Optimal. Leaf size=87 \[ \frac{(7 A+6 C) \tan ^5(c+d x)}{35 d}+\frac{2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac{(7 A+6 C) \tan (c+d x)}{7 d}+\frac{C \tan (c+d x) \sec ^6(c+d x)}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0559017, antiderivative size = 87, 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 = {4046, 3767} \[ \frac{(7 A+6 C) \tan ^5(c+d x)}{35 d}+\frac{2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac{(7 A+6 C) \tan (c+d x)}{7 d}+\frac{C \tan (c+d x) \sec ^6(c+d x)}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4046
Rule 3767
Rubi steps
\begin{align*} \int \sec ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{7} (7 A+6 C) \int \sec ^6(c+d x) \, dx\\ &=\frac{C \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac{(7 A+6 C) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d}\\ &=\frac{(7 A+6 C) \tan (c+d x)}{7 d}+\frac{C \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{2 (7 A+6 C) \tan ^3(c+d x)}{21 d}+\frac{(7 A+6 C) \tan ^5(c+d x)}{35 d}\\ \end{align*}
Mathematica [A] time = 0.302337, size = 81, normalized size = 0.93 \[ \frac{A \left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{C \left (\frac{1}{7} \tan ^7(c+d x)+\frac{3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.031, size = 78, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -A \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) -C \left ( -{\frac{16}{35}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.936541, size = 81, normalized size = 0.93 \begin{align*} \frac{15 \, C \tan \left (d x + c\right )^{7} + 21 \,{\left (A + 3 \, C\right )} \tan \left (d x + c\right )^{5} + 35 \,{\left (2 \, A + 3 \, C\right )} \tan \left (d x + c\right )^{3} + 105 \,{\left (A + C\right )} \tan \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.908206, size = 188, normalized size = 2.16 \begin{align*} \frac{{\left (8 \,{\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} + 15 \, C\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\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 \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{6}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17973, size = 107, normalized size = 1.23 \begin{align*} \frac{15 \, C \tan \left (d x + c\right )^{7} + 21 \, A \tan \left (d x + c\right )^{5} + 63 \, C \tan \left (d x + c\right )^{5} + 70 \, A \tan \left (d x + c\right )^{3} + 105 \, C \tan \left (d x + c\right )^{3} + 105 \, A \tan \left (d x + c\right ) + 105 \, C \tan \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]