Optimal. Leaf size=33 \[ \frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4377, 12, 2606, 30} \[ \frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2606
Rule 4377
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx &=a \int \sec ^2(c+d x) \tan (c+d x) \, dx+\int b \sec ^3(c+d x) \tan (c+d x) \, dx\\ &=b \int \sec ^3(c+d x) \tan (c+d x) \, dx+\frac {a \operatorname {Subst}(\int x \, dx,x,\sec (c+d x))}{d}\\ &=\frac {a \sec ^2(c+d x)}{2 d}+\frac {b \operatorname {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 1.00 \[ \frac {a \sec ^2(c+d x)}{2 d}+\frac {b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 26, normalized size = 0.79 \[ \frac {3 \, a \cos \left (d x + c\right ) + 2 \, b}{6 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 97, normalized size = 2.94 \[ \frac {2 \, {\left (b - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{3 \, d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.85 \[ \frac {\frac {\left (\sec ^{3}\left (d x +c \right )\right ) b}{3}+\frac {\left (\sec ^{2}\left (d x +c \right )\right ) a}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 32, normalized size = 0.97 \[ -\frac {\frac {3 \, a}{\sin \left (d x + c\right )^{2} - 1} - \frac {2 \, b}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 29, normalized size = 0.88 \[ \frac {a}{2\,d\,{\cos \left (c+d\,x\right )}^2}+\frac {b}{3\,d\,{\cos \left (c+d\,x\right )}^3} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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