Optimal. Leaf size=387 \[ \frac{12267 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{10240 a^4 d}-\frac{8171 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12288 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8192 a^2 d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac{16363 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{8192 \sqrt{2} a^{3/2} d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{128 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{768 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{3072 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{2048 a^4 d} \]
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Rubi [A] time = 0.3678, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{12267 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{10240 a^4 d}-\frac{8171 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{12288 a^3 d}-\frac{21 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{8192 a^2 d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac{16363 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{8192 \sqrt{2} a^{3/2} d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{128 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{768 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{3072 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{2048 a^4 d} \]
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 ^6(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^5} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^4 d}\\ &=-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{3 a-13 a^2 x^2}{x^6 \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 )}{8 a^5 d}\\ &=-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{-127 a^2-319 a^3 x^2}{x^6 \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 )}{96 a^6 d}\\ &=-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{-3063 a^3-4599 a^4 x^2}{x^6 \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 )}{768 a^7 d}\\ &=-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{-36801 a^4-42945 a^5 x^2}{x^6 \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 )}{3072 a^8 d}\\ &=\frac{12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{-122565 a^5-184005 a^6 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 )}{30720 a^8 d}\\ &=-\frac{8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac{12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{945 a^6-367695 a^7 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 )}{184320 a^8 d}\\ &=-\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{8192 a^2 d}-\frac{8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac{12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{738225 a^7+945 a^8 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 )}{368640 a^8 d}\\ &=-\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{8192 a^2 d}-\frac{8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac{12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}-\frac{16363 \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 )}{8192 a 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{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^{3/2} d}+\frac{16363 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{8192 \sqrt{2} a^{3/2} d}-\frac{21 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{8192 a^2 d}-\frac{8171 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{12288 a^3 d}+\frac{12267 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{10240 a^4 d}-\frac{2045 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{2048 a^4 d}-\frac{511 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{3072 a^4 d}-\frac{29 \cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{768 a^4 d}-\frac{\cos ^4(c+d x) \cot ^5(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{128 a^4 d}\\ \end{align*}
Mathematica [C] time = 23.6231, size = 5672, normalized size = 14.66 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.379, size = 1240, 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(-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(-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 = 10.2234, size = 559, normalized size = 1.44 \begin{align*} \frac{5 \,{\left (2 \,{\left (4 \,{\left (\frac{6 \, \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{65 \, \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{1451 \, \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{13503 \, \sqrt{2}}{a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \frac{256 \, \sqrt{2}{\left (555 \,{\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 )}^{8} - 1950 \,{\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 )}^{6} a + 2780 \,{\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} a^{2} - 1810 \,{\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^{3} + 473 \, a^{4}\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 )}^{5} \sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{245760 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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