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 = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 697
Rule 2668
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.46, size = 78, normalized size = 1.24 \[ \frac {b^{2} \cos \left (d x + c\right )^{2} - a b \sin \left (d x + c\right ) + a^{2} - 2 \, b^{2} + 2 \, {\left (a b \sin \left (d x + c\right ) + a^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{4} d \sin \left (d x + c\right ) + a b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.69, size = 91, normalized size = 1.44 \[ -\frac {\frac {2 \, a \log \left (\frac {{\left | b \sin \left (d x + c\right ) + a \right |}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {b \sin \left (d x + c\right ) + a}{b^{3}} - \frac {a^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{3}} + \frac {1}{{\left (b \sin \left (d x + c\right ) + a\right )} b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 78, normalized size = 1.24 \[ -\frac {\sin \left (d x +c \right )}{b^{2} d}+\frac {2 a \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{3} d}+\frac {a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {1}{b d \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 61, normalized size = 0.97 \[ \frac {\frac {a^{2} - b^{2}}{b^{4} \sin \left (d x + c\right ) + a b^{3}} + \frac {2 \, a \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{3}} - \frac {\sin \left (d x + c\right )}{b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 69, normalized size = 1.10 \[ \frac {2\,a\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{b^3\,d}-\frac {\sin \left (c+d\,x\right )}{b^2\,d}+\frac {a^2-b^2}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^3+a\,b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.88, size = 221, normalized size = 3.51 \[ \begin {cases} \frac {x \cos ^{3}{\relax (c )}}{a^{2}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\frac {2 \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {\sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d}}{a^{2}} & \text {for}\: b = 0 \\\frac {x \cos ^{3}{\relax (c )}}{\left (a + b \sin {\relax (c )}\right )^{2}} & \text {for}\: d = 0 \\\frac {2 a^{2} \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{a b^{3} d + b^{4} d \sin {\left (c + d x \right )}} + \frac {2 a^{2}}{a b^{3} d + b^{4} d \sin {\left (c + d x \right )}} + \frac {2 a b \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )} \sin {\left (c + d x \right )}}{a b^{3} d + b^{4} d \sin {\left (c + d x \right )}} - \frac {2 b^{2} \sin ^{2}{\left (c + d x \right )}}{a b^{3} d + b^{4} d \sin {\left (c + d x \right )}} - \frac {b^{2} \cos ^{2}{\left (c + d x \right )}}{a b^{3} d + b^{4} d \sin {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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