Optimal. Leaf size=171 \[ \frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}-\frac {71 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} d}+\frac {7 a^2}{32 d \sqrt {a \sec (c+d x)+a}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.14, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {7 a^2}{32 d \sqrt {a \sec (c+d x)+a}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a \sec (c+d x)+a}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a \sec (c+d x)+a}}+\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}-\frac {71 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} d} \]
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 \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)^{3/2}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a+a \sec (c+d x)}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {4 a^2+\frac {5 a^2 x}{2}}{x (-a+a x)^2 (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{4 d}\\ &=-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a+a \sec (c+d x)}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a+a \sec (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {8 a^4+\frac {39 a^4 x}{4}}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{8 d}\\ &=\frac {7 a^2}{32 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a+a \sec (c+d x)}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a+a \sec (c+d x)}}-\frac {\operatorname {Subst}\left (\int \frac {-8 a^6-\frac {7 a^6 x}{8}}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{8 a^3 d}\\ &=\frac {7 a^2}{32 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a+a \sec (c+d x)}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a+a \sec (c+d x)}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}+\frac {\left (71 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{64 d}\\ &=\frac {7 a^2}{32 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a+a \sec (c+d x)}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a+a \sec (c+d x)}}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d}+\frac {\left (71 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{32 d}\\ &=\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {71 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{32 \sqrt {2} d}+\frac {7 a^2}{32 d \sqrt {a+a \sec (c+d x)}}-\frac {a^2}{4 d (1-\sec (c+d x))^2 \sqrt {a+a \sec (c+d x)}}-\frac {13 a^2}{16 d (1-\sec (c+d x)) \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 104, normalized size = 0.61 \[ \frac {a^2 \left (71 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {1}{2} (\sec (c+d x)+1)\right )-64 (\sec (c+d x)-1)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\sec (c+d x)+1\right )+26 \sec (c+d x)-34\right )}{32 d (\sec (c+d x)-1)^2 \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 589, normalized size = 3.44 \[ \left [\frac {64 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\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 ) + 71 \, {\left (\sqrt {2} a \cos \left (d x + c\right )^{3} - \sqrt {2} a \cos \left (d x + c\right )^{2} - \sqrt {2} a \cos \left (d x + c\right ) + \sqrt {2} a\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 ) - 4 \, {\left (27 \, a \cos \left (d x + c\right )^{3} - 12 \, a \cos \left (d x + c\right )^{2} - 7 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{128 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )}}, \frac {71 \, {\left (\sqrt {2} a \cos \left (d x + c\right )^{3} - \sqrt {2} a \cos \left (d x + c\right )^{2} - \sqrt {2} a \cos \left (d x + c\right ) + \sqrt {2} a\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 ) - 64 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + a\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 (27 \, a \cos \left (d x + c\right )^{3} - 12 \, a \cos \left (d x + c\right )^{2} - 7 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{64 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.16, size = 211, normalized size = 1.23 \[ \frac {\frac {71 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{\sqrt {-a}} - \frac {128 \, a^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{\sqrt {-a}} - 8 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - \frac {17 \, \sqrt {2} {\left (-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 15 \, \sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.27, size = 502, normalized size = 2.94 \[ \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 )^{2} \left (64 \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 )+71 \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 )-64 \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}\, \left (\cos ^{2}\left (d x +c \right )\right )-71 \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 ) \left (\cos ^{2}\left (d x +c \right )\right )-64 \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 )+54 \left (\cos ^{3}\left (d x +c \right )\right )-71 \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 )+64 \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}-24 \left (\cos ^{2}\left (d x +c \right )\right )+71 \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 )-14 \cos \left (d x +c \right )\right ) a}{64 d \sin \left (d x +c \right )^{6}} \]
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 = -0.01 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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