Optimal. Leaf size=135 \[ \frac {A b-a B}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac {(A+B) \log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac {(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
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Rubi [A] time = 0.19, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2837, 801} \[ \frac {A b-a B}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac {(A+B) \log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac {(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 801
Rule 2837
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {A+\frac {B x}{b}}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {A+B}{2 b (a+b)^2 (b-x)}+\frac {-A b+a B}{(a-b) b (a+b) (a+x)^2}+\frac {-2 a A b+a^2 B+b^2 B}{(a-b)^2 b (a+b)^2 (a+x)}+\frac {A-B}{2 (a-b)^2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {(A+B) \log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {(A-B) \log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {\left (2 a A b-a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^2 d}+\frac {A b-a B}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.37, size = 178, normalized size = 1.32 \[ \frac {b \left (A-\frac {a B}{b}\right ) \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )-\frac {B ((b-a) \log (1-\sin (c+d x))+(a+b) \log (\sin (c+d x)+1)-2 b \log (a+b \sin (c+d x)))}{2 b (b-a) (a+b)}}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 283, normalized size = 2.10 \[ -\frac {2 \, B a^{3} - 2 \, A a^{2} b - 2 \, B a b^{2} + 2 \, A b^{3} - 2 \, {\left (B a^{3} - 2 \, A a^{2} b + B a b^{2} + {\left (B a^{2} b - 2 \, A a b^{2} + B b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left ({\left (A - B\right )} a^{3} + 2 \, {\left (A - B\right )} a^{2} b + {\left (A - B\right )} a b^{2} + {\left ({\left (A - B\right )} a^{2} b + 2 \, {\left (A - B\right )} a b^{2} + {\left (A - B\right )} b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (A + B\right )} a^{3} - 2 \, {\left (A + B\right )} a^{2} b + {\left (A + B\right )} a b^{2} + {\left ({\left (A + B\right )} a^{2} b - 2 \, {\left (A + B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, {\left ({\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 205, normalized size = 1.52 \[ \frac {\frac {2 \, {\left (B a^{2} b - 2 \, A a b^{2} + B b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} - \frac {{\left (A + B\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {{\left (A - B\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {2 \, {\left (B a^{2} b \sin \left (d x + c\right ) - 2 \, A a b^{2} \sin \left (d x + c\right ) + B b^{3} \sin \left (d x + c\right ) + 2 \, B a^{3} - 3 \, A a^{2} b + A b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.69, size = 240, normalized size = 1.78 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right ) A}{2 d \left (a +b \right )^{2}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) B}{2 d \left (a +b \right )^{2}}+\frac {A b}{d \left (a +b \right ) \left (a -b \right ) \left (a +b \sin \left (d x +c \right )\right )}-\frac {a B}{d \left (a +b \right ) \left (a -b \right ) \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 \ln \left (a +b \sin \left (d x +c \right )\right ) A a b}{d \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) B \,a^{2}}{d \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) B \,b^{2}}{d \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) A}{2 d \left (a -b \right )^{2}}-\frac {\ln \left (1+\sin \left (d x +c \right )\right ) B}{2 d \left (a -b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 147, normalized size = 1.09 \[ \frac {\frac {2 \, {\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {{\left (A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {2 \, {\left (B a - A b\right )}}{a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 131, normalized size = 0.97 \[ \frac {A\,b-B\,a}{d\,\left (a^2-b^2\right )\,\left (a+b\,\sin \left (c+d\,x\right )\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {A}{2}+\frac {B}{2}\right )}{d\,{\left (a+b\right )}^2}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (B\,a^2-2\,A\,a\,b+B\,b^2\right )}{d\,{\left (a^2-b^2\right )}^2}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {A}{2}-\frac {B}{2}\right )}{d\,{\left (a-b\right )}^2} \]
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 \[ \int \frac {\left (A + B \sin {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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