Optimal. Leaf size=215 \[ \frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )} \]
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Rubi [A]
time = 0.35, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3902, 4105,
3872, 2715, 8, 2717} \begin {gather*} -\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}-\frac {3832 \sin (c+d x) \cos (c+d x)}{315 d \left (a^5 \sec (c+d x)+a^5\right )}+\frac {31 x}{2 a^5}-\frac {577 \sin (c+d x) \cos (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac {28 \sin (c+d x) \cos (c+d x)}{45 a^2 d (a \sec (c+d x)+a)^3}-\frac {17 \sin (c+d x) \cos (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac {\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 3902
Rule 4105
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {\int \frac {\cos ^2(c+d x) (-11 a+6 a \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {\int \frac {\cos ^2(c+d x) \left (-111 a^2+85 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {\int \frac {\cos ^2(c+d x) \left (-947 a^3+784 a^3 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {\int \frac {\cos ^2(c+d x) \left (-6303 a^4+5193 a^4 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {\int \cos ^2(c+d x) \left (-29295 a^5+22992 a^5 \sec (c+d x)\right ) \, dx}{945 a^{10}}\\ &=-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}-\frac {7664 \int \cos (c+d x) \, dx}{315 a^5}+\frac {31 \int \cos ^2(c+d x) \, dx}{a^5}\\ &=-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}+\frac {31 \int 1 \, dx}{2 a^5}\\ &=\frac {31 x}{2 a^5}-\frac {7664 \sin (c+d x)}{315 a^5 d}+\frac {31 \cos (c+d x) \sin (c+d x)}{2 a^5 d}-\frac {\cos (c+d x) \sin (c+d x)}{9 d (a+a \sec (c+d x))^5}-\frac {17 \cos (c+d x) \sin (c+d x)}{63 a d (a+a \sec (c+d x))^4}-\frac {28 \cos (c+d x) \sin (c+d x)}{45 a^2 d (a+a \sec (c+d x))^3}-\frac {577 \cos (c+d x) \sin (c+d x)}{315 a^3 d (a+a \sec (c+d x))^2}-\frac {3832 \cos (c+d x) \sin (c+d x)}{315 d \left (a^5+a^5 \sec (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.76, size = 345, normalized size = 1.60 \begin {gather*} \frac {\sec \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \left (4921560 d x \cos \left (\frac {d x}{2}\right )+4921560 d x \cos \left (c+\frac {d x}{2}\right )+3281040 d x \cos \left (c+\frac {3 d x}{2}\right )+3281040 d x \cos \left (2 c+\frac {3 d x}{2}\right )+1406160 d x \cos \left (2 c+\frac {5 d x}{2}\right )+1406160 d x \cos \left (3 c+\frac {5 d x}{2}\right )+351540 d x \cos \left (3 c+\frac {7 d x}{2}\right )+351540 d x \cos \left (4 c+\frac {7 d x}{2}\right )+39060 d x \cos \left (4 c+\frac {9 d x}{2}\right )+39060 d x \cos \left (5 c+\frac {9 d x}{2}\right )-9163224 \sin \left (\frac {d x}{2}\right )+7194600 \sin \left (c+\frac {d x}{2}\right )-7472241 \sin \left (c+\frac {3 d x}{2}\right )+3432975 \sin \left (2 c+\frac {3 d x}{2}\right )-3871989 \sin \left (2 c+\frac {5 d x}{2}\right )+801675 \sin \left (3 c+\frac {5 d x}{2}\right )-1186056 \sin \left (3 c+\frac {7 d x}{2}\right )-17640 \sin \left (4 c+\frac {7 d x}{2}\right )-175184 \sin \left (4 c+\frac {9 d x}{2}\right )-45360 \sin \left (5 c+\frac {9 d x}{2}\right )-3465 \sin \left (5 c+\frac {11 d x}{2}\right )-3465 \sin \left (6 c+\frac {11 d x}{2}\right )+315 \sin \left (6 c+\frac {13 d x}{2}\right )+315 \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{1290240 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 127, normalized size = 0.59
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(127\) |
default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {-176 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+496 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{5}}\) | \(127\) |
norman | \(\frac {\frac {31 x}{2 a}-\frac {495 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}-\frac {207 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {1303 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 a d}+\frac {141 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 a d}-\frac {2159 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5040 a d}+\frac {19 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{252 a d}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 a d}+\frac {31 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {31 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a^{4}}\) | \(192\) |
risch | \(\frac {31 x}{2 a^{5}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{5} d}+\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{5} d}-\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{5} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{5} d}-\frac {2 i \left (11025 \,{\mathrm e}^{8 i \left (d x +c \right )}+77175 \,{\mathrm e}^{7 i \left (d x +c \right )}+247695 \,{\mathrm e}^{6 i \left (d x +c \right )}+465255 \,{\mathrm e}^{5 i \left (d x +c \right )}+557109 \,{\mathrm e}^{4 i \left (d x +c \right )}+433881 \,{\mathrm e}^{3 i \left (d x +c \right )}+214929 \,{\mathrm e}^{2 i \left (d x +c \right )}+62001 \,{\mathrm e}^{i \left (d x +c \right )}+8114\right )}{315 d \,a^{5} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}\) | \(192\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 224, normalized size = 1.04 \begin {gather*} -\frac {\frac {5040 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{5} + \frac {2 \, a^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {110565 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15750 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3024 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {450 \, \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}}}{a^{5}} - \frac {156240 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{5}}}{5040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.49, size = 207, normalized size = 0.96 \begin {gather*} \frac {9765 \, d x \cos \left (d x + c\right )^{5} + 48825 \, d x \cos \left (d x + c\right )^{4} + 97650 \, d x \cos \left (d x + c\right )^{3} + 97650 \, d x \cos \left (d x + c\right )^{2} + 48825 \, d x \cos \left (d x + c\right ) + 9765 \, d x + {\left (315 \, \cos \left (d x + c\right )^{6} - 1575 \, \cos \left (d x + c\right )^{5} - 28828 \, \cos \left (d x + c\right )^{4} - 87440 \, \cos \left (d x + c\right )^{3} - 112119 \, \cos \left (d x + c\right )^{2} - 66875 \, \cos \left (d x + c\right ) - 15328\right )} \sin \left (d x + c\right )}{630 \, {\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 )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\cos ^{2}{\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 145, normalized size = 0.67 \begin {gather*} \frac {\frac {78120 \, {\left (d x + c\right )}}{a^{5}} - \frac {5040 \, {\left (11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{5}} - \frac {35 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 450 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3024 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15750 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 110565 \, a^{40} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{45}}}{5040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.95, size = 181, normalized size = 0.84 \begin {gather*} -\frac {35\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-590\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+4584\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-23288\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+129824\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+55440\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-10080\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-78120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )}{5040\,a^5\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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