Optimal. Leaf size=145 \[ \frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}+\frac {2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}-\frac {2 a^2 \sqrt {a \sec (c+d x)+a}}{d}-\frac {2 (a \sec (c+d x)+a)^{7/2}}{7 a d}-\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 d}-\frac {2 a (a \sec (c+d x)+a)^{3/2}}{3 d} \]
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Rubi [A] time = 0.12, antiderivative size = 145, 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, 80, 50, 63, 207} \[ \frac {2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}-\frac {2 a^2 \sqrt {a \sec (c+d x)+a}}{d}+\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}-\frac {2 (a \sec (c+d x)+a)^{7/2}}{7 a d}-\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 d}-\frac {2 a (a \sec (c+d x)+a)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 207
Rule 3880
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{5/2} \tan ^3(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+a x) (a+a x)^{7/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{7/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {a \operatorname {Subst}\left (\int \frac {(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}-\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}-\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d}\\ &=\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}-\frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}-\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 102, normalized size = 0.70 \[ \frac {2 (a (\sec (c+d x)+1))^{5/2} \left (\sqrt {\sec (c+d x)+1} \left (35 \sec ^4(c+d x)+95 \sec ^3(c+d x)+12 \sec ^2(c+d x)-226 \sec (c+d x)-493\right )+315 \tanh ^{-1}\left (\sqrt {\sec (c+d x)+1}\right )\right )}{315 d (\sec (c+d x)+1)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 334, normalized size = 2.30 \[ \left [\frac {315 \, a^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) - 4 \, {\left (493 \, a^{2} \cos \left (d x + c\right )^{4} + 226 \, a^{2} \cos \left (d x + c\right )^{3} - 12 \, a^{2} \cos \left (d x + c\right )^{2} - 95 \, a^{2} \cos \left (d x + c\right ) - 35 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{630 \, d \cos \left (d x + c\right )^{4}}, -\frac {315 \, \sqrt {-a} a^{2} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{4} + 2 \, {\left (493 \, a^{2} \cos \left (d x + c\right )^{4} + 226 \, a^{2} \cos \left (d x + c\right )^{3} - 12 \, a^{2} \cos \left (d x + c\right )^{2} - 95 \, a^{2} \cos \left (d x + c\right ) - 35 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.92, size = 198, normalized size = 1.37 \[ -\frac {\sqrt {2} {\left (\frac {315 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, {\left (315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} a^{2} - 210 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} a^{3} + 252 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} a^{4} - 360 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a^{5} - 560 \, a^{6}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}\right )} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.24, size = 362, normalized size = 2.50 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (315 \sqrt {2}\, \left (\cos ^{4}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}}+1260 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}\, \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right )+1890 \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}}+1260 \sqrt {2}\, \cos \left (d x +c \right ) \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}}+315 \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}}+15776 \left (\cos ^{4}\left (d x +c \right )\right )+7232 \left (\cos ^{3}\left (d x +c \right )\right )-384 \left (\cos ^{2}\left (d x +c \right )\right )-3040 \cos \left (d x +c \right )-1120\right ) a^{2}}{5040 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 143, normalized size = 0.99 \[ -\frac {315 \, a^{\frac {5}{2}} \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 126 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}} - \frac {70 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {9}{2}}}{a^{2}} + \frac {90 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {7}{2}}}{a} + 210 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} a + 630 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}} a^{2}}{315 \, d} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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