Optimal. Leaf size=265 \[ \frac{63 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 a^3 d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac{319 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{128 \sqrt{2} a^{5/2} d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{192 a^3 d}-\frac{191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{384 a^3 d} \]
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Rubi [A] time = 0.23319, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{63 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 a^3 d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac{319 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{128 \sqrt{2} a^{5/2} d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{19 \cos ^2(c+d x) \cot (c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{192 a^3 d}-\frac{191 \cos (c+d x) \cot (c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{a \sec (c+d x)+a}}{384 a^3 d} \]
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))^{5/2}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )^4} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^3 d}\\ &=-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{5 a-7 a^2 x^2}{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 )}{6 a^4 d}\\ &=-\frac{19 \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)}}{192 a^3 d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{a^2-95 a^3 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 )}{48 a^5 d}\\ &=-\frac{191 \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)}}{384 a^3 d}-\frac{19 \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)}}{192 a^3 d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{-189 a^3-573 a^4 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 )}{192 a^6 d}\\ &=\frac{63 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{128 a^3 d}-\frac{191 \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)}}{384 a^3 d}-\frac{19 \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)}}{192 a^3 d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{48 a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{579 a^4-189 a^5 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 )}{384 a^6 d}\\ &=\frac{63 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{128 a^3 d}-\frac{191 \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)}}{384 a^3 d}-\frac{19 \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)}}{192 a^3 d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{48 a^3 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^2 d}-\frac{319 \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 )}{128 a^2 d}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac{319 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{128 \sqrt{2} a^{5/2} d}+\frac{63 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{128 a^3 d}-\frac{191 \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)}}{384 a^3 d}-\frac{19 \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)}}{192 a^3 d}-\frac{\cos ^3(c+d x) \cot (c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) \sqrt{a+a \sec (c+d x)}}{48 a^3 d}\\ \end{align*}
Mathematica [C] time = 23.627, size = 5614, normalized size = 21.18 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.259, size = 714, normalized size = 2.7 \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] 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{5}{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(-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 [A] time = 9.28286, size = 277, normalized size = 1.05 \begin{align*} -\frac{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left (2 \,{\left (\frac{4 \, \sqrt{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{31 \, \sqrt{2}}{a^{3} \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{291 \, \sqrt{2}}{a^{3} \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{96 \, \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} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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