Optimal. Leaf size=63 \[ \frac{a^2-b^2}{b^3 d (a+b \sin (c+d x))}+\frac{2 a \log (a+b \sin (c+d x))}{b^3 d}-\frac{\sin (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.0651441, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2668, 697} \[ \frac{a^2-b^2}{b^3 d (a+b \sin (c+d x))}+\frac{2 a \log (a+b \sin (c+d x))}{b^3 d}-\frac{\sin (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+\frac{-a^2+b^2}{(a+x)^2}+\frac{2 a}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{2 a \log (a+b \sin (c+d x))}{b^3 d}-\frac{\sin (c+d x)}{b^2 d}+\frac{a^2-b^2}{b^3 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0376845, size = 52, normalized size = 0.83 \[ \frac{\frac{(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 78, normalized size = 1.2 \begin{align*} -{\frac{\sin \left ( dx+c \right ) }{{b}^{2}d}}+2\,{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{3}d}}+{\frac{{a}^{2}}{{b}^{3}d \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{1}{bd \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943103, size = 82, normalized size = 1.3 \begin{align*} \frac{\frac{a^{2} - b^{2}}{b^{4} \sin \left (d x + c\right ) + a b^{3}} + \frac{2 \, a \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{3}} - \frac{\sin \left (d x + c\right )}{b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.805, size = 188, normalized size = 2.98 \begin{align*} \frac{b^{2} \cos \left (d x + c\right )^{2} - a b \sin \left (d x + c\right ) + a^{2} - 2 \, b^{2} + 2 \,{\left (a b \sin \left (d x + c\right ) + a^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4} d \sin \left (d x + c\right ) + a b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.13483, size = 316, normalized size = 5.02 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \cos ^{3}{\left (c \right )}}{\sin ^{2}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{- \frac{2 \sin{\left (c + d x \right )}}{d} - \frac{\cos ^{2}{\left (c + d x \right )}}{d \sin{\left (c + d x \right )}}}{b^{2}} & \text{for}\: a = 0 \\\frac{\frac{2 \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{\sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d}}{a^{2}} & \text{for}\: b = 0 \\\frac{x \cos ^{3}{\left (c \right )}}{\left (a + b \sin{\left (c \right )}\right )^{2}} & \text{for}\: d = 0 \\\frac{2 a^{3} \log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )}}{a^{2} b^{3} d + a b^{4} d \sin{\left (c + d x \right )}} + \frac{2 a^{3}}{a^{2} b^{3} d + a b^{4} d \sin{\left (c + d x \right )}} + \frac{2 a^{2} b \log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )} \sin{\left (c + d x \right )}}{a^{2} b^{3} d + a b^{4} d \sin{\left (c + d x \right )}} - \frac{a b^{2} \sin ^{2}{\left (c + d x \right )}}{a^{2} b^{3} d + a b^{4} d \sin{\left (c + d x \right )}} + \frac{b^{3} \sin ^{3}{\left (c + d x \right )}}{a^{2} b^{3} d + a b^{4} d \sin{\left (c + d x \right )}} + \frac{b^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{a^{2} b^{3} d + a b^{4} d \sin{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10733, size = 123, normalized size = 1.95 \begin{align*} -\frac{\frac{2 \, a \log \left (\frac{{\left | b \sin \left (d x + c\right ) + a \right |}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{b \sin \left (d x + c\right ) + a}{b^{3}} - \frac{a^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{3}} + \frac{1}{{\left (b \sin \left (d x + c\right ) + a\right )} b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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