Optimal. Leaf size=223 \[ \frac{2 a (99 A+80 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (99 A+80 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.515776, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4089, 4016, 3803, 3800, 4001, 3792} \[ \frac{2 a (99 A+80 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (99 A+80 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 4089
Rule 4016
Rule 3803
Rule 3800
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac{2 \int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{1}{2} a (11 A+8 C)+\frac{1}{2} a C \sec (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac{1}{99} (99 A+80 C) \int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (99 A+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac{1}{231} (2 (99 A+80 C)) \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a (99 A+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac{4 (99 A+80 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac{(4 (99 A+80 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}{1155 a}\\ &=\frac{2 a (99 A+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt{a+a \sec (c+d x)}}-\frac{8 (99 A+80 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac{2 C \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac{4 (99 A+80 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}+\frac{1}{495} (2 (99 A+80 C)) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{4 a (99 A+80 C) \tan (c+d x)}{495 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (99 A+80 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a C \sec ^4(c+d x) \tan (c+d x)}{99 d \sqrt{a+a \sec (c+d x)}}-\frac{8 (99 A+80 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac{2 C \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{11 d}+\frac{4 (99 A+80 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 a d}\\ \end{align*}
Mathematica [A] time = 1.06486, size = 143, normalized size = 0.64 \[ \frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} ((2871 A+3020 C) \cos (c+d x)+13 (99 A+80 C) \cos (2 (c+d x))+1287 A \cos (3 (c+d x))+198 A \cos (4 (c+d x))+198 A \cos (5 (c+d x))+1089 A+1040 C \cos (3 (c+d x))+160 C \cos (4 (c+d x))+160 C \cos (5 (c+d x))+1510 C)}{3465 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.406, size = 151, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -2+2\,\cos \left ( dx+c \right ) \right ) \left ( 1584\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1280\,C \left ( \cos \left ( dx+c \right ) \right ) ^{5}+792\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+640\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+594\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+480\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+495\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+400\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+350\,C\cos \left ( dx+c \right ) +315\,C \right ) }{3465\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}\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.
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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.
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Fricas [A] time = 0.505868, size = 352, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (16 \,{\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{5} + 8 \,{\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{4} + 6 \,{\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{3} + 5 \,{\left (99 \, A + 80 \, C\right )} \cos \left (d x + c\right )^{2} + 350 \, C \cos \left (d x + c\right ) + 315 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3465 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}} \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 [A] time = 4.7097, size = 424, normalized size = 1.9 \begin{align*} -\frac{2 \,{\left (3465 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 3465 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (10395 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 5775 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (15246 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 16170 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (14058 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 8910 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (6633 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 5885 \, \sqrt{2} C a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (891 \, \sqrt{2} A a^{6} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 755 \, \sqrt{2} C a^{6} \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 )^{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 )}{3465 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{5} \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.
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