Optimal. Leaf size=57 \[ \frac{b^2}{a^3 d (a \cos (c+d x)+b)}+\frac{2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac{\cos (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.112259, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2833, 12, 43} \[ \frac{b^2}{a^3 d (a \cos (c+d x)+b)}+\frac{2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac{\cos (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \frac{\sin (c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{b^2}{(b-x)^2}-\frac{2 b}{b-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac{\cos (c+d x)}{a^2 d}+\frac{b^2}{a^3 d (b+a \cos (c+d x))}+\frac{2 b \log (b+a \cos (c+d x))}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.132602, size = 76, normalized size = 1.33 \[ \frac{-a^2 \cos ^2(c+d x)+b^2 (2 \log (a \cos (c+d x)+b)+1)+a b \cos (c+d x) (2 \log (a \cos (c+d x)+b)-1)}{a^3 d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 75, normalized size = 1.3 \begin{align*} -{\frac{b}{d{a}^{2} \left ( a+b\sec \left ( dx+c \right ) \right ) }}+2\,{\frac{b\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{d{a}^{2}\sec \left ( dx+c \right ) }}-2\,{\frac{b\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00148, size = 74, normalized size = 1.3 \begin{align*} \frac{\frac{b^{2}}{a^{4} \cos \left (d x + c\right ) + a^{3} b} - \frac{\cos \left (d x + c\right )}{a^{2}} + \frac{2 \, b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{3}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75957, size = 178, normalized size = 3.12 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right )^{2} + a b \cos \left (d x + c\right ) - b^{2} - 2 \,{\left (a b \cos \left (d x + c\right ) + b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4} d \cos \left (d x + c\right ) + a^{3} b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2496, size = 82, normalized size = 1.44 \begin{align*} -\frac{\cos \left (d x + c\right )}{a^{2} d} + \frac{2 \, b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{3} d} + \frac{b^{2}}{{\left (a \cos \left (d x + c\right ) + b\right )} a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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