Optimal. Leaf size=66 \[ -\frac {B (a-a \sin (c+d x))^2}{2 a^4 d}-\frac {(A-B) \sin (c+d x)}{a^2 d}+\frac {2 (A-B) \log (\sin (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2836, 77} \[ -\frac {(A-B) \sin (c+d x)}{a^2 d}+\frac {2 (A-B) \log (\sin (c+d x)+1)}{a^2 d}-\frac {B (a-a \sin (c+d x))^2}{2 a^4 d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x) \left (A+\frac {B x}{a}\right )}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-A+B+\frac {B (a-x)}{a}+\frac {2 a (A-B)}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {2 (A-B) \log (1+\sin (c+d x))}{a^2 d}-\frac {(A-B) \sin (c+d x)}{a^2 d}-\frac {B (a-a \sin (c+d x))^2}{2 a^4 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 51, normalized size = 0.77 \[ -\frac {2 (A-2 B) \sin (c+d x)-4 (A-B) \log (\sin (c+d x)+1)+B \sin ^2(c+d x)+B}{2 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 48, normalized size = 0.73 \[ \frac {B \cos \left (d x + c\right )^{2} + 4 \, {\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (A - 2 \, B\right )} \sin \left (d x + c\right )}{2 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 92, normalized size = 1.39 \[ -\frac {\frac {4 \, {\left (A - B\right )} \log \left (\frac {{\left | a \sin \left (d x + c\right ) + a \right |}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left | a \right |}}\right )}{a^{2}} + \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{2} {\left (B + \frac {2 \, {\left (A a^{2} - 3 \, B a^{2}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )} a}\right )}}{a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 85, normalized size = 1.29 \[ -\frac {B \left (\sin ^{2}\left (d x +c \right )\right )}{2 d \,a^{2}}-\frac {A \sin \left (d x +c \right )}{d \,a^{2}}+\frac {2 B \sin \left (d x +c \right )}{d \,a^{2}}+\frac {2 \ln \left (1+\sin \left (d x +c \right )\right ) A}{d \,a^{2}}-\frac {2 B \ln \left (1+\sin \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 54, normalized size = 0.82 \[ \frac {\frac {4 \, {\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {B \sin \left (d x + c\right )^{2} + 2 \, {\left (A - 2 \, B\right )} \sin \left (d x + c\right )}{a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 61, normalized size = 0.92 \[ -\frac {2\,A\,\sin \left (c+d\,x\right )-4\,B\,\sin \left (c+d\,x\right )+B\,{\sin \left (c+d\,x\right )}^2-4\,A\,\ln \left (\sin \left (c+d\,x\right )+1\right )+4\,B\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.01, size = 1096, normalized size = 16.61 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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