Optimal. Leaf size=251 \[ \frac{43 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{96 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{16 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{32 a^2 d}+\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{64 a d}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}-\frac{107 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{64 \sqrt{2} \sqrt{a} d} \]
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Rubi [A] time = 0.225917, antiderivative size = 251, 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} \[ \frac{43 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{96 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{16 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{32 a^2 d}+\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{64 a d}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}-\frac{107 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{64 \sqrt{2} \sqrt{a} d} \]
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 ^4(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \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 ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{a-7 a^2 x^2}{x^4 \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{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{-43 a^2-75 a^3 x^2}{x^4 \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{43 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{96 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{16 a^2 d}+\frac{\operatorname{Subst}\left (\int \frac{63 a^3-129 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 )}{96 a^4 d}\\ &=\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{64 a d}+\frac{43 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{96 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{447 a^4+63 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 )}{192 a^4 d}\\ &=\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{64 a d}+\frac{43 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{96 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{16 a^2 d}+\frac{107 \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 )}{64 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 )}{d}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{\sqrt{a} d}-\frac{107 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{64 \sqrt{2} \sqrt{a} d}+\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{64 a d}+\frac{43 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{96 a^2 d}-\frac{15 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{32 a^2 d}-\frac{\cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{16 a^2 d}\\ \end{align*}
Mathematica [C] time = 23.7249, size = 5584, normalized size = 22.25 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.304, size = 722, normalized size = 2.9 \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 )^{4}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{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 ^{4}{\left (c + d x \right )}}{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 9.98379, size = 346, normalized size = 1.38 \begin{align*} -\frac{\sqrt{2}{\left (3 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left (\frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{21}{a \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{32 \,{\left (9 \,{\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 )}^{4} \sqrt{-a} - 15 \,{\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} \sqrt{-a} a + 8 \, \sqrt{-a} a^{2}\right )}}{{\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 )}^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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