Optimal. Leaf size=200 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {13 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} a^{5/2} d}+\frac {51}{32 a^2 d \sqrt {a \sec (c+d x)+a}}-\frac {5 a}{28 d (a \sec (c+d x)+a)^{7/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{7/2}}+\frac {3}{40 d (a \sec (c+d x)+a)^{5/2}}+\frac {19}{48 a d (a \sec (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.17, antiderivative size = 200, normalized size of antiderivative = 1.00, 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.
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Rule 63
Rule 103
Rule 152
Rule 156
Rule 207
Rule 3880
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*}
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Mathematica [C] time = 0.21, size = 90, normalized size = 0.45 \[ \frac {a \left (-13 (\sec (c+d x)-1) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\frac {1}{2} (\sec (c+d x)+1)\right )+8 (\sec (c+d x)-1) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};\sec (c+d x)+1\right )-14\right )}{28 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{7/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 748, normalized size = 3.74 \[ \left [\frac {1365 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 ) + 6720 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 (8017 \, \cos \left (d x + c\right )^{5} + 12640 \, \cos \left (d x + c\right )^{4} - 1582 \, \cos \left (d x + c\right )^{3} - 12040 \, \cos \left (d x + c\right )^{2} - 5355 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{13440 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}}, -\frac {1365 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 ) - 6720 \, {\left (\cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} - 3 \, \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 (8017 \, \cos \left (d x + c\right )^{5} + 12640 \, \cos \left (d x + c\right )^{4} - 1582 \, \cos \left (d x + c\right )^{3} - 12040 \, \cos \left (d x + c\right )^{2} - 5355 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{6720 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.02, size = 323, normalized size = 1.62 \[ \frac {\frac {1365 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {13440 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} + \frac {105 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} + \frac {2 \, \sqrt {2} {\left (15 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{36} - 84 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{37} - 385 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{38} - 2730 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{39}\right )}}{a^{42} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.57, size = 744, normalized size = 3.72 \[ \text {result too large to display} \]
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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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