Optimal. Leaf size=223 \[ \frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(75 A-19 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.699583, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2955, 4020, 4013, 3808, 206} \[ \frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(75 A-19 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2955
Rule 4020
Rule 4013
Rule 3808
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx\\ &=-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} a (9 A-B)-2 a (A-B) \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} a^2 (49 A-9 B)-\frac{1}{2} a^2 (13 A-5 B) \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}-\frac{\left ((75 A-19 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx}{32 a^2}\\ &=-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{\left ((75 A-19 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{16 a^2 d}\\ &=-\frac{(75 A-19 B) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.46083, size = 228, normalized size = 1.02 \[ \frac{\tan (c+d x) \sqrt{1-\sec (c+d x)} \sec (c+d x) \left (2 \left (73 A^2+76 A B-13 B^2\right ) \cos (c+d x)+16 A^2 \cos (3 (c+d x))+85 A^2+A (85 A+19 B) \cos (2 (c+d x))+117 A B-18 B^2\right )+8 \sqrt{2} (75 A-19 B) \sin \left (\frac{1}{2} (c+d x)\right ) \cos ^5\left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right ) (A \cos (c+d x)+B)}{32 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{5/2} (A \cos (c+d x)+B)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.444, size = 365, normalized size = 1.6 \begin{align*} -{\frac{ \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}{16\,d{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}} \left ( 32\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+75\,A\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) +53\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}-19\,B\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) -13\,B \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+75\,A\sin \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) -36\,A\cos \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}-19\,B\sin \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) +4\,B\cos \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}-49\,A\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}+9\,B\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) \sqrt{\cos \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}{\frac{1}{\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}}}} \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 = 0.546412, size = 1338, normalized size = 6. \begin{align*} \left [-\frac{\sqrt{2}{\left ({\left (75 \, A - 19 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (75 \, A - 19 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (75 \, A - 19 \, B\right )} \cos \left (d x + c\right ) + 75 \, A - 19 \, B\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) - 4 \,{\left (32 \, A \cos \left (d x + c\right )^{2} +{\left (85 \, A - 13 \, B\right )} \cos \left (d x + c\right ) + 49 \, A - 9 \, B\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{64 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, \frac{\sqrt{2}{\left ({\left (75 \, A - 19 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (75 \, A - 19 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (75 \, A - 19 \, B\right )} \cos \left (d x + c\right ) + 75 \, A - 19 \, B\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )}}{a \sin \left (d x + c\right )}\right ) + 2 \,{\left (32 \, A \cos \left (d x + c\right )^{2} +{\left (85 \, A - 13 \, B\right )} \cos \left (d x + c\right ) + 49 \, A - 9 \, B\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{32 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \sec \left (d x + c\right ) + A\right )} \sqrt{\cos \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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