Optimal. Leaf size=200 \[ \frac{51}{32 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{a^{5/2} d}+\frac{13 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{32 \sqrt{2} a^{5/2} d}-\frac{5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac{3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac{19}{48 a d (a \sec (c+d x)+a)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.174887, antiderivative size = 200, 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 = {3880, 103, 152, 156, 63, 207} \[ \frac{51}{32 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{a^{5/2} d}+\frac{13 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{32 \sqrt{2} a^{5/2} d}-\frac{5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac{3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac{19}{48 a d (a \sec (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3880
Rule 103
Rule 152
Rule 156
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx &=\frac{a^4 \operatorname{Subst}\left (\int \frac{1}{x (-a+a x)^2 (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}-\frac{a \operatorname{Subst}\left (\int \frac{2 a^2+\frac{9 a^2 x}{2}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac{5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac{\operatorname{Subst}\left (\int \frac{-14 a^4-\frac{35 a^4 x}{4}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{14 a^2 d}\\ &=-\frac{5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac{3}{40 d (a+a \sec (c+d x))^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{70 a^6-\frac{105 a^6 x}{8}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{70 a^5 d}\\ &=-\frac{5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac{3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac{19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-210 a^8+\frac{1995 a^8 x}{16}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{210 a^8 d}\\ &=-\frac{5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac{3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac{19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac{51}{32 a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{210 a^{10}-\frac{5355 a^{10} x}{32}}{x (-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{210 a^{11} d}\\ &=-\frac{5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac{3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac{19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac{51}{32 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac{13 \operatorname{Subst}\left (\int \frac{1}{(-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{64 a d}\\ &=-\frac{5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac{3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac{19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac{51}{32 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{a^3 d}-\frac{13 \operatorname{Subst}\left (\int \frac{1}{-2 a+x^2} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{32 a^2 d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{a^{5/2} d}+\frac{13 \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{32 \sqrt{2} a^{5/2} d}-\frac{5 a}{28 d (a+a \sec (c+d x))^{7/2}}+\frac{a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac{3}{40 d (a+a \sec (c+d x))^{5/2}}+\frac{19}{48 a d (a+a \sec (c+d x))^{3/2}}+\frac{51}{32 a^2 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.18076, size = 90, normalized size = 0.45 \[ \frac{a \left (-13 (\sec (c+d x)-1) \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},\frac{1}{2} (\sec (c+d x)+1)\right )+8 (\sec (c+d x)-1) \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},\sec (c+d x)+1\right )-14\right )}{28 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.318, size = 744, normalized size = 3.7 \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 [B] time = 10.1288, size = 448, normalized size = 2.24 \begin{align*} \frac{\frac{1365 \, \sqrt{2} \arctan \left (\frac{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{13440 \, \arctan \left (\frac{\sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} + \frac{105 \, \sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{a^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}} + \frac{2 \,{\left (15 \, \sqrt{2}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{36} - 84 \, \sqrt{2}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{37} - 385 \, \sqrt{2}{\left (-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}} a^{38} - 2730 \, \sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{39}\right )}}{a^{42} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}}{6720 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]