Optimal. Leaf size=26 \[ \frac {(a \sin (c+d x)+b \sec (c+d x))^2}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {4385} \[ \frac {(a \sin (c+d x)+b \sec (c+d x))^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 4385
Rubi steps
\begin {align*} \int (b \sec (c+d x)+a \sin (c+d x)) (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx &=\frac {(b \sec (c+d x)+a \sin (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 67, normalized size = 2.58 \[ -\frac {a^2 \cos ^2(c+d x)}{2 d}-\frac {a b \tan ^{-1}(\tan (c+d x))}{d}+\frac {a b \tan (c+d x)}{d}+a b x+\frac {b^2 \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 61, normalized size = 2.35 \[ -\frac {2 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{2} - 4 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, b^{2}}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.67, size = 45, normalized size = 1.73 \[ \frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) - \frac {a^{2}}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 57, normalized size = 2.19 \[ \frac {-\frac {\left (\cos ^{2}\left (d x +c \right )\right ) a^{2}}{2}+a b \left (\tan \left (d x +c \right )-d x -c \right )+\left (d x +c \right ) a b +\frac {b^{2}}{2 \cos \left (d x +c \right )^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 24, normalized size = 0.92 \[ \frac {{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{2}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.18, size = 61, normalized size = 2.35 \[ -\frac {\frac {a^2\,\left (2\,{\sin \left (2\,c+2\,d\,x\right )}^2-1\right )}{16}+\frac {a^2}{16}+\frac {b^2}{2}+\frac {a\,b\,\sin \left (2\,c+2\,d\,x\right )}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.46, size = 73, normalized size = 2.81 \[ \begin {cases} \frac {a^{2} \sin ^{2}{\left (c + d x \right )}}{2 d} + \frac {a b \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}}{d} + \frac {b^{2} \sec ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + b \sec {\relax (c )}\right ) \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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