Optimal. Leaf size=139 \[ \frac {2 \tan (c+d x)}{45 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac {2 \tan (c+d x)}{45 a^3 d (a \sec (c+d x)+a)^2}+\frac {\tan (c+d x)}{15 a^2 d (a \sec (c+d x)+a)^3}-\frac {2 \tan (c+d x)}{9 a d (a \sec (c+d x)+a)^4}+\frac {\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
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Rubi [A] time = 0.18, antiderivative size = 139, 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 = {3799, 4000, 3796, 3794} \[ \frac {2 \tan (c+d x)}{45 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac {2 \tan (c+d x)}{45 a^3 d (a \sec (c+d x)+a)^2}+\frac {\tan (c+d x)}{15 a^2 d (a \sec (c+d x)+a)^3}-\frac {2 \tan (c+d x)}{9 a d (a \sec (c+d x)+a)^4}+\frac {\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5} \]
Antiderivative was successfully verified.
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Rule 3794
Rule 3796
Rule 3799
Rule 4000
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+a \sec (c+d x))^5} \, dx &=\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}+\frac {\int \frac {\sec (c+d x) (-5 a+9 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac {\int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{3 a^2}\\ &=\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac {\tan (c+d x)}{15 a^2 d (a+a \sec (c+d x))^3}+\frac {2 \int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{15 a^3}\\ &=\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac {\tan (c+d x)}{15 a^2 d (a+a \sec (c+d x))^3}+\frac {2 \tan (c+d x)}{45 a^3 d (a+a \sec (c+d x))^2}+\frac {2 \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{45 a^4}\\ &=\frac {\tan (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {2 \tan (c+d x)}{9 a d (a+a \sec (c+d x))^4}+\frac {\tan (c+d x)}{15 a^2 d (a+a \sec (c+d x))^3}+\frac {2 \tan (c+d x)}{45 a^3 d (a+a \sec (c+d x))^2}+\frac {2 \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 = 0.25, size = 110, normalized size = 0.79 \[ \frac {\sec \left (\frac {c}{2}\right ) \left (-45 \sin \left (c+\frac {d x}{2}\right )+54 \sin \left (c+\frac {3 d x}{2}\right )-30 \sin \left (2 c+\frac {3 d x}{2}\right )+36 \sin \left (2 c+\frac {5 d x}{2}\right )+9 \sin \left (3 c+\frac {7 d x}{2}\right )+\sin \left (4 c+\frac {9 d x}{2}\right )+81 \sin \left (\frac {d x}{2}\right )\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right )}{5760 a^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 123, normalized size = 0.88 \[ \frac {{\left (2 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 21 \, \cos \left (d x + c\right )^{2} + 10 \, \cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{45 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 46, normalized size = 0.33 \[ \frac {5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 18 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{720 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 45, normalized size = 0.32 \[ \frac {\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 67, normalized size = 0.48 \[ \frac {\frac {45 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {18 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{720 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 45, normalized size = 0.32 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+45\right )}{720\,a^5\,d} \]
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 \[ \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{\sec ^{5}{\left (c + d x \right )} + 5 \sec ^{4}{\left (c + d x \right )} + 10 \sec ^{3}{\left (c + d x \right )} + 10 \sec ^{2}{\left (c + d x \right )} + 5 \sec {\left (c + d x \right )} + 1}\, dx}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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