Optimal. Leaf size=41 \[ \frac {\sec ^2(c+d x) (a \sin (c+d x)+b)}{2 d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2668, 639, 206} \[ \frac {\sec ^2(c+d x) (a \sin (c+d x)+b)}{2 d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 639
Rule 2668
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sin (c+d x)) \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {a+x}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^2(c+d x) (b+a \sin (c+d x))}{2 d}+\frac {(a b) \operatorname {Subst}\left (\int \frac {1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {\sec ^2(c+d x) (b+a \sin (c+d x))}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 1.27 \[ \frac {a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a \tan (c+d x) \sec (c+d x)}{2 d}+\frac {b \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 67, normalized size = 1.63 \[ \frac {a \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - a \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, a \sin \left (d x + c\right ) + 2 \, b}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 55, normalized size = 1.34 \[ \frac {a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (a \sin \left (d x + c\right ) + b\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 54, normalized size = 1.32 \[ \frac {a \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {b}{2 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 53, normalized size = 1.29 \[ \frac {a \log \left (\sin \left (d x + c\right ) + 1\right ) - a \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (a \sin \left (d x + c\right ) + b\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 44, normalized size = 1.07 \[ \frac {a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{2\,d}-\frac {\frac {b}{2}+\frac {a\,\sin \left (c+d\,x\right )}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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