Optimal. Leaf size=71 \[ -\frac {A+B}{4 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {(A+B) \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {A-B}{4 d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 77, 206} \[ -\frac {A+B}{4 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {(A+B) \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {A-B}{4 d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x) (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {A-B}{2 a (a+x)^3}+\frac {A+B}{4 a^2 (a+x)^2}+\frac {A+B}{4 a^2 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {A-B}{4 d (a+a \sin (c+d x))^2}-\frac {A+B}{4 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {(A+B) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{4 a d}\\ &=\frac {(A+B) \tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac {A-B}{4 d (a+a \sin (c+d x))^2}-\frac {A+B}{4 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 69, normalized size = 0.97 \[ \frac {a \left (\frac {(A+B) \tanh ^{-1}(\sin (c+d x))}{4 a^3}-\frac {A+B}{4 a^2 (a \sin (c+d x)+a)}-\frac {A-B}{4 a (a \sin (c+d x)+a)^2}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 134, normalized size = 1.89 \[ \frac {{\left ({\left (A + B\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (A + B\right )} \sin \left (d x + c\right ) - 2 \, A - 2 \, B\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (A + B\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (A + B\right )} \sin \left (d x + c\right ) - 2 \, A - 2 \, B\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A + B\right )} \sin \left (d x + c\right ) + 4 \, A}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{2} d \sin \left (d x + c\right ) - 2 \, a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 104, normalized size = 1.46 \[ \frac {\frac {2 \, {\left (A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2}} - \frac {3 \, A \sin \left (d x + c\right )^{2} + 3 \, B \sin \left (d x + c\right )^{2} + 10 \, A \sin \left (d x + c\right ) + 10 \, B \sin \left (d x + c\right ) + 11 \, A + 3 \, B}{a^{2} {\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 150, normalized size = 2.11 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right ) A}{8 d \,a^{2}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) B}{8 d \,a^{2}}-\frac {A}{4 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {B}{4 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {A}{4 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}-\frac {B}{4 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) A}{8 d \,a^{2}}+\frac {B \ln \left (1+\sin \left (d x +c \right )\right )}{8 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 84, normalized size = 1.18 \[ -\frac {\frac {2 \, {\left ({\left (A + B\right )} \sin \left (d x + c\right ) + 2 \, A\right )}}{a^{2} \sin \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) + a^{2}} - \frac {{\left (A + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {{\left (A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.25, size = 71, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (A+B\right )}{4\,a^2\,d}-\frac {\frac {A}{2}+\sin \left (c+d\,x\right )\,\left (\frac {A}{4}+\frac {B}{4}\right )}{d\,\left (a^2\,{\sin \left (c+d\,x\right )}^2+2\,a^2\,\sin \left (c+d\,x\right )+a^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A \sec {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sin {\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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