Optimal. Leaf size=28 \[ \frac {a \sec (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4377, 12, 2606, 30, 8} \[ \frac {a \sec (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 2606
Rule 4377
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \sec (c+d x) \tan (c+d x) \, dx+\int b \sec ^2(c+d x) \tan (c+d x) \, dx\\ &=b \int \sec ^2(c+d x) \tan (c+d x) \, dx+\frac {a \operatorname {Subst}(\int 1 \, dx,x,\sec (c+d x))}{d}\\ &=\frac {a \sec (c+d x)}{d}+\frac {b \operatorname {Subst}(\int x \, dx,x,\sec (c+d x))}{d}\\ &=\frac {a \sec (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 1.00 \[ \frac {a \sec (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 24, normalized size = 0.86 \[ \frac {2 \, a \cos \left (d x + c\right ) + b}{2 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 71, normalized size = 2.54 \[ \frac {2 \, {\left (a + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 0.89 \[ \frac {\frac {b \left (\sec ^{2}\left (d x +c \right )\right )}{2}+a \sec \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 27, normalized size = 0.96 \[ \frac {b \tan \left (d x + c\right )^{2} + \frac {2 \, a}{\cos \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 24, normalized size = 0.86 \[ \frac {\frac {b}{2}+a\,\cos \left (c+d\,x\right )}{d\,{\cos \left (c+d\,x\right )}^2} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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