Optimal. Leaf size=39 \[ \frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2667, 44, 206} \[ \frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 44
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {1}{2 a (a-x)^2}+\frac {1}{2 a \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^2}{2 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 1.33 \[ \frac {a \sec ^2(c+d x)}{2 d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 67, normalized size = 1.72 \[ \frac {{\left (a \sin \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a \sin \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, a}{4 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 54, normalized size = 1.38 \[ \frac {a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac {a \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right ) - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 54, normalized size = 1.38 \[ \frac {a}{2 d \cos \left (d x +c \right )^{2}}+\frac {a \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 42, normalized size = 1.08 \[ \frac {a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, a}{\sin \left (d x + c\right ) - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 30, normalized size = 0.77 \[ \frac {a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{2\,d}-\frac {a}{2\,d\,\left (\sin \left (c+d\,x\right )-1\right )} \]
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 \[ a \left (\int \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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