Optimal. Leaf size=146 \[ \frac{5 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac{7 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right )}{\sqrt{2} a^{3/2} d}+\frac{2 A \sin (c+d x)}{a d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.354706, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4020, 4022, 3920, 3774, 203, 3795} \[ \frac{5 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac{7 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right )}{\sqrt{2} a^{3/2} d}+\frac{2 A \sin (c+d x)}{a d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4020
Rule 4022
Rule 3920
Rule 3774
Rule 203
Rule 3795
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx &=-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{\int \frac{\cos (c+d x) (4 a A+3 a A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{2 A \sin (c+d x)}{a d \sqrt{a-a \sec (c+d x)}}-\frac{\int \frac{-5 a^2 A-2 a^2 A \sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx}{2 a^3}\\ &=-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{2 A \sin (c+d x)}{a d \sqrt{a-a \sec (c+d x)}}+\frac{(5 A) \int \sqrt{a-a \sec (c+d x)} \, dx}{2 a^2}+\frac{(7 A) \int \frac{\sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx}{2 a}\\ &=-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{2 A \sin (c+d x)}{a d \sqrt{a-a \sec (c+d x)}}+\frac{(5 A) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,\frac{a \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right )}{a d}-\frac{(7 A) \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 )}{a d}\\ &=\frac{5 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac{7 A \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right )}{\sqrt{2} a^{3/2} d}-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{2 A \sin (c+d x)}{a d \sqrt{a-a \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.61696, size = 361, normalized size = 2.47 \[ A \left (\frac{\sin ^3\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^2(c+d x) \left (-\frac{2 \sin \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )}{d}+\frac{2 \sin \left (\frac{3 c}{2}\right ) \sin \left (\frac{3 d x}{2}\right )}{d}+\frac{2 \cos \left (\frac{c}{2}\right ) \cos \left (\frac{d x}{2}\right )}{d}-\frac{2 \cos \left (\frac{3 c}{2}\right ) \cos \left (\frac{3 d x}{2}\right )}{d}-\frac{2 \cot \left (\frac{c}{2}\right ) \csc \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}+\frac{2 \csc \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \csc ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{d}\right )}{(a-a \sec (c+d x))^{3/2}}+\frac{\sqrt{2} e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^3\left (\frac{c}{2}+\frac{d x}{2}\right ) \sec ^{\frac{3}{2}}(c+d x) \left (-5 \sinh ^{-1}\left (e^{i (c+d x)}\right )+7 \sqrt{2} \tanh ^{-1}\left (\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right )-5 \tanh ^{-1}\left (\sqrt{1+e^{2 i (c+d x)}}\right )\right )}{d (a-a \sec (c+d x))^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.283, size = 462, normalized size = 3.2 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (A \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )}{{\left (-a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.541201, size = 1345, normalized size = 9.21 \begin{align*} \left [-\frac{7 \, \sqrt{2}{\left (A \cos \left (d x + c\right ) - A\right )} \sqrt{-a} \log \left (\frac{2 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} +{\left (3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 10 \,{\left (A \cos \left (d x + c\right ) - A\right )} \sqrt{-a} \log \left (\frac{2 \,{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}} -{\left (2 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right )}{\sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 4 \,{\left (A \cos \left (d x + c\right )^{3} - A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{4 \,{\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )}, \frac{7 \, \sqrt{2}{\left (A \cos \left (d x + c\right ) - A\right )} \sqrt{a} \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 ) \sin \left (d x + c\right ) - 10 \,{\left (A \cos \left (d x + c\right ) - A\right )} \sqrt{a} \arctan \left (\frac{\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 ) \sin \left (d x + c\right ) - 2 \,{\left (A \cos \left (d x + c\right )^{3} - A \cos \left (d x + c\right )^{2} - 2 \, A \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}}{2 \,{\left (a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\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*} A \left (\int \frac{\cos{\left (c + d x \right )}}{- a \sqrt{- a \sec{\left (c + d x \right )} + a} \sec{\left (c + d x \right )} + a \sqrt{- a \sec{\left (c + d x \right )} + a}}\, dx + \int \frac{\cos{\left (c + d x \right )} \sec{\left (c + d x \right )}}{- a \sqrt{- a \sec{\left (c + d x \right )} + a} \sec{\left (c + d x \right )} + a \sqrt{- a \sec{\left (c + d x \right )} + a}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.06882, size = 344, normalized size = 2.36 \begin{align*} -\frac{A{\left (\frac{7 \, \sqrt{2} \arctan \left (\frac{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} - \frac{\sqrt{2}{\left (3 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{\frac{3}{2}} + 4 \, \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a} a\right )}}{{\left ({\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} + 3 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a + 2 \, a^{2}\right )} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} - \frac{10 \, \arctan \left (\frac{\sqrt{2} \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a}}{2 \, \sqrt{a}}\right )}{a^{\frac{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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