Optimal. Leaf size=214 \[ \frac{87 a^2}{160 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}+\frac{23 a}{192 d (a \sec (c+d x)+a)^{3/2}}-\frac{105}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{\sqrt{a} d}-\frac{151 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{128 \sqrt{2} \sqrt{a} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.177853, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {3880, 103, 151, 152, 156, 63, 207} \[ \frac{87 a^2}{160 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}+\frac{23 a}{192 d (a \sec (c+d x)+a)^{3/2}}-\frac{105}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{\sqrt{a} d}-\frac{151 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{128 \sqrt{2} \sqrt{a} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3880
Rule 103
Rule 151
Rule 152
Rule 156
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\cot ^5(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx &=\frac{a^6 \operatorname{Subst}\left (\int \frac{1}{x (-a+a x)^3 (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{4 a^2+\frac{9 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{4 d}\\ &=-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{8 a^4+\frac{119 a^4 x}{4}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{8 d}\\ &=\frac{87 a^2}{160 d (a+a \sec (c+d x))^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{-40 a^6-\frac{435 a^6 x}{8}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{40 a^3 d}\\ &=\frac{87 a^2}{160 d (a+a \sec (c+d x))^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac{23 a}{192 d (a+a \sec (c+d x))^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{120 a^8+\frac{345 a^8 x}{16}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{120 a^6 d}\\ &=\frac{87 a^2}{160 d (a+a \sec (c+d x))^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac{23 a}{192 d (a+a \sec (c+d x))^{3/2}}-\frac{105}{128 d \sqrt{a+a \sec (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{-120 a^{10}+\frac{1575 a^{10} x}{32}}{x (-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{120 a^9 d}\\ &=\frac{87 a^2}{160 d (a+a \sec (c+d x))^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac{23 a}{192 d (a+a \sec (c+d x))^{3/2}}-\frac{105}{128 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 )}{d}+\frac{(151 a) \operatorname{Subst}\left (\int \frac{1}{(-a+a x) \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{256 d}\\ &=\frac{87 a^2}{160 d (a+a \sec (c+d x))^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac{23 a}{192 d (a+a \sec (c+d x))^{3/2}}-\frac{105}{128 d \sqrt{a+a \sec (c+d x)}}+\frac{151 \operatorname{Subst}\left (\int \frac{1}{-2 a+x^2} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{128 d}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{a d}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}-\frac{151 \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{128 \sqrt{2} \sqrt{a} d}+\frac{87 a^2}{160 d (a+a \sec (c+d x))^{5/2}}-\frac{a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{5/2}}-\frac{17 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac{23 a}{192 d (a+a \sec (c+d x))^{3/2}}-\frac{105}{128 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.255118, size = 102, normalized size = 0.48 \[ \frac{\cot ^4(c+d x) \left (151 (\sec (c+d x)-1)^2 \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},\frac{1}{2} (\sec (c+d x)+1)\right )-2 \left (32 (\sec (c+d x)-1)^2 \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},\sec (c+d x)+1\right )-85 \sec (c+d x)+105\right )\right )}{160 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.382, size = 746, normalized size = 3.5 \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 = 10.1823, size = 400, normalized size = 1.87 \begin{align*} \frac{\sqrt{2}{\left (\frac{3840 \, \sqrt{2} \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} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{2265 \, \arctan \left (\frac{\sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} + \frac{15 \,{\left (25 \,{\left (-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}} - 23 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a\right )}}{a^{2} \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 )^{4}} + \frac{8 \,{\left (3 \,{\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^{12} + 25 \,{\left (-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a\right )}^{\frac{3}{2}} a^{13} + 240 \, \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a^{14}\right )}}{a^{15} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )}}{3840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]