Optimal. Leaf size=159 \[ \frac {4 \tan (c+d x)}{45 d \left (a^5 \sec (c+d x)+a^5\right )}-\frac {32 \tan (c+d x)}{315 a d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac {\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac {4 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4} \]
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Rubi [A] time = 0.22, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3810, 3799, 4000, 3794} \[ \frac {4 \tan (c+d x)}{45 d \left (a^5 \sec (c+d x)+a^5\right )}-\frac {32 \tan (c+d x)}{315 a d \left (a^2 \sec (c+d x)+a^2\right )^2}+\frac {4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac {\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac {4 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 3794
Rule 3799
Rule 3810
Rule 4000
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+a \sec (c+d x))^5} \, dx &=\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {4 \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx}{9 a}\\ &=\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {4 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {4 \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx}{21 a^2}\\ &=\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {4 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {4 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}+\frac {4 \int \frac {\sec (c+d x) (-3 a+5 a \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{105 a^4}\\ &=\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {4 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {4 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {32 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac {4 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{45 a^4}\\ &=\frac {\sec ^4(c+d x) \tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {4 \sec ^3(c+d x) \tan (c+d x)}{63 a d (a+a \sec (c+d x))^4}+\frac {4 \tan (c+d x)}{105 a^2 d (a+a \sec (c+d x))^3}-\frac {32 \tan (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}+\frac {4 \tan (c+d x)}{45 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 97, normalized size = 0.61 \[ \frac {\left (126 \sin \left (\frac {1}{2} (c+d x)\right )+84 \sin \left (\frac {3}{2} (c+d x)\right )+36 \sin \left (\frac {5}{2} (c+d x)\right )+9 \sin \left (\frac {7}{2} (c+d x)\right )+\sin \left (\frac {9}{2} (c+d x)\right )\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \sec ^5(c+d x)}{315 a^5 d (\sec (c+d x)+1)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.70, size = 123, normalized size = 0.77 \[ \frac {{\left (8 \, \cos \left (d x + c\right )^{4} + 40 \, \cos \left (d x + c\right )^{3} + 84 \, \cos \left (d x + c\right )^{2} + 100 \, \cos \left (d x + c\right ) + 83\right )} \sin \left (d x + c\right )}{315 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.46, size = 72, normalized size = 0.45 \[ \frac {35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 180 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 378 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 420 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{5040 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 71, normalized size = 0.45 \[ \frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {4 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 107, normalized size = 0.67 \[ \frac {\frac {315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {420 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {180 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{5040 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.76, size = 127, normalized size = 0.80 \[ \frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+420\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+180\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \]
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 \[ \frac {\int \frac {\sec ^{5}{\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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