Optimal. Leaf size=78 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a^4 d}-\frac {6 \sqrt {a \sec (c+d x)+a}}{a^3 d} \]
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Rubi [A] time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3880, 88, 63, 207} \[ \frac {2 (a \sec (c+d x)+a)^{3/2}}{3 a^4 d}-\frac {6 \sqrt {a \sec (c+d x)+a}}{a^3 d}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{a^{5/2} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 88
Rule 207
Rule 3880
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+a x)^2}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {3 a^2}{\sqrt {a+a x}}+\frac {a^2}{x \sqrt {a+a x}}+a \sqrt {a+a x}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac {6 \sqrt {a+a \sec (c+d x)}}{a^3 d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {6 \sqrt {a+a \sec (c+d x)}}{a^3 d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 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 )}{a^3 d}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {6 \sqrt {a+a \sec (c+d x)}}{a^3 d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 a^4 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 69, normalized size = 0.88 \[ \frac {2 \left (\sec ^2(c+d x)-7 \sec (c+d x)-3 \sqrt {\sec (c+d x)+1} \tanh ^{-1}\left (\sqrt {\sec (c+d x)+1}\right )-8\right )}{3 a^2 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 241, normalized size = 3.09 \[ \left [\frac {3 \, \sqrt {a} \cos \left (d x + c\right ) \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) - 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (8 \, \cos \left (d x + c\right ) - 1\right )}}{6 \, a^{3} d \cos \left (d x + c\right )}, \frac {3 \, \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right ) - 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (8 \, \cos \left (d x + c\right ) - 1\right )}}{3 \, a^{3} d \cos \left (d x + c\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.51, size = 140, normalized size = 1.79 \[ -\frac {2 \, {\left (\frac {3 \, \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {\sqrt {2} {\left (9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, a\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.19, size = 155, normalized size = 1.99 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (3 \cos \left (d x +c \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}+3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}+32 \cos \left (d x +c \right )-4\right )}{6 d \cos \left (d x +c \right ) a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 163, normalized size = 2.09 \[ \frac {\frac {3 \, \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}}}{a^{4}} - \frac {18 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}}}{a^{3}} + \frac {2 \, {\left (4 \, a + \frac {3 \, a}{\cos \left (d x + c\right )}\right )}}{{\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} a^{2}} - \frac {6}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} a^{2}} - \frac {2}{{\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} a}}{3 \, d} \]
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 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^5}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\tan ^{5}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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