Optimal. Leaf size=25 \[ \frac {b \sec (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4377, 12, 2606, 8, 3475} \[ \frac {b \sec (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2606
Rule 3475
Rule 4377
Rubi steps
\begin {align*} \int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \tan (c+d x) \, dx+\int b \sec (c+d x) \tan (c+d x) \, dx\\ &=-\frac {a \log (\cos (c+d x))}{d}+b \int \sec (c+d x) \tan (c+d x) \, dx\\ &=-\frac {a \log (\cos (c+d x))}{d}+\frac {b \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=-\frac {a \log (\cos (c+d x))}{d}+\frac {b \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {b \sec (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 34, normalized size = 1.36 \[ -\frac {a \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - b}{d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 107, normalized size = 4.28 \[ \frac {a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {a + 2 \, b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.00 \[ \frac {b \sec \left (d x +c \right )}{d}+\frac {a \ln \left (\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 32, normalized size = 1.28 \[ -\frac {a \log \left (-\sin \left (d x + c\right )^{2} + 1\right ) - \frac {2 \, b}{\cos \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 40, normalized size = 1.60 \[ \frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,b}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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