Optimal. Leaf size=120 \[ \frac {x}{125}+\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}-\frac {8361 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{256000 d}+\frac {9 \tan (c+d x)}{160 d (5+3 \sec (c+d x))^2}+\frac {963 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, 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 = {3870, 4145,
4004, 3916, 2738, 212} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2738
Rule 3870
Rule 3916
Rule 4004
Rule 4145
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 \text {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*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(241\) vs. \(2(120)=240\).
time = 0.33, size = 241, normalized size = 2.01 \begin {gather*} \frac {88064 c+88064 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 (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \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 (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-359523 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+115560 \sin (c+d x)+110700 \sin (2 (c+d x))}{512000 d (3+5 \cos (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 106, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {-\frac {27}{2560 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {1323}{25600 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{256000}+\frac {27}{2560 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {1323}{25600 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{256000}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{125}}{d}\) | \(106\) |
default | \(\frac {-\frac {27}{2560 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {1323}{25600 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{256000}+\frac {27}{2560 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {1323}{25600 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{256000}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{125}}{d}\) | \(106\) |
risch | \(\frac {x}{125}+\frac {27 i \left (695 \,{\mathrm e}^{3 i \left (d x +c \right )}+1763 \,{\mathrm e}^{2 i \left (d x +c \right )}+1765 \,{\mathrm e}^{i \left (d x +c \right )}+1025\right )}{32000 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{2}}+\frac {8361 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{256000 d}-\frac {8361 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{256000 d}\) | \(110\) |
norman | \(\frac {\frac {16 x}{125}+\frac {1053 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3200 d}-\frac {1323 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12800 d}-\frac {8 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{125}+\frac {x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{125}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-4\right )^{2}}+\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{256000 d}-\frac {8361 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{256000 d}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.48, size = 155, normalized size = 1.29 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.82, size = 155, normalized size = 1.29 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 \sec {\left (c + d x \right )} + 5\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.45, size = 85, normalized size = 0.71 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.95, size = 78, normalized size = 0.65 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________