Optimal. Leaf size=176 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {3 a}{20 d (a \sec (c+d x)+a)^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}+\frac {5}{24 d (a \sec (c+d x)+a)^{3/2}}+\frac {21}{16 a d \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.16, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {3 a}{20 d (a \sec (c+d x)+a)^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a \sec (c+d x)+a)^{5/2}}+\frac {5}{24 d (a \sec (c+d x)+a)^{3/2}}+\frac {21}{16 a d \sqrt {a \sec (c+d x)+a}} \]
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))^{3/2}} \, dx &=\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{x (-a+a x)^2 (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}-\frac {a \operatorname {Subst}\left (\int \frac {2 a^2+\frac {7 a^2 x}{2}}{x (-a+a x) (a+a x)^{7/2}} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {-10 a^4-\frac {15 a^4 x}{4}}{x (-a+a x) (a+a x)^{5/2}} \, dx,x,\sec (c+d x)\right )}{10 a^2 d}\\ &=-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {30 a^6-\frac {75 a^6 x}{8}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{30 a^5 d}\\ &=-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {-30 a^8+\frac {315 a^8 x}{16}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{30 a^8 d}\\ &=-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}}-\frac {11 \operatorname {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{32 d}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 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 {11 \operatorname {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{16 a d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} a^{3/2} d}-\frac {3 a}{20 d (a+a \sec (c+d x))^{5/2}}+\frac {a}{2 d (1-\sec (c+d x)) (a+a \sec (c+d x))^{5/2}}+\frac {5}{24 d (a+a \sec (c+d x))^{3/2}}+\frac {21}{16 a d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 90, normalized size = 0.51 \[ \frac {a \left (-11 (\sec (c+d x)-1) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {1}{2} (\sec (c+d x)+1)\right )+8 (\sec (c+d x)-1) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\sec (c+d x)+1\right )-10\right )}{20 d (\sec (c+d x)-1) (a (\sec (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 592, normalized size = 3.36 \[ \left [\frac {165 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 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 ) + 480 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 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 (449 \, \cos \left (d x + c\right )^{4} + 351 \, \cos \left (d x + c\right )^{3} - 365 \, \cos \left (d x + c\right )^{2} - 315 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{960 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )}}, -\frac {165 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 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 ) - 480 \, {\left (\cos \left (d x + c\right )^{4} + 2 \, \cos \left (d x + c\right )^{3} - 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 (449 \, \cos \left (d x + c\right )^{4} + 351 \, \cos \left (d x + c\right )^{3} - 365 \, \cos \left (d x + c\right )^{2} - 315 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{480 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} + 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 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 = 1.63, size = 281, normalized size = 1.60 \[ \frac {\frac {165 \, \sqrt {2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {960 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} + \frac {15 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right ) \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {2 \, \sqrt {2} {\left (3 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{16} + 20 \, {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{17} + 165 \, \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{18}\right )}}{a^{20} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.36, size = 514, normalized size = 2.92 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{3} \left (480 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+165 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+960 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+330 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )+898 \left (\cos ^{4}\left (d x +c \right )\right )-960 \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+702 \left (\cos ^{3}\left (d x +c \right )\right )-330 \cos \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-480 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sqrt {2}-730 \left (\cos ^{2}\left (d x +c \right )\right )-165 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {1}{\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-630 \cos \left (d x +c \right )\right )}{480 d \sin \left (d x +c \right )^{8} a^{2}} \]
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 \[ \int \frac {\cot \left (d x + c\right )^{3}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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