Optimal. Leaf size=355 \[ \frac{5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac{1491 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4096 a^3 d}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{5/2} d}-\frac{9683 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{4096 \sqrt{2} a^{5/2} d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac{9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac{145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac{1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.329425, antiderivative size = 355, 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} \[ \frac{5587 \cot ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{6144 a^4 d}-\frac{1491 \cot (c+d x) \sqrt{a \sec (c+d x)+a}}{4096 a^3 d}+\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{5/2} d}-\frac{9683 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{4096 \sqrt{2} a^{5/2} d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{128 a^4 d}-\frac{9 \cos ^3(c+d x) \cot ^3(c+d x) \sec ^6\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{256 a^4 d}-\frac{145 \cos ^2(c+d x) \cot ^3(c+d x) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{1024 a^4 d}-\frac{1527 \cos (c+d x) \cot ^3(c+d x) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \sec (c+d x)+a)^{3/2}}{2048 a^4 d} \]
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 ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \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 ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{5 a-11 a^2 x^2}{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 )}{8 a^5 d}\\ &=-\frac{9 \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}}{256 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{-51 a^2-243 a^3 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 )}{96 a^6 d}\\ &=-\frac{145 \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}}{1024 a^4 d}-\frac{9 \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}}{256 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{-1509 a^3-3045 a^4 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 )}{768 a^7 d}\\ &=-\frac{1527 \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}}{2048 a^4 d}-\frac{145 \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}}{1024 a^4 d}-\frac{9 \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}}{256 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{-16761 a^4-22905 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 )}{3072 a^8 d}\\ &=\frac{5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac{1527 \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}}{2048 a^4 d}-\frac{145 \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}}{1024 a^4 d}-\frac{9 \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}}{256 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{-13419 a^5-50283 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 )}{18432 a^8 d}\\ &=-\frac{1491 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{4096 a^3 d}+\frac{5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac{1527 \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}}{2048 a^4 d}-\frac{145 \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}}{1024 a^4 d}-\frac{9 \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}}{256 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}-\frac{\operatorname{Subst}\left (\int \frac{60309 a^6-13419 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 )}{36864 a^8 d}\\ &=-\frac{1491 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{4096 a^3 d}+\frac{5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac{1527 \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}}{2048 a^4 d}-\frac{145 \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}}{1024 a^4 d}-\frac{9 \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}}{256 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 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^2 d}+\frac{9683 \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 )}{4096 a^2 d}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^{5/2} d}-\frac{9683 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{4096 \sqrt{2} a^{5/2} d}-\frac{1491 \cot (c+d x) \sqrt{a+a \sec (c+d x)}}{4096 a^3 d}+\frac{5587 \cot ^3(c+d x) (a+a \sec (c+d x))^{3/2}}{6144 a^4 d}-\frac{1527 \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}}{2048 a^4 d}-\frac{145 \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}}{1024 a^4 d}-\frac{9 \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}}{256 a^4 d}-\frac{\cos ^4(c+d x) \cot ^3(c+d x) \sec ^8\left (\frac{1}{2} (c+d x)\right ) (a+a \sec (c+d x))^{3/2}}{128 a^4 d}\\ \end{align*}
Mathematica [C] time = 23.6848, size = 5656, normalized size = 15.93 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.356, size = 1066, normalized size = 3. \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 = 10.1262, size = 447, normalized size = 1.26 \begin{align*} -\frac{3 \,{\left (2 \,{\left (4 \,{\left (\frac{2 \, \sqrt{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{a^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{19 \, \sqrt{2}}{a^{3} \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{369 \, \sqrt{2}}{a^{3} \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{2989 \, \sqrt{2}}{a^{3} \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{512 \, \sqrt{2}{\left (12 \,{\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} - 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 )}^{2} a + 11 \, 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} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{24576 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]