Optimal. Leaf size=147 \[ \frac {2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac {6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac {2 \sqrt {a \sec (c+d x)+a}}{d}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3880, 88, 50, 63, 207} \[ \frac {2 (a \sec (c+d x)+a)^{9/2}}{9 a^4 d}-\frac {6 (a \sec (c+d x)+a)^{7/2}}{7 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 a^2 d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a d}+\frac {2 \sqrt {a \sec (c+d x)+a}}{d}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 207
Rule 3880
Rubi steps
\begin {align*} \int \sqrt {a+a \sec (c+d x)} \tan ^5(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+a x)^2 (a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-3 a^2 (a+a x)^{5/2}+\frac {a^2 (a+a x)^{5/2}}{x}+a (a+a x)^{7/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d}\\ &=-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a^2 d}-\frac {6 (a+a \sec (c+d x))^{7/2}}{7 a^3 d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^4 d}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 102, normalized size = 0.69 \[ \frac {2 \sqrt {a (\sec (c+d x)+1)} \left (\sqrt {\sec (c+d x)+1} \left (35 \sec ^4(c+d x)+5 \sec ^3(c+d x)-132 \sec ^2(c+d x)-34 \sec (c+d x)+383\right )-315 \tanh ^{-1}\left (\sqrt {\sec (c+d x)+1}\right )\right )}{315 d \sqrt {\sec (c+d x)+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 299, normalized size = 2.03 \[ \left [\frac {315 \, \sqrt {a} \cos \left (d x + c\right )^{4} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (383 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{630 \, d \cos \left (d x + c\right )^{4}}, \frac {315 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{4} + 2 \, {\left (383 \, \cos \left (d x + c\right )^{4} - 34 \, \cos \left (d x + c\right )^{3} - 132 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 35\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.82, size = 193, normalized size = 1.31 \[ \frac {\sqrt {2} {\left (\frac {315 \, \sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, {\left (315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} a - 210 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} a^{2} + 252 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} a^{3} + 1080 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a^{4} + 560 \, a^{5}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.38, size = 359, normalized size = 2.44 \[ \frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (315 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}\, \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+1260 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}\, \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+1890 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}\, \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+1260 \cos \left (d x +c \right ) \sqrt {2}\, \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+315 \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}}+12256 \left (\cos ^{4}\left (d x +c \right )\right )-1088 \left (\cos ^{3}\left (d x +c \right )\right )-4224 \left (\cos ^{2}\left (d x +c \right )\right )+160 \cos \left (d x +c \right )+1120\right )}{5040 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 145, normalized size = 0.99 \[ \frac {315 \, \sqrt {a} \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 630 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \frac {70 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {9}{2}}}{a^{4}} - \frac {270 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {7}{2}}}{a^{3}} + \frac {126 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}}}{a^{2}} + \frac {210 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}}}{a}}{315 \, d} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^5\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )} \tan ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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