Optimal. Leaf size=303 \[ \frac{277 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{384 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{256 a^2 d}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}-\frac{533 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{256 \sqrt{2} a^{3/2} d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{48 a^3 d}-\frac{7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{64 a^3 d}-\frac{81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^3 d} \]
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Rubi [A] time = 0.28155, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{277 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{384 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{256 a^2 d}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}-\frac{533 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{256 \sqrt{2} a^{3/2} d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{48 a^3 d}-\frac{7 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{64 a^3 d}-\frac{81 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^3 d} \]
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 ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \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 ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{3 a-9 a^2 x^2}{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 )}{6 a^4 d}\\ &=-\frac{7 \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}}{64 a^3 d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{-51 a^2-147 a^3 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 )}{48 a^5 d}\\ &=-\frac{81 \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}}{128 a^3 d}-\frac{7 \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}}{64 a^3 d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{-831 a^3-1215 a^4 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 )}{192 a^6 d}\\ &=\frac{277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac{81 \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}}{128 a^3 d}-\frac{7 \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}}{64 a^3 d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{-189 a^4-2493 a^5 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 )}{1152 a^6 d}\\ &=-\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{256 a^2 d}+\frac{277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac{81 \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}}{128 a^3 d}-\frac{7 \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}}{64 a^3 d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{4419 a^5-189 a^6 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 )}{2304 a^6 d}\\ &=-\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{256 a^2 d}+\frac{277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac{81 \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}}{128 a^3 d}-\frac{7 \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}}{64 a^3 d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{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 d}+\frac{533 \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 )}{256 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{533 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{256 \sqrt{2} a^{3/2} d}-\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{256 a^2 d}+\frac{277 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{384 a^3 d}-\frac{81 \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}}{128 a^3 d}-\frac{7 \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}}{64 a^3 d}-\frac{\cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{48 a^3 d}\\ \end{align*}
Mathematica [C] time = 23.679, size = 5630, normalized size = 18.58 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.326, size = 732, normalized size = 2.4 \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(-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 [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 )}}{\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 = 10.0694, size = 390, normalized size = 1.29 \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^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{37 \, \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 )^{2} + \frac{417 \, \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{32 \, \sqrt{2}{\left (21 \,{\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} - 36 \,{\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 + 19 \, 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} \sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{1536 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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