Optimal. Leaf size=193 \[ -\frac {2 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 a^2 \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 a (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {2 (a+a \sec (c+d x))^{5/2}}{5 d}+\frac {2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac {2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3965, 90, 52,
65, 213} \begin {gather*} -\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)^{13/2}}{13 a^4 d}-\frac {6 (a \sec (c+d x)+a)^{11/2}}{11 a^3 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 90
Rule 213
Rule 3965
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{5/2} \tan ^5(c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {(-a+a x)^2 (a+a x)^{9/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac {\text {Subst}\left (\int \left (-3 a^2 (a+a x)^{9/2}+\frac {a^2 (a+a x)^{9/2}}{x}+a (a+a x)^{11/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\text {Subst}\left (\int \frac {(a+a x)^{9/2}}{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 {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\text {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 {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\text {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 {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {a \text {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 {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {a^2 \text {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 {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {a^3 \text {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 {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}+\frac {\left (2 a^2\right ) \text {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}-\frac {6 (a+a \sec (c+d x))^{11/2}}{11 a^3 d}+\frac {2 (a+a \sec (c+d x))^{13/2}}{13 a^4 d}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 156, normalized size = 0.81 \begin {gather*} \frac {(a (1+\sec (c+d x)))^{5/2} \left (-2 \tanh ^{-1}\left (\sqrt {1+\sec (c+d x)}\right )+2 \sqrt {1+\sec (c+d x)}+\frac {2}{3} (1+\sec (c+d x))^{3/2}+\frac {2}{5} (1+\sec (c+d x))^{5/2}+\frac {2}{7} (1+\sec (c+d x))^{7/2}+\frac {2}{9} (1+\sec (c+d x))^{9/2}-\frac {6}{11} (1+\sec (c+d x))^{11/2}+\frac {2}{13} (1+\sec (c+d x))^{13/2}\right )}{d (1+\sec (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(499\) vs.
\(2(161)=322\).
time = 0.19, size = 500, normalized size = 2.59
method | result | size |
default | \(\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (45045 \left (\cos ^{6}\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 {13}{2}} \sqrt {2}+270270 \left (\cos ^{5}\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 {13}{2}} \sqrt {2}+675675 \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 {13}{2}} \sqrt {2}+900900 \left (\cos ^{3}\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 {13}{2}} \sqrt {2}+675675 \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 {13}{2}} \sqrt {2}+270270 \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 {13}{2}} \sqrt {2}+45045 \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 {13}{2}}+9176192 \left (\cos ^{6}\left (d x +c \right )\right )+4060544 \left (\cos ^{5}\left (d x +c \right )\right )-1603968 \left (\cos ^{4}\left (d x +c \right )\right )-3468160 \left (\cos ^{3}\left (d x +c \right )\right )-568960 \left (\cos ^{2}\left (d x +c \right )\right )+1088640 \cos \left (d x +c \right )+443520\right ) a^{2}}{2882880 d \cos \left (d x +c \right )^{6}}\) | \(500\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 181, normalized size = 0.94 \begin {gather*} \frac {45045 \, 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 ) + 18018 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {5}{2}} + \frac {6930 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {13}{2}}}{a^{4}} - \frac {24570 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {11}{2}}}{a^{3}} + \frac {10010 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {9}{2}}}{a^{2}} + \frac {12870 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {7}{2}}}{a} + 30030 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} a + 90090 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}} a^{2}}{45045 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.93, size = 386, normalized size = 2.00 \begin {gather*} \left [\frac {45045 \, a^{\frac {5}{2}} \cos \left (d x + c\right )^{6} \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 (71689 \, a^{2} \cos \left (d x + c\right )^{6} + 31723 \, a^{2} \cos \left (d x + c\right )^{5} - 12531 \, a^{2} \cos \left (d x + c\right )^{4} - 27095 \, a^{2} \cos \left (d x + c\right )^{3} - 4445 \, a^{2} \cos \left (d x + c\right )^{2} + 8505 \, a^{2} \cos \left (d x + c\right ) + 3465 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{90090 \, d \cos \left (d x + c\right )^{6}}, \frac {45045 \, \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 )^{6} + 2 \, {\left (71689 \, a^{2} \cos \left (d x + c\right )^{6} + 31723 \, a^{2} \cos \left (d x + c\right )^{5} - 12531 \, a^{2} \cos \left (d x + c\right )^{4} - 27095 \, a^{2} \cos \left (d x + c\right )^{3} - 4445 \, a^{2} \cos \left (d x + c\right )^{2} + 8505 \, a^{2} \cos \left (d x + c\right ) + 3465 \, a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{45045 \, d \cos \left (d x + c\right )^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.68, size = 244, normalized size = 1.26 \begin {gather*} \frac {\sqrt {2} {\left (\frac {45045 \, \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 (45045 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{6} a - 30030 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{5} a^{2} + 36036 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} a^{3} - 51480 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} a^{4} + 80080 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{2} a^{5} + 393120 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )} a^{6} + 221760 \, a^{7}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{6} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}\right )} a^{2} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{45045 \, d} \end {gather*}
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 \begin {gather*} \int {\mathrm {tan}\left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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