Optimal. Leaf size=212 \[ \frac{(19 A+163 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{5 (3 A+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(A+17 C) \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.659367, antiderivative size = 212, 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 = {4085, 4019, 4010, 4001, 3795, 203} \[ \frac{(19 A+163 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}+\frac{5 (3 A+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(A+17 C) \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4085
Rule 4019
Rule 4010
Rule 4001
Rule 3795
Rule 203
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{5/2}} \, dx &=-\frac{(A+C) \sec ^3(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{\int \frac{\sec ^3(c+d x) \left (-a (A-3 C)-\frac{1}{2} a (3 A+11 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{(A+C) \sec ^3(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(A+17 C) \sec ^2(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac{\int \frac{\sec ^2(c+d x) \left (a^2 (A+17 C)-\frac{5}{4} a^2 (3 A+19 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A+C) \sec ^3(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(A+17 C) \sec ^2(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac{5 (3 A+19 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{48 a^3 d}-\frac{\int \frac{\sec (c+d x) \left (-\frac{5}{8} a^3 (3 A+19 C)+\frac{1}{4} a^3 (21 A+197 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{12 a^5}\\ &=-\frac{(A+C) \sec ^3(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(A+17 C) \sec ^2(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac{(21 A+197 C) \tan (c+d x)}{24 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{5 (3 A+19 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{48 a^3 d}+\frac{(19 A+163 C) \int \frac{\sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx}{32 a^2}\\ &=-\frac{(A+C) \sec ^3(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(A+17 C) \sec ^2(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac{(21 A+197 C) \tan (c+d x)}{24 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{5 (3 A+19 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{48 a^3 d}-\frac{(19 A+163 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 )}{16 a^2 d}\\ &=\frac{(19 A+163 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A+C) \sec ^3(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(A+17 C) \sec ^2(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}-\frac{(21 A+197 C) \tan (c+d x)}{24 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{5 (3 A+19 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{48 a^3 d}\\ \end{align*}
Mathematica [A] time = 2.33556, size = 196, normalized size = 0.92 \[ \frac{\tan (c+d x) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (-(81 A+1537 C) \cos (c+d x)-2 (39 A+503 C) \cos (2 (c+d x))-\frac{6 \sqrt{2} (19 A+163 C) \cos ^2(c+d x) (\cos (c+d x)+1)^2 \sqrt{\sec (c+d x)-1} \tan ^{-1}\left (\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right )}{\cos (c+d x)-1}-27 A \cos (3 (c+d x))-78 A-299 C \cos (3 (c+d x))-878 C\right )}{96 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.365, size = 786, normalized size = 3.7 \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 = 0.619588, size = 1447, normalized size = 6.83 \begin{align*} \text{result too large to display} \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 + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 9.61947, size = 419, normalized size = 1.98 \begin{align*} -\frac{\frac{{\left ({\left (3 \,{\left (\frac{2 \, \sqrt{2}{\left (A a^{5} + C a^{5}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{6} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} + \frac{\sqrt{2}{\left (7 \, A a^{5} + 23 \, C a^{5}\right )}}{a^{6} \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{4 \, \sqrt{2}{\left (15 \, A a^{5} + 167 \, C a^{5}\right )}}{a^{6} \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{3 \, \sqrt{2}{\left (11 \, A a^{5} + 155 \, C a^{5}\right )}}{a^{6} \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 )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}} - \frac{3 \, \sqrt{2}{\left (19 \, A + 163 \, 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} a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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