Optimal. Leaf size=123 \[ \frac {A+B}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3 A+B}{16 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {(2 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac {a (A-B)}{12 d (a \sin (c+d x)+a)^3}-\frac {A}{8 d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.15, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2836, 77, 206} \[ \frac {A+B}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3 A+B}{16 d \left (a^2 \sin (c+d x)+a^2\right )}+\frac {(2 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac {a (A-B)}{12 d (a \sin (c+d x)+a)^3}-\frac {A}{8 d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 206
Rule 2836
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x) (A+B \sin (c+d x))}{(a+a \sin (c+d x))^2} \, dx &=\frac {a^3 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^2 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {A+B}{16 a^4 (a-x)^2}+\frac {A-B}{4 a^2 (a+x)^4}+\frac {A}{4 a^3 (a+x)^3}+\frac {3 A+B}{16 a^4 (a+x)^2}+\frac {2 A+B}{8 a^4 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a (A-B)}{12 d (a+a \sin (c+d x))^3}-\frac {A}{8 d (a+a \sin (c+d x))^2}+\frac {A+B}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3 A+B}{16 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {(2 A+B) \operatorname {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 a d}\\ &=\frac {(2 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac {a (A-B)}{12 d (a+a \sin (c+d x))^3}-\frac {A}{8 d (a+a \sin (c+d x))^2}+\frac {A+B}{16 d \left (a^2-a^2 \sin (c+d x)\right )}-\frac {3 A+B}{16 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.74, size = 87, normalized size = 0.71 \[ -\frac {\frac {3 (A+B)}{\sin (c+d x)-1}+\frac {3 (3 A+B)}{\sin (c+d x)+1}+\frac {4 (A-B)}{(\sin (c+d x)+1)^3}-6 (2 A+B) \tanh ^{-1}(\sin (c+d x))+\frac {6 A}{(\sin (c+d x)+1)^2}}{48 a^2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 230, normalized size = 1.87 \[ \frac {12 \, {\left (2 \, A + B\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left ({\left (2 \, A + B\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (2 \, A + B\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, {\left (2 \, A + B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (2 \, A + B\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (2 \, A + B\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, {\left (2 \, A + B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (3 \, {\left (2 \, A + B\right )} \cos \left (d x + c\right )^{2} - 8 \, A - 4 \, B\right )} \sin \left (d x + c\right ) - 8 \, A - 16 \, B}{48 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 169, normalized size = 1.37 \[ \frac {\frac {6 \, {\left (2 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2}} - \frac {6 \, {\left (2 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (2 \, A \sin \left (d x + c\right ) + B \sin \left (d x + c\right ) - 3 \, A - 2 \, B\right )}}{a^{2} {\left (\sin \left (d x + c\right ) - 1\right )}} - \frac {22 \, A \sin \left (d x + c\right )^{3} + 11 \, B \sin \left (d x + c\right )^{3} + 84 \, A \sin \left (d x + c\right )^{2} + 39 \, B \sin \left (d x + c\right )^{2} + 114 \, A \sin \left (d x + c\right ) + 45 \, B \sin \left (d x + c\right ) + 60 \, A + 9 \, B}{a^{2} {\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 207, normalized size = 1.68 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right ) A}{8 d \,a^{2}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) B}{16 d \,a^{2}}-\frac {A}{16 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {B}{16 d \,a^{2} \left (\sin \left (d x +c \right )-1\right )}-\frac {A}{8 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {A}{12 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {B}{12 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) A}{8 d \,a^{2}}+\frac {B \ln \left (1+\sin \left (d x +c \right )\right )}{16 a^{2} d}-\frac {3 A}{16 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )}-\frac {B}{16 d \,a^{2} \left (1+\sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 139, normalized size = 1.13 \[ -\frac {\frac {2 \, {\left (3 \, {\left (2 \, A + B\right )} \sin \left (d x + c\right )^{3} + 6 \, {\left (2 \, A + B\right )} \sin \left (d x + c\right )^{2} + {\left (2 \, A + B\right )} \sin \left (d x + c\right ) - 8 \, A + 2 \, B\right )}}{a^{2} \sin \left (d x + c\right )^{4} + 2 \, a^{2} \sin \left (d x + c\right )^{3} - 2 \, a^{2} \sin \left (d x + c\right ) - a^{2}} - \frac {3 \, {\left (2 \, A + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2}} + \frac {3 \, {\left (2 \, A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 121, normalized size = 0.98 \[ \frac {\left (\frac {A}{4}+\frac {B}{8}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {A}{2}+\frac {B}{4}\right )\,{\sin \left (c+d\,x\right )}^2+\left (\frac {A}{12}+\frac {B}{24}\right )\,\sin \left (c+d\,x\right )-\frac {A}{3}+\frac {B}{12}}{d\,\left (-a^2\,{\sin \left (c+d\,x\right )}^4-2\,a^2\,{\sin \left (c+d\,x\right )}^3+2\,a^2\,\sin \left (c+d\,x\right )+a^2\right )}+\frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (2\,A+B\right )}{8\,a^2\,d} \]
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 \[ \frac {\int \frac {A \sec ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sin {\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} + 2 \sin {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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