Optimal. Leaf size=120 \[ \frac {963 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)}+\frac {9 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}+\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}-\frac {8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}+\frac {x}{125} \]
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Rubi [A] time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3785, 4060, 3919, 3831, 2659, 206} \[ \frac {963 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)}+\frac {9 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}+\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}-\frac {8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}+\frac {x}{125} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2659
Rule 3785
Rule 3831
Rule 3919
Rule 4060
Rubi steps
\begin {align*} \int \frac {1}{(5+3 \sec (c+d x))^3} \, dx &=\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}-\frac {1}{160} \int \frac {-32+30 \sec (c+d x)-9 \sec ^2(c+d x)}{(5+3 \sec (c+d x))^2} \, dx\\ &=\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}+\frac {\int \frac {512-1365 \sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{12800}\\ &=\frac {x}{125}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}-\frac {8361 \int \frac {\sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{64000}\\ &=\frac {x}{125}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}-\frac {2787 \int \frac {1}{1+\frac {5}{3} \cos (c+d x)} \, dx}{64000}\\ &=\frac {x}{125}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}-\frac {2787 \operatorname {Subst}\left (\int \frac {1}{\frac {8}{3}-\frac {2 x^2}{3}} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32000 d}\\ &=\frac {x}{125}+\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}-\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.32, size = 241, normalized size = 2.01 \[ \frac {115560 \sin (c+d x)+110700 \sin (2 (c+d x))+359523 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \cos (c+d x) \left (2048 (c+d x)+8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+25 \cos (2 (c+d x)) \left (2048 (c+d x)+8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-8361 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-359523 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )+88064 c+88064 d x}{512000 d (5 \cos (c+d x)+3)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 155, normalized size = 1.29 \[ \frac {102400 \, d x \cos \left (d x + c\right )^{2} + 122880 \, d x \cos \left (d x + c\right ) + 36864 \, d x - 8361 \, {\left (25 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 8361 \, {\left (25 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 1080 \, {\left (205 \, \cos \left (d x + c\right ) + 107\right )} \sin \left (d x + c\right )}{512000 \, {\left (25 \, d \cos \left (d x + c\right )^{2} + 30 \, d \cos \left (d x + c\right ) + 9 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 85, normalized size = 0.71 \[ \frac {2048 \, d x + 2048 \, c - \frac {540 \, {\left (49 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 156 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{2}} - 8361 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + 8361 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{256000 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 123, normalized size = 1.02 \[ -\frac {27}{2560 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {1323}{25600 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{256000 d}+\frac {27}{2560 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {1323}{25600 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{256000 d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{125 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 155, normalized size = 1.29 \[ -\frac {\frac {540 \, {\left (\frac {156 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {49 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{\frac {8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 16} - 4096 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 8361 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 8361 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{256000 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 78, normalized size = 0.65 \[ \frac {x}{125}-\frac {8361\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{128000\,d}+\frac {\frac {1053\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3200}-\frac {1323\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12800}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+16\right )} \]
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 \[ \int \frac {1}{\left (3 \sec {\left (c + d x \right )} + 5\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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