Optimal. Leaf size=51 \[ -\frac {\tanh ^{-1}(\sin (c+d x))}{a d}+\frac {\tan (c+d x)}{a d}+\frac {\tan (c+d x)}{d (a+a \sec (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3875, 3874,
3855, 3879} \begin {gather*} \frac {\tan (c+d x)}{a d}-\frac {\tanh ^{-1}(\sin (c+d x))}{a d}+\frac {\tan (c+d x)}{d (a \sec (c+d x)+a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 3874
Rule 3875
Rule 3879
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx &=\frac {\tan (c+d x)}{a d}-\int \frac {\sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx\\ &=\frac {\tan (c+d x)}{a d}-\frac {\int \sec (c+d x) \, dx}{a}+\int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx\\ &=-\frac {\tanh ^{-1}(\sin (c+d x))}{a d}+\frac {\tan (c+d x)}{a d}+\frac {\tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(194\) vs. \(2(51)=102\).
time = 0.83, size = 194, normalized size = 3.80 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (\sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {\sin (d x)}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )\right )}{a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 74, normalized size = 1.45
method | result | size |
derivativedivides | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(74\) |
default | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}\) | \(74\) |
risch | \(\frac {2 i \left ({\mathrm e}^{2 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}+2\right )}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a d}\) | \(98\) |
norman | \(\frac {\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (51) = 102\).
time = 0.30, size = 119, normalized size = 2.33 \begin {gather*} -\frac {\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.72, size = 97, normalized size = 1.90 \begin {gather*} -\frac {{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right )\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 )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 84, normalized size = 1.65 \begin {gather*} -\frac {\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.68, size = 67, normalized size = 1.31 \begin {gather*} \frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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