Optimal. Leaf size=47 \[ \frac {b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec (c+d x) \tan (c+d x)}{2 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3872, 3852, 8,
3853, 3855} \begin {gather*} \frac {a \tan (c+d x)}{d}+\frac {b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b \tan (c+d x) \sec (c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3872
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sec (c+d x)) \, dx &=a \int \sec ^2(c+d x) \, dx+b \int \sec ^3(c+d x) \, dx\\ &=\frac {b \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} b \int \sec (c+d x) \, dx-\frac {a \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 47, normalized size = 1.00 \begin {gather*} \frac {b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a \tan (c+d x)}{d}+\frac {b \sec (c+d x) \tan (c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 47, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {a \tan \left (d x +c \right )+b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(47\) |
default | \(\frac {a \tan \left (d x +c \right )+b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(47\) |
norman | \(\frac {\frac {\left (2 a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {\left (2 a -b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(96\) |
risch | \(-\frac {i \left (b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 a \,{\mathrm e}^{2 i \left (d x +c \right )}-b \,{\mathrm e}^{i \left (d x +c \right )}-2 a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 58, normalized size = 1.23 \begin {gather*} -\frac {b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 4 \, a \tan \left (d x + c\right )}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.58, size = 74, normalized size = 1.57 \begin {gather*} \frac {b \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - b \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \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*} \int \left (a + b \sec {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs.
\(2 (43) = 86\).
time = 0.44, size = 107, normalized size = 2.28 \begin {gather*} \frac {b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \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} - 1\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.48, size = 85, normalized size = 1.81 \begin {gather*} \frac {b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a-b\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a+b\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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