Optimal. Leaf size=214 \[ \frac {87 a^2}{160 d (a \sec (c+d x)+a)^{5/2}}-\frac {17 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{5/2}}+\frac {23 a}{192 d (a \sec (c+d x)+a)^{3/2}}-\frac {105}{128 d \sqrt {a \sec (c+d x)+a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {151 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} \sqrt {a} d} \]
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Rubi [A] time = 0.18, antiderivative size = 214, normalized size of antiderivative = 1.00, 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.
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Rule 63
Rule 103
Rule 151
Rule 152
Rule 156
Rule 207
Rule 3880
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*}
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Mathematica [C] time = 0.28, size = 102, normalized size = 0.48 \[ \frac {\cot ^4(c+d x) \left (151 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {1}{2} (\sec (c+d x)+1)\right )-2 \left (32 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\sec (c+d x)+1\right )-85 \sec (c+d x)+105\right )\right )}{160 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 705, normalized size = 3.29 \[ \left [\frac {2265 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) - 1}\right ) + 3840 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) - 4 \, {\left (2821 \, \cos \left (d x + c\right )^{5} + 278 \, \cos \left (d x + c\right )^{4} - 3964 \, \cos \left (d x + c\right )^{3} - 230 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{7680 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )}}, \frac {2265 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{a \cos \left (d x + c\right ) + a}\right ) - 3840 \, {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, {\left (2821 \, \cos \left (d x + c\right )^{5} + 278 \, \cos \left (d x + c\right )^{4} - 3964 \, \cos \left (d x + c\right )^{3} - 230 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3840 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 8.55, size = 296, normalized size = 1.38 \[ \frac {\sqrt {2} {\left (\frac {3840 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {2265 \, \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} + \frac {15 \, {\left (25 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 23 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {8 \, {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{12} + 25 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{13} + 240 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{14}\right )}}{a^{15} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )}}{3840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.85, size = 746, normalized size = 3.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \left (d x + c\right )^{5}}{\sqrt {a \sec \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{5}{\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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