Optimal. Leaf size=280 \[ \frac{87 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{160 a^2 d}-\frac{\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{16 a^2 d}-\frac{17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{32 a^2 d}-\frac{23 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{192 a d}-\frac{105 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 d}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{151 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{128 \sqrt{2} d} \]
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Rubi [A] time = 0.271305, antiderivative size = 280, 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} \[ \frac{87 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{160 a^2 d}-\frac{\cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{16 a^2 d}-\frac{17 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{32 a^2 d}-\frac{23 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{192 a d}-\frac{105 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{128 d}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{151 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{128 \sqrt{2} d} \]
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 \cot ^6(c+d x) \sqrt{a+a \sec (c+d x)} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{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 )}{a^2 d}\\ &=-\frac{\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}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{-a-9 a^2 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 )}{4 a^3 d}\\ &=-\frac{17 \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}}{32 a^2 d}-\frac{\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}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{-87 a^2-119 a^3 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 )}{16 a^4 d}\\ &=\frac{87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac{17 \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}}{32 a^2 d}-\frac{\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}}{16 a^2 d}+\frac{\operatorname{Subst}\left (\int \frac{-115 a^3-435 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 )}{160 a^4 d}\\ &=-\frac{23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac{87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac{17 \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}}{32 a^2 d}-\frac{\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}}{16 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{1575 a^4-345 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 )}{960 a^4 d}\\ &=-\frac{105 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{128 d}-\frac{23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac{87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac{17 \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}}{32 a^2 d}-\frac{\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}}{16 a^2 d}+\frac{\operatorname{Subst}\left (\int \frac{5415 a^5+1575 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 )}{1920 a^4 d}\\ &=-\frac{105 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{128 d}-\frac{23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac{87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac{17 \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}}{32 a^2 d}-\frac{\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}}{16 a^2 d}-\frac{(151 a) \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 d}+\frac{(2 a) \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 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{151 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{128 \sqrt{2} d}-\frac{105 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{128 d}-\frac{23 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{192 a d}+\frac{87 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{160 a^2 d}-\frac{17 \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}}{32 a^2 d}-\frac{\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}}{16 a^2 d}\\ \end{align*}
Mathematica [C] time = 23.6775, size = 5604, normalized size = 20.01 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.371, size = 573, normalized size = 2.1 \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 [A] time = 2.86668, size = 1781, normalized size = 6.36 \begin{align*} \text{result too large to display} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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