Optimal. Leaf size=262 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a \sec (c+d x)+a)^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac {761}{512 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {135 a}{448 d (a \sec (c+d x)+a)^{7/2}}+\frac {7}{640 d (a \sec (c+d x)+a)^{5/2}}-\frac {83}{256 a d (a \sec (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.23, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 13, 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} \[ \frac {199 a^2}{288 d (a \sec (c+d x)+a)^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{9/2}}-\frac {761}{512 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {135 a}{448 d (a \sec (c+d x)+a)^{7/2}}+\frac {7}{640 d (a \sec (c+d x)+a)^{5/2}}-\frac {83}{256 a d (a \sec (c+d x)+a)^{3/2}} \]
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)}{(a+a \sec (c+d x))^{5/2}} \, dx &=\frac {a^6 \operatorname {Subst}\left (\int \frac {1}{x (-a+a x)^3 (a+a x)^{11/2}} \, 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))^{9/2}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {4 a^2+\frac {13 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{11/2}} \, 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))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {\operatorname {Subst}\left (\int \frac {8 a^4+\frac {231 a^4 x}{4}}{x (-a+a x) (a+a x)^{11/2}} \, dx,x,\sec (c+d x)\right )}{8 d}\\ &=\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}-\frac {\operatorname {Subst}\left (\int \frac {-72 a^6-\frac {1791 a^6 x}{8}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{72 a^3 d}\\ &=\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {\operatorname {Subst}\left (\int \frac {504 a^8+\frac {8505 a^8 x}{16}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{504 a^6 d}\\ &=\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {-2520 a^{10}-\frac {2205 a^{10} x}{32}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{2520 a^9 d}\\ &=\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {7560 a^{12}-\frac {235305 a^{12} x}{64}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{7560 a^{12} d}\\ &=\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {\operatorname {Subst}\left (\int \frac {-7560 a^{14}+\frac {719145 a^{14} x}{128}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{7560 a^{15} d}\\ &=\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 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 {263 \operatorname {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{1024 a d}\\ &=\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 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 {263 \operatorname {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{512 a^2 d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {263 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{512 \sqrt {2} a^{5/2} d}+\frac {199 a^2}{288 d (a+a \sec (c+d x))^{9/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{9/2}}-\frac {21 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{9/2}}+\frac {135 a}{448 d (a+a \sec (c+d x))^{7/2}}+\frac {7}{640 d (a+a \sec (c+d x))^{5/2}}-\frac {83}{256 a d (a+a \sec (c+d x))^{3/2}}-\frac {761}{512 a^2 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 99, normalized size = 0.38 \[ \frac {\cot ^4(c+d x) \left (263 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {9}{2},1;-\frac {7}{2};\frac {1}{2} (\sec (c+d x)+1)\right )-64 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {9}{2},1;-\frac {7}{2};\sec (c+d x)+1\right )+378 \sec (c+d x)-450\right )}{288 d (a (\sec (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 905, normalized size = 3.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.67, size = 392, normalized size = 1.50 \[ -\frac {\frac {82845 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {645120 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {315 \, {\left (33 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 31 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{4} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} - \frac {8 \, \sqrt {2} {\left (35 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{56} - 225 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{57} + 1008 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{58} + 4410 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{59} + 31185 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{60}\right )}}{a^{63} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{322560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.49, size = 986, normalized size = 3.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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