Optimal. Leaf size=215 \[ \frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac{71 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{32 \sqrt{2} a^{3/2} d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{16 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{32 a^2 d} \]
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Rubi [A] time = 0.195117, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3887, 472, 579, 583, 522, 203} \[ \frac{7 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{32 a^2 d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac{71 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{32 \sqrt{2} a^{3/2} d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{16 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{32 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3887
Rule 472
Rule 579
Rule 583
Rule 522
Rule 203
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^2 d}\\ &=-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{3 a-5 a^2 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{4 a^3 d}\\ &=-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{-7 a^2-39 a^3 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{16 a^4 d}\\ &=\frac{7 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{32 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{16 a^2 d}+\frac{\operatorname{Subst}\left (\int \frac{57 a^3-7 a^4 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{32 a^4 d}\\ &=\frac{7 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{32 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{16 a^2 d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a d}-\frac{71 \operatorname{Subst}\left (\int \frac{1}{2+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{32 a d}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^{3/2} d}+\frac{71 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{32 \sqrt{2} a^{3/2} d}+\frac{7 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{32 a^2 d}-\frac{13 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{16 a^2 d}\\ \end{align*}
Mathematica [C] time = 23.5864, size = 5588, normalized size = 25.99 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.238, size = 542, normalized size = 2.5 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (d x + c\right )^{2}}{{\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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{\cot ^{2}{\left (c + d x \right )}}{\left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 9.45942, size = 221, normalized size = 1.03 \begin{align*} \frac{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left (\frac{2 \, \sqrt{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{17 \, \sqrt{2}}{a^{2} \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 ) + \frac{16 \, \sqrt{2}}{{\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 )}^{2} - a\right )} \sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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