Optimal. Leaf size=28 \[ \frac {a \tan (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.06, 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 = {3090, 3767, 8, 2606, 30} \[ \frac {a \tan (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2606
Rule 3090
Rule 3767
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\int \left (a \sec ^2(c+d x)+b \sec ^2(c+d x) \tan (c+d x)\right ) \, dx\\ &=a \int \sec ^2(c+d x) \, dx+b \int \sec ^2(c+d x) \tan (c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac {b \operatorname {Subst}(\int x \, dx,x,\sec (c+d x))}{d}\\ &=\frac {b \sec ^2(c+d x)}{2 d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {a \tan (c+d x)}{d}+\frac {b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 30, normalized size = 1.07 \[ \frac {2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b}{2 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 25, normalized size = 0.89 \[ \frac {b \tan \left (d x + c\right )^{2} + 2 \, a \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.70, size = 25, normalized size = 0.89 \[ \frac {a \tan \left (d x +c \right )+\frac {b}{2 \cos \left (d x +c \right )^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 30, normalized size = 1.07 \[ \frac {2 \, a \tan \left (d x + c\right ) - \frac {b}{\sin \left (d x + c\right )^{2} - 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 23, normalized size = 0.82 \[ \frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{2\,d} \]
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 \[ \int \left (a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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