Optimal. Leaf size=24 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3090, 3770, 2606, 8} \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2606
Rule 3090
Rule 3770
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\int (a \sec (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx\\ &=a \int \sec (c+d x) \, dx+b \int \sec (c+d x) \tan (c+d x) \, dx\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.00 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{d}+\frac {b \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 54, normalized size = 2.25 \[ \frac {a \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, b}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 54, normalized size = 2.25 \[ \frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.77, size = 34, normalized size = 1.42 \[ \frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b}{d \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 40, normalized size = 1.67 \[ \frac {a {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, b}{\cos \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 38, normalized size = 1.58 \[ \frac {2\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,b}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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