Optimal. Leaf size=238 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a \sec (c+d x)+a)^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}+\frac {15 a}{64 d (a \sec (c+d x)+a)^{5/2}}-\frac {53}{384 d (a \sec (c+d x)+a)^{3/2}}-\frac {309}{256 a d \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.20, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \frac {139 a^2}{224 d (a \sec (c+d x)+a)^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{7/2}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {15 a}{64 d (a \sec (c+d x)+a)^{5/2}}-\frac {53}{384 d (a \sec (c+d x)+a)^{3/2}}-\frac {309}{256 a d \sqrt {a \sec (c+d x)+a}} \]
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))^{3/2}} \, dx &=\frac {a^6 \operatorname {Subst}\left (\int \frac {1}{x (-a+a x)^3 (a+a x)^{9/2}} \, 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))^{7/2}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {4 a^2+\frac {11 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{9/2}} \, 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))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {\operatorname {Subst}\left (\int \frac {8 a^4+\frac {171 a^4 x}{4}}{x (-a+a x) (a+a x)^{9/2}} \, dx,x,\sec (c+d x)\right )}{8 d}\\ &=\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}-\frac {\operatorname {Subst}\left (\int \frac {-56 a^6-\frac {973 a^6 x}{8}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{56 a^3 d}\\ &=\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {280 a^8+\frac {2625 a^8 x}{16}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{280 a^6 d}\\ &=\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {-840 a^{10}+\frac {5565 a^{10} x}{32}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{840 a^9 d}\\ &=\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {840 a^{12}-\frac {32445 a^{12} x}{64}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{840 a^{12} d}\\ &=\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}+\frac {203 \operatorname {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{512 d}-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a 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^2 d}+\frac {203 \operatorname {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{256 a d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {203 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}+\frac {139 a^2}{224 d (a+a \sec (c+d x))^{7/2}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 (a+a \sec (c+d x))^{7/2}}-\frac {19 a^2}{16 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{7/2}}+\frac {15 a}{64 d (a+a \sec (c+d x))^{5/2}}-\frac {53}{384 d (a+a \sec (c+d x))^{3/2}}-\frac {309}{256 a d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 99, normalized size = 0.42 \[ \frac {\cot ^4(c+d x) \left (203 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\frac {1}{2} (\sec (c+d x)+1)\right )-64 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\sec (c+d x)+1\right )+266 \sec (c+d x)-322\right )}{224 d (a (\sec (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 837, normalized size = 3.52 \[ \left [\frac {4263 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 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 ) + 10752 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 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 (10363 \, \cos \left (d x + c\right )^{6} + 8037 \, \cos \left (d x + c\right )^{5} - 16538 \, \cos \left (d x + c\right )^{4} - 14238 \, \cos \left (d x + c\right )^{3} + 7231 \, \cos \left (d x + c\right )^{2} + 6489 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{21504 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, \frac {4263 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 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 ) - 10752 \, {\left (\cos \left (d x + c\right )^{6} + 2 \, \cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 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 (10363 \, \cos \left (d x + c\right )^{6} + 8037 \, \cos \left (d x + c\right )^{5} - 16538 \, \cos \left (d x + c\right )^{4} - 14238 \, \cos \left (d x + c\right )^{3} + 7231 \, \cos \left (d x + c\right )^{2} + 6489 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{10752 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} + 2 \, a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{4} - 4 \, a^{2} d \cos \left (d x + c\right )^{3} - a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.90, size = 350, normalized size = 1.47 \[ -\frac {\frac {4263 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {21504 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {21 \, {\left (29 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} - 27 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a\right )}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {8 \, \sqrt {2} {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{30} - 21 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{31} - 112 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{32} - 882 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{33}\right )}}{a^{35} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{10752 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.38, size = 866, normalized size = 3.64 \[ \text {result too large to display} \]
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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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