Optimal. Leaf size=30 \[ \frac {(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)} \]
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Rubi [A] time = 0.06, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {4385} \[ \frac {(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)} \]
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))^n (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))^{1+n}}{d (1+n)}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 51, normalized size = 1.70 \[ \frac {\sec (c+d x) (a \sin (2 (c+d x))+2 b) (a \sin (c+d x)+b \sec (c+d x))^n}{2 d (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.05, size = 59, normalized size = 1.97 \[ \frac {{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b\right )} \left (\frac {a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b}{\cos \left (d x + c\right )}\right )^{n}}{{\left (d n + d\right )} \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) \tan \left (d x + c\right ) + a \cos \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 31, normalized size = 1.03 \[ \frac {\left (b \sec \left (d x +c \right )+a \sin \left (d x +c \right )\right )^{n +1}}{d \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 30, normalized size = 1.00 \[ \frac {{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{n + 1}}{d {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.54, size = 63, normalized size = 2.10 \[ \left \{\begin {array}{cl} \frac {\ln \left (a\,\sin \left (c+d\,x\right )+\frac {b}{\cos \left (c+d\,x\right )}\right )}{d} & \text {\ if\ \ }n=-1\\ \frac {{\left (a\,\sin \left (c+d\,x\right )+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{n+1}}{d\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 74.07, size = 138, normalized size = 4.60 \[ \begin {cases} \frac {x \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right )}{a \sin {\relax (c )} + b \sec {\relax (c )}} & \text {for}\: d = 0 \wedge n = -1 \\x \left (a \sin {\relax (c )} + b \sec {\relax (c )}\right )^{n} \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right ) & \text {for}\: d = 0 \\\frac {\log {\left (\sin {\left (c + d x \right )} + \frac {b \sec {\left (c + d x \right )}}{a} \right )}}{d} & \text {for}\: n = -1 \\\frac {a \left (a \sin {\left (c + d x \right )} + b \sec {\left (c + d x \right )}\right )^{n} \sin {\left (c + d x \right )}}{d n + d} + \frac {b \left (a \sin {\left (c + d x \right )} + b \sec {\left (c + d x \right )}\right )^{n} \sec {\left (c + d x \right )}}{d n + d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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