Optimal. Leaf size=26 \[ \frac {(a \sin (c+d x)+b \sec (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.04, antiderivative size = 26, 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} \[ \frac {(a \sin (c+d x)+b \sec (c+d x))^3}{3 d} \]
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))^2 (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))^3}{3 d}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 31, normalized size = 1.19 \[ \frac {\sec ^3(c+d x) (a \sin (2 (c+d x))+2 b)^3}{24 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 92, normalized size = 3.54 \[ -\frac {3 \, a^{2} b \cos \left (d x + c\right )^{4} - 3 \, a^{2} b \cos \left (d x + c\right )^{2} - b^{3} + {\left (a^{3} \cos \left (d x + c\right )^{5} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a b^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.56, size = 118, normalized size = 4.54 \[ \frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{2} b \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {a^{2} b \left (\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{d}+\frac {a \,b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {a \,b^{2} \sin \left (d x +c \right )}{d}+\frac {b^{3}}{3 d \cos \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 24, normalized size = 0.92 \[ \frac {{\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{3}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.22, size = 100, normalized size = 3.85 \[ \frac {a^3\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a^2\,b\,{\cos \left (c+d\,x\right )}^2+\sin \left (c+d\,x\right )\,a\,b^2\,\cos \left (c+d\,x\right )+\frac {b^3}{3}}{d\,{\cos \left (c+d\,x\right )}^3}-\frac {a^3\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}-\frac {a^2\,b\,\cos \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.16, size = 100, normalized size = 3.85 \[ \begin {cases} \frac {a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} b \sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{d} + \frac {a b^{2} \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{d} + \frac {b^{3} \sec ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + b \sec {\relax (c )}\right )^{2} \left (a \cos {\relax (c )} + b \tan {\relax (c )} \sec {\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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