Optimal. Leaf size=144 \[ \frac{2 (7 B+6 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 B+6 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (7 B+6 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.359372, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {4072, 4016, 3800, 4001, 3792} \[ \frac{2 (7 B+6 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 B+6 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (7 B+6 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4072
Rule 4016
Rule 3800
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} (B+C \sec (c+d x)) \, dx\\ &=\frac{2 a C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{1}{7} (7 B+6 C) \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (7 B+6 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 a d}+\frac{(2 (7 B+6 C)) \int \sec (c+d x) \left (\frac{3 a}{2}-a \sec (c+d x)\right ) \sqrt{a+a \sec (c+d x)} \, dx}{35 a}\\ &=\frac{2 a C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}-\frac{4 (7 B+6 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 B+6 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 a d}+\frac{1}{15} (7 B+6 C) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (7 B+6 C) \tan (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}-\frac{4 (7 B+6 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac{2 (7 B+6 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 a d}\\ \end{align*}
Mathematica [A] time = 0.281068, size = 81, normalized size = 0.56 \[ \frac{2 a \tan (c+d x) \left (3 (7 B+6 C) \sec ^2(c+d x)+4 (7 B+6 C) \sec (c+d x)+8 (7 B+6 C)+15 C \sec ^3(c+d x)\right )}{105 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.317, size = 116, normalized size = 0.8 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 56\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+48\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+28\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+24\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+21\,B\cos \left ( dx+c \right ) +18\,C\cos \left ( dx+c \right ) +15\,C \right ) }{105\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.49372, size = 265, normalized size = 1.84 \begin{align*} \frac{2 \,{\left (8 \,{\left (7 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (7 \, B + 6 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (7 \, B + 6 \, C\right )} \cos \left (d x + c\right ) + 15 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \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 \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \left (B + C \sec{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 4.45322, size = 300, normalized size = 2.08 \begin{align*} -\frac{2 \,{\left (105 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 105 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (175 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 105 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (119 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 147 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (49 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 27 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{105 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]