Optimal. Leaf size=335 \[ \frac{579 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{640 a^3 d}-\frac{323 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{768 a^2 d}-\frac{\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{48 a^3 d}-\frac{23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{192 a^3 d}-\frac{101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{128 a^3 d}-\frac{189 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{512 a d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{835 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{512 \sqrt{2} \sqrt{a} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.321512, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{579 \cot ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{640 a^3 d}-\frac{323 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{768 a^2 d}-\frac{\cos ^3(c+d x) \cot ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{48 a^3 d}-\frac{23 \cos ^2(c+d x) \cot ^5(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{192 a^3 d}-\frac{101 \cos (c+d x) \cot ^5(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{5/2}}{128 a^3 d}-\frac{189 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{512 a d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{835 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{512 \sqrt{2} \sqrt{a} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3887
Rule 472
Rule 579
Rule 583
Rule 522
Rule 203
Rubi steps
\begin{align*} \int \frac{\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 )^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 ^5(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{5/2}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{a-11 a^2 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 )}{6 a^4 d}\\ &=-\frac{23 \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}}{192 a^3 d}-\frac{\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}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{-111 a^2-207 a^3 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 )}{48 a^5 d}\\ &=-\frac{101 \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}}{128 a^3 d}-\frac{23 \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}}{192 a^3 d}-\frac{\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}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{-1737 a^3-2121 a^4 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 )}{192 a^6 d}\\ &=\frac{579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac{101 \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}}{128 a^3 d}-\frac{23 \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}}{192 a^3 d}-\frac{\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}}{48 a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{-4845 a^4-8685 a^5 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 )}{1920 a^6 d}\\ &=-\frac{323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac{579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac{101 \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}}{128 a^3 d}-\frac{23 \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}}{192 a^3 d}-\frac{\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}}{48 a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{8505 a^5-14535 a^6 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 )}{11520 a^6 d}\\ &=-\frac{189 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{512 a d}-\frac{323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac{579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac{101 \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}}{128 a^3 d}-\frac{23 \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}}{192 a^3 d}-\frac{\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}}{48 a^3 d}+\frac{\operatorname{Subst}\left (\int \frac{54585 a^6+8505 a^7 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 )}{23040 a^6 d}\\ &=-\frac{189 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{512 a d}-\frac{323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac{579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac{101 \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}}{128 a^3 d}-\frac{23 \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}}{192 a^3 d}-\frac{\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}}{48 a^3 d}-\frac{835 \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 )}{512 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{835 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{512 \sqrt{2} \sqrt{a} d}-\frac{189 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{512 a d}-\frac{323 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{768 a^2 d}+\frac{579 \cot ^5(c+d x) (a+a \sec (c+d x))^{5/2}}{640 a^3 d}-\frac{101 \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}}{128 a^3 d}-\frac{23 \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}}{192 a^3 d}-\frac{\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}}{48 a^3 d}\\ \end{align*}
Mathematica [C] time = 23.5984, size = 5628, normalized size = 16.8 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.292, size = 1068, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 10.0916, size = 522, normalized size = 1.56 \begin{align*} -\frac{\sqrt{2}{\left (5 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}{\left (2 \,{\left (\frac{4 \, \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{43}{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 )^{2} + \frac{567}{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{96 \,{\left (145 \,{\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} \sqrt{-a} - 500 \,{\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} \sqrt{-a} a + 710 \,{\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} a^{2} - 460 \,{\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^{3} + 121 \, \sqrt{-a} 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} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )}}{15360 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]