Optimal. Leaf size=224 \[ \frac {31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac {7664 \tan (c+d x)}{315 a^5 d}+\frac {31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )} \]
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Rubi [A]
time = 0.34, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2845, 3057,
2827, 3853, 3855, 3852, 8} \begin {gather*} -\frac {7664 \tan (c+d x)}{315 a^5 d}+\frac {31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac {31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac {3832 \tan (c+d x) \sec (c+d x)}{315 d \left (a^5 \cos (c+d x)+a^5\right )}-\frac {577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac {28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac {17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac {\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2827
Rule 2845
Rule 3057
Rule 3852
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}+\frac {\int \frac {(11 a-6 a \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx}{9 a^2}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}+\frac {\int \frac {\left (111 a^2-85 a^2 \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{63 a^4}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}+\frac {\int \frac {\left (947 a^3-784 a^3 \cos (c+d x)\right ) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{315 a^6}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\left (6303 a^4-5193 a^4 \cos (c+d x)\right ) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx}{945 a^8}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac {\int \left (29295 a^5-22992 a^5 \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{945 a^{10}}\\ &=-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}-\frac {7664 \int \sec ^2(c+d x) \, dx}{315 a^5}+\frac {31 \int \sec ^3(c+d x) \, dx}{a^5}\\ &=\frac {31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}+\frac {31 \int \sec (c+d x) \, dx}{2 a^5}+\frac {7664 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{315 a^5 d}\\ &=\frac {31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac {7664 \tan (c+d x)}{315 a^5 d}+\frac {31 \sec (c+d x) \tan (c+d x)}{2 a^5 d}-\frac {\sec (c+d x) \tan (c+d x)}{9 d (a+a \cos (c+d x))^5}-\frac {17 \sec (c+d x) \tan (c+d x)}{63 a d (a+a \cos (c+d x))^4}-\frac {28 \sec (c+d x) \tan (c+d x)}{45 a^2 d (a+a \cos (c+d x))^3}-\frac {577 \sec (c+d x) \tan (c+d x)}{315 a^3 d (a+a \cos (c+d x))^2}-\frac {3832 \sec (c+d x) \tan (c+d x)}{315 d \left (a^5+a^5 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(507\) vs. \(2(224)=448\).
time = 6.36, size = 507, normalized size = 2.26 \begin {gather*} -\frac {496 \cos ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^5}+\frac {496 \cos ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))^5}+\frac {\cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (1472562 \sin \left (\frac {d x}{2}\right )-2822886 \sin \left (\frac {3 d x}{2}\right )+3057654 \sin \left (c-\frac {d x}{2}\right )-1885854 \sin \left (c+\frac {d x}{2}\right )+2644362 \sin \left (2 c+\frac {d x}{2}\right )+867048 \sin \left (c+\frac {3 d x}{2}\right )-1868436 \sin \left (2 c+\frac {3 d x}{2}\right )+1821498 \sin \left (3 c+\frac {3 d x}{2}\right )-2083537 \sin \left (c+\frac {5 d x}{2}\right )+339885 \sin \left (2 c+\frac {5 d x}{2}\right )-1456687 \sin \left (3 c+\frac {5 d x}{2}\right )+966735 \sin \left (4 c+\frac {5 d x}{2}\right )-1195641 \sin \left (2 c+\frac {7 d x}{2}\right )+46515 \sin \left (3 c+\frac {7 d x}{2}\right )-874341 \sin \left (4 c+\frac {7 d x}{2}\right )+367815 \sin \left (5 c+\frac {7 d x}{2}\right )-494579 \sin \left (3 c+\frac {9 d x}{2}\right )-31815 \sin \left (4 c+\frac {9 d x}{2}\right )-374879 \sin \left (5 c+\frac {9 d x}{2}\right )+87885 \sin \left (6 c+\frac {9 d x}{2}\right )-128187 \sin \left (4 c+\frac {11 d x}{2}\right )-18585 \sin \left (5 c+\frac {11 d x}{2}\right )-99837 \sin \left (6 c+\frac {11 d x}{2}\right )+9765 \sin \left (7 c+\frac {11 d x}{2}\right )-15328 \sin \left (5 c+\frac {13 d x}{2}\right )-3150 \sin \left (6 c+\frac {13 d x}{2}\right )-12178 \sin \left (7 c+\frac {13 d x}{2}\right )\right )}{40320 d (a+a \cos (c+d x))^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 161, normalized size = 0.72
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d \,a^{5}}\) | \(161\) |
default | \(\frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-\frac {10 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {48 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-50 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-351 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}-248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {88}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+248 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d \,a^{5}}\) | \(161\) |
norman | \(\frac {-\frac {495 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}+\frac {207 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {1303 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 a d}-\frac {141 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{70 a d}-\frac {2159 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5040 a d}-\frac {19 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{252 a d}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{144 a d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2} a^{4}}-\frac {31 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{5} d}+\frac {31 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{5} d}\) | \(193\) |
risch | \(-\frac {i \left (9765 \,{\mathrm e}^{12 i \left (d x +c \right )}+87885 \,{\mathrm e}^{11 i \left (d x +c \right )}+367815 \,{\mathrm e}^{10 i \left (d x +c \right )}+966735 \,{\mathrm e}^{9 i \left (d x +c \right )}+1821498 \,{\mathrm e}^{8 i \left (d x +c \right )}+2644362 \,{\mathrm e}^{7 i \left (d x +c \right )}+3057654 \,{\mathrm e}^{6 i \left (d x +c \right )}+2822886 \,{\mathrm e}^{5 i \left (d x +c \right )}+2083537 \,{\mathrm e}^{4 i \left (d x +c \right )}+1195641 \,{\mathrm e}^{3 i \left (d x +c \right )}+494579 \,{\mathrm e}^{2 i \left (d x +c \right )}+128187 \,{\mathrm e}^{i \left (d x +c \right )}+15328\right )}{315 d \,a^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9}}-\frac {31 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{5} d}+\frac {31 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{5} d}\) | \(213\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 251, normalized size = 1.12 \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 {78120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{5}} + \frac {78120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 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 = 0.40, size = 294, normalized size = 1.31 \begin {gather*} \frac {9765 \, {\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9765 \, {\left (\cos \left (d x + c\right )^{7} + 5 \, \cos \left (d x + c\right )^{6} + 10 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15328 \, \cos \left (d x + c\right )^{6} + 66875 \, \cos \left (d x + c\right )^{5} + 112119 \, \cos \left (d x + c\right )^{4} + 87440 \, \cos \left (d x + c\right )^{3} + 28828 \, \cos \left (d x + c\right )^{2} + 1575 \, \cos \left (d x + c\right ) - 315\right )} \sin \left (d x + c\right )}{1260 \, {\left (a^{5} d \cos \left (d x + c\right )^{7} + 5 \, a^{5} d \cos \left (d x + c\right )^{6} + 10 \, a^{5} d \cos \left (d x + c\right )^{5} + 10 \, a^{5} d \cos \left (d x + c\right )^{4} + 5 \, a^{5} d \cos \left (d x + c\right )^{3} + a^{5} d \cos \left (d x + c\right )^{2}\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 {\sec ^{3}{\left (c + d x \right )}}{\cos ^{5}{\left (c + d x \right )} + 5 \cos ^{4}{\left (c + d x \right )} + 10 \cos ^{3}{\left (c + d x \right )} + 10 \cos ^{2}{\left (c + d x \right )} + 5 \cos {\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 = 0.53, size = 171, normalized size = 0.76 \begin {gather*} \frac {\frac {78120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{5}} - \frac {78120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\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 = 0.48, size = 179, normalized size = 0.80 \begin {gather*} \frac {31\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^5\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5\,a^5\,d}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{56\,a^5\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{144\,a^5\,d}-\frac {25\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,a^5\,d}-\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^5\right )}-\frac {351\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a^5\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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