Optimal. Leaf size=95 \[ \frac {9 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}+\frac {123 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{1600 d}-\frac {123 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{1600 d}+\frac {x}{25} \]
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3785, 3919, 3831, 2659, 206} \[ \frac {9 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}+\frac {123 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{1600 d}-\frac {123 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{1600 d}+\frac {x}{25} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2659
Rule 3785
Rule 3831
Rule 3919
Rubi steps
\begin {align*} \int \frac {1}{(5+3 \sec (c+d x))^2} \, dx &=\frac {9 \tan (c+d x)}{80 d (5+3 \sec (c+d x))}-\frac {1}{80} \int \frac {-16+15 \sec (c+d x)}{5+3 \sec (c+d x)} \, dx\\ &=\frac {x}{25}+\frac {9 \tan (c+d x)}{80 d (5+3 \sec (c+d x))}-\frac {123}{400} \int \frac {\sec (c+d x)}{5+3 \sec (c+d x)} \, dx\\ &=\frac {x}{25}+\frac {9 \tan (c+d x)}{80 d (5+3 \sec (c+d x))}-\frac {41}{400} \int \frac {1}{1+\frac {5}{3} \cos (c+d x)} \, dx\\ &=\frac {x}{25}+\frac {9 \tan (c+d x)}{80 d (5+3 \sec (c+d x))}-\frac {41 \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 )}{200 d}\\ &=\frac {x}{25}+\frac {123 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{1600 d}-\frac {123 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{1600 d}+\frac {9 \tan (c+d x)}{80 d (5+3 \sec (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 162, normalized size = 1.71 \[ \frac {5 \cos (c+d x) \left (64 (c+d x)+123 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-123 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+3 \left (60 \sin (c+d x)+123 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-123 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )+64 c+64 d x\right )}{1600 d (5 \cos (c+d x)+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 102, normalized size = 1.07 \[ \frac {640 \, d x \cos \left (d x + c\right ) + 384 \, d x - 123 \, {\left (5 \, \cos \left (d x + c\right ) + 3\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 123 \, {\left (5 \, \cos \left (d x + c\right ) + 3\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 360 \, \sin \left (d x + c\right )}{3200 \, {\left (5 \, d \cos \left (d x + c\right ) + 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 69, normalized size = 0.73 \[ \frac {64 \, d x + 64 \, c - \frac {180 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4} - 123 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + 123 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{1600 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 87, normalized size = 0.92 \[ -\frac {9}{160 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {123 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{1600 d}-\frac {9}{160 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {123 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{1600 d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{25 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 111, normalized size = 1.17 \[ -\frac {\frac {180 \, \sin \left (d x + c\right )}{{\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 4\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - 128 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 123 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 123 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{1600 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 52, normalized size = 0.55 \[ \frac {x}{25}-\frac {\frac {123\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{800}+\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{80\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\right )}}{d} \]
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 \[ \int \frac {1}{\left (3 \sec {\left (c + d x \right )} + 5\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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