tan ^7(a+b x)}{7 b}+\frac{3 \tan ^5(a+b x)}{5 b}+\frac{\tan ^3(a+b x)}{b}+\frac{\tan (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0167995, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3767} \[ \frac{\tan ^7(a+b x)}{7 b}+\frac{3 \tan ^5(a+b x)}{5 b}+\frac{\tan ^3(a+b x)}{b}+\frac{\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3767
Rubi steps
\begin{align*} \int \sec ^8(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (a+b x)\right )}{b}\\ &=\frac{\tan (a+b x)}{b}+\frac{\tan ^3(a+b x)}{b}+\frac{3 \tan ^5(a+b x)}{5 b}+\frac{\tan ^7(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.218945, size = 43, normalized size = 0.81 \[ \frac{\frac{1}{7} \tan ^7(a+b x)+\frac{3}{5} \tan ^5(a+b x)+\tan ^3(a+b x)+\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 44, normalized size = 0.8 \begin{align*} -{\frac{\tan \left ( bx+a \right ) }{b} \left ( -{\frac{16}{35}}-{\frac{ \left ( \sec \left ( bx+a \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sec \left ( bx+a \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \sec \left ( bx+a \right ) \right ) ^{2}}{35}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07473, size = 59, normalized size = 1.11 \begin{align*} \frac{5 \, \tan \left (b x + a\right )^{7} + 21 \, \tan \left (b x + a\right )^{5} + 35 \, \tan \left (b x + a\right )^{3} + 35 \, \tan \left (b x + a\right )}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.39288, size = 135, normalized size = 2.55 \begin{align*} \frac{{\left (16 \, \cos \left (b x + a\right )^{6} + 8 \, \cos \left (b x + a\right )^{4} + 6 \, \cos \left (b x + a\right )^{2} + 5\right )} \sin \left (b x + a\right )}{35 \, b \cos \left (b x + a\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec ^{8}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27431, size = 59, normalized size = 1.11 \begin{align*} \frac{5 \, \tan \left (b x + a\right )^{7} + 21 \, \tan \left (b x + a\right )^{5} + 35 \, \tan \left (b x + a\right )^{3} + 35 \, \tan \left (b x + a\right )}{35 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]