Optimal. Leaf size=61 \[ -\frac {\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^3 d}+\frac {a \sin (c+d x)}{b^2 d}-\frac {\sin ^2(c+d x)}{2 b d} \]
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Rubi [A] time = 0.07, antiderivative size = 61, 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} \[ -\frac {\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^3 d}+\frac {a \sin (c+d x)}{b^2 d}-\frac {\sin ^2(c+d x)}{2 b d} \]
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)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a-x+\frac {-a^2+b^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac {\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^3 d}+\frac {a \sin (c+d x)}{b^2 d}-\frac {\sin ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 54, normalized size = 0.89 \[ \frac {-\left (a^2-b^2\right ) \log (a+b \sin (c+d x))+a b \sin (c+d x)-\frac {1}{2} b^2 \sin ^2(c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 53, normalized size = 0.87 \[ \frac {b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \sin \left (d x + c\right ) - 2 \, {\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{2 \, b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.80, size = 56, normalized size = 0.92 \[ -\frac {\frac {b \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{b^{2}} + \frac {2 \, {\left (a^{2} - b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 72, normalized size = 1.18 \[ -\frac {\sin ^{2}\left (d x +c \right )}{2 b d}+\frac {a \sin \left (d x +c \right )}{b^{2} d}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) a^{2}}{d \,b^{3}}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 55, normalized size = 0.90 \[ -\frac {\frac {b \sin \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{b^{2}} + \frac {2 \, {\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 55, normalized size = 0.90 \[ -\frac {\frac {{\sin \left (c+d\,x\right )}^2}{2\,b}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{b^3}-\frac {a\,\sin \left (c+d\,x\right )}{b^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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