Optimal. Leaf size=70 \[ \frac {3 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20 d}-\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{20 d}+\frac {x}{5} \]
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Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3783, 2659, 206} \[ \frac {3 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20 d}-\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{20 d}+\frac {x}{5} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2659
Rule 3783
Rubi steps
\begin {align*} \int \frac {1}{5+3 \sec (c+d x)} \, dx &=\frac {x}{5}-\frac {1}{5} \int \frac {1}{1+\frac {5}{3} \cos (c+d x)} \, dx\\ &=\frac {x}{5}-\frac {2 \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 )}{5 d}\\ &=\frac {x}{5}+\frac {3 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20 d}-\frac {3 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 69, normalized size = 0.99 \[ \frac {4 (c+d x)+3 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{20 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 52, normalized size = 0.74 \[ \frac {8 \, d x - 3 \, \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 3 \, \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right )}{40 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 43, normalized size = 0.61 \[ \frac {4 \, d x + 4 \, c - 3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + 3 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 51, normalized size = 0.73 \[ \frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{20 d}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{20 d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 70, normalized size = 1.00 \[ \frac {8 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) - 3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) + 3 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{20 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 21, normalized size = 0.30 \[ \frac {x}{5}-\frac {3\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{10\,d} \]
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 \[ \int \frac {1}{3 \sec {\left (c + d x \right )} + 5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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