Optimal. Leaf size=243 \[ \frac{2 a^2 (99 A+110 B+84 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d} \]
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Rubi [A] time = 0.694628, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {4088, 4018, 4016, 3800, 4001, 3792} \[ \frac{2 a^2 (99 A+110 B+84 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d} \]
Antiderivative was successfully verified.
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Rule 4088
Rule 4018
Rule 4016
Rule 3800
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{2 \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{1}{2} a (11 A+6 C)+\frac{1}{2} a (11 B+3 C) \sec (c+d x)\right ) \, dx}{11 a}\\ &=\frac{2 a (11 B+3 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{4 \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{3}{4} a^2 (33 A+22 B+24 C)+\frac{1}{4} a^2 (99 A+110 B+84 C) \sec (c+d x)\right ) \, dx}{99 a}\\ &=\frac{2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (11 B+3 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{1}{231} (a (429 A+374 B+336 C)) \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a (11 B+3 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{(2 (429 A+374 B+336 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}\\ &=\frac{2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}-\frac{4 a (429 A+374 B+336 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac{2 a (11 B+3 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}+\frac{1}{495} (a (429 A+374 B+336 C)) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (99 A+110 B+84 C) \sec ^3(c+d x) \tan (c+d x)}{693 d \sqrt{a+a \sec (c+d x)}}-\frac{4 a (429 A+374 B+336 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{3465 d}+\frac{2 a (11 B+3 C) \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{99 d}+\frac{2 (429 A+374 B+336 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{1155 d}+\frac{2 C \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 2.03983, size = 185, normalized size = 0.76 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} ((12441 A+12386 B+12684 C) \cos (c+d x)+(4422 A+4862 B+4368 C) \cos (2 (c+d x))+5577 A \cos (3 (c+d x))+858 A \cos (4 (c+d x))+858 A \cos (5 (c+d x))+3564 A+4862 B \cos (3 (c+d x))+748 B \cos (4 (c+d x))+748 B \cos (5 (c+d x))+4114 B+4368 C \cos (3 (c+d x))+672 C \cos (4 (c+d x))+672 C \cos (5 (c+d x))+4956 C)}{6930 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.36, size = 205, normalized size = 0.8 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 3432\,A \left ( \cos \left ( dx+c \right ) \right ) ^{5}+2992\,B \left ( \cos \left ( dx+c \right ) \right ) ^{5}+2688\,C \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1716\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1496\,B \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1344\,C \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1287\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1122\,B \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1008\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+495\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+935\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}+840\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+385\,B\cos \left ( dx+c \right ) +735\,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 = 0.523222, size = 435, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (8 \,{\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{5} + 4 \,{\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{4} + 3 \,{\left (429 \, A + 374 \, B + 336 \, C\right )} a \cos \left (d x + c\right )^{3} + 5 \,{\left (99 \, A + 187 \, B + 168 \, C\right )} a \cos \left (d x + c\right )^{2} + 35 \,{\left (11 \, B + 21 \, C\right )} a \cos \left (d x + c\right ) + 315 \, C a\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 = 5.19118, size = 554, normalized size = 2.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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