Optimal. Leaf size=208 \[ \frac{\sqrt{2} (A-B+C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 (35 A-7 B+31 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 a d}-\frac{4 (35 A-49 B+37 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 B-C) \tan (c+d x) \sec ^2(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.637046, antiderivative size = 208, 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, 4021, 4010, 4001, 3795, 203} \[ \frac{\sqrt{2} (A-B+C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 (35 A-7 B+31 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 a d}-\frac{4 (35 A-49 B+37 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 B-C) \tan (c+d x) \sec ^2(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 4088
Rule 4021
Rule 4010
Rule 4001
Rule 3795
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx &=\frac{2 C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{2 \int \frac{\sec ^3(c+d x) \left (\frac{1}{2} a (7 A+6 C)+\frac{1}{2} a (7 B-C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{7 a}\\ &=\frac{2 (7 B-C) \sec ^2(c+d x) \tan (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{4 \int \frac{\sec ^2(c+d x) \left (a^2 (7 B-C)+\frac{1}{4} a^2 (35 A-7 B+31 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{35 a^2}\\ &=\frac{2 (7 B-C) \sec ^2(c+d x) \tan (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (35 A-7 B+31 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 a d}+\frac{8 \int \frac{\sec (c+d x) \left (\frac{1}{8} a^3 (35 A-7 B+31 C)-\frac{1}{4} a^3 (35 A-49 B+37 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{105 a^3}\\ &=-\frac{4 (35 A-49 B+37 C) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (7 B-C) \sec ^2(c+d x) \tan (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (35 A-7 B+31 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 a d}+(A-B+C) \int \frac{\sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx\\ &=-\frac{4 (35 A-49 B+37 C) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (7 B-C) \sec ^2(c+d x) \tan (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (35 A-7 B+31 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 a d}-\frac{(2 (A-B+C)) \operatorname{Subst}\left (\int \frac{1}{2 a+x^2} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{\sqrt{a} d}-\frac{4 (35 A-49 B+37 C) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (7 B-C) \sec ^2(c+d x) \tan (c+d x)}{35 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^3(c+d x) \tan (c+d x)}{7 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (35 A-7 B+31 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 a d}\\ \end{align*}
Mathematica [C] time = 29.5461, size = 7134, normalized size = 34.3 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.39, size = 1144, normalized size = 5.5 \begin{align*} \text{result too large to display} \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 = 0.633937, size = 1176, normalized size = 5.65 \begin{align*} \left [\frac{105 \, \sqrt{2}{\left ({\left (A - B + C\right )} a \cos \left (d x + c\right )^{4} +{\left (A - B + C\right )} a \cos \left (d x + c\right )^{3}\right )} \sqrt{-\frac{1}{a}} \log \left (-\frac{2 \, \sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{-\frac{1}{a}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, \cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 4 \,{\left ({\left (35 \, A - 91 \, B + 43 \, C\right )} \cos \left (d x + c\right )^{3} -{\left (35 \, A - 7 \, B + 31 \, C\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (7 \, B - 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 )}{210 \,{\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\right )}}, -\frac{2 \,{\left ({\left (35 \, A - 91 \, B + 43 \, C\right )} \cos \left (d x + c\right )^{3} -{\left (35 \, A - 7 \, B + 31 \, C\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (7 \, B - 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 ) + \frac{105 \, \sqrt{2}{\left ({\left (A - B + C\right )} a \cos \left (d x + c\right )^{4} +{\left (A - B + C\right )} a \cos \left (d x + c\right )^{3}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right )}{\sqrt{a}}}{105 \,{\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \sec{\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 9.26495, size = 412, normalized size = 1.98 \begin{align*} \frac{\frac{105 \, \sqrt{2}{\left (A - B + C\right )} \log \left ({\left | -\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} + \frac{2 \,{\left (\frac{105 \, \sqrt{2} B a^{3}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} +{\left ({\left (\frac{\sqrt{2}{\left (70 \, A a^{3} - 119 \, B a^{3} + 92 \, C a^{3}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{7 \, \sqrt{2}{\left (20 \, A a^{3} - 37 \, B a^{3} + 16 \, C a^{3}\right )}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{35 \, \sqrt{2}{\left (2 \, A a^{3} - 7 \, B a^{3} + 4 \, C a^{3}\right )}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\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 )}{{\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}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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