Optimal. Leaf size=169 \[ -\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}+\frac {2 (a \sec (c+d x)+a)^{11/2}}{11 a^4 d}-\frac {2 (a \sec (c+d x)+a)^{9/2}}{3 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 a d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac {2 a \sqrt {a \sec (c+d x)+a}}{d} \]
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Rubi [A] time = 0.14, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {2 (a \sec (c+d x)+a)^{11/2}}{11 a^4 d}-\frac {2 (a \sec (c+d x)+a)^{9/2}}{3 a^3 d}+\frac {2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}+\frac {2 (a \sec (c+d x)+a)^{5/2}}{5 a d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac {2 a \sqrt {a \sec (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 88
Rule 207
Rule 3880
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{3/2} \tan ^5(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-a+a x)^2 (a+a x)^{7/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-3 a^2 (a+a x)^{7/2}+\frac {a^2 (a+a x)^{7/2}}{x}+a (a+a x)^{9/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac {2 (a+a \sec (c+d x))^{9/2}}{3 a^3 d}+\frac {2 (a+a \sec (c+d x))^{11/2}}{11 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{7/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac {2 (a+a \sec (c+d x))^{9/2}}{3 a^3 d}+\frac {2 (a+a \sec (c+d x))^{11/2}}{11 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac {2 (a+a \sec (c+d x))^{9/2}}{3 a^3 d}+\frac {2 (a+a \sec (c+d x))^{11/2}}{11 a^4 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac {2 (a+a \sec (c+d x))^{9/2}}{3 a^3 d}+\frac {2 (a+a \sec (c+d x))^{11/2}}{11 a^4 d}+\frac {a \operatorname {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 a \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac {2 (a+a \sec (c+d x))^{9/2}}{3 a^3 d}+\frac {2 (a+a \sec (c+d x))^{11/2}}{11 a^4 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {2 a \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac {2 (a+a \sec (c+d x))^{9/2}}{3 a^3 d}+\frac {2 (a+a \sec (c+d x))^{11/2}}{11 a^4 d}+\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 {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 a \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac {2 (a+a \sec (c+d x))^{9/2}}{3 a^3 d}+\frac {2 (a+a \sec (c+d x))^{11/2}}{11 a^4 d}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 112, normalized size = 0.66 \[ \frac {2 (a (\sec (c+d x)+1))^{3/2} \left (\sqrt {\sec (c+d x)+1} \left (105 \sec ^5(c+d x)+140 \sec ^4(c+d x)-325 \sec ^3(c+d x)-534 \sec ^2(c+d x)+327 \sec (c+d x)+1656\right )-1155 \tanh ^{-1}\left (\sqrt {\sec (c+d x)+1}\right )\right )}{1155 d (\sec (c+d x)+1)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.69, size = 334, normalized size = 1.98 \[ \left [\frac {1155 \, a^{\frac {3}{2}} \cos \left (d x + c\right )^{5} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (1656 \, a \cos \left (d x + c\right )^{5} + 327 \, a \cos \left (d x + c\right )^{4} - 534 \, a \cos \left (d x + c\right )^{3} - 325 \, a \cos \left (d x + c\right )^{2} + 140 \, a \cos \left (d x + c\right ) + 105 \, a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{2310 \, d \cos \left (d x + c\right )^{5}}, \frac {1155 \, \sqrt {-a} a \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{5} + 2 \, {\left (1656 \, a \cos \left (d x + c\right )^{5} + 327 \, a \cos \left (d x + c\right )^{4} - 534 \, a \cos \left (d x + c\right )^{3} - 325 \, a \cos \left (d x + c\right )^{2} + 140 \, a \cos \left (d x + c\right ) + 105 \, a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{1155 \, d \cos \left (d x + c\right )^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.76, size = 218, normalized size = 1.29 \[ \frac {\sqrt {2} {\left (\frac {1155 \, \sqrt {2} a \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} + \frac {2 \, {\left (1155 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{5} a - 770 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} a^{2} + 924 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} a^{3} - 1320 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} a^{4} - 6160 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a^{5} - 3360 \, a^{6}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{5} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}\right )} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{1155 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.21, size = 429, normalized size = 2.54 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (1155 \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 {11}{2}} \left (\cos ^{5}\left (d x +c \right )\right )+5775 \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 {11}{2}} \left (\cos ^{4}\left (d x +c \right )\right )+11550 \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 {11}{2}} \left (\cos ^{3}\left (d x +c \right )\right )+11550 \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 {11}{2}} \left (\cos ^{2}\left (d x +c \right )\right )+5775 \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 {11}{2}} \cos \left (d x +c \right )+1155 \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 {11}{2}}-105984 \left (\cos ^{5}\left (d x +c \right )\right )-20928 \left (\cos ^{4}\left (d x +c \right )\right )+34176 \left (\cos ^{3}\left (d x +c \right )\right )+20800 \left (\cos ^{2}\left (d x +c \right )\right )-8960 \cos \left (d x +c \right )-6720\right ) a}{36960 d \cos \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 162, normalized size = 0.96 \[ \frac {1155 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 770 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} + \frac {210 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {11}{2}}}{a^{4}} - \frac {770 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {9}{2}}}{a^{3}} + \frac {330 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {7}{2}}}{a^{2}} + \frac {462 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}}}{a} + 2310 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}} a}{1155 \, d} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \tan ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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