Optimal. Leaf size=104 \[ \frac {b}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {2 a b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2668, 710, 801} \[ \frac {b}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {2 a b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 710
Rule 801
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {1}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {b \operatorname {Subst}\left (\int \frac {a-x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=\frac {b}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {b \operatorname {Subst}\left (\int \left (\frac {a-b}{2 b (a+b) (b-x)}-\frac {2 a}{(a-b) (a+b) (a+x)}+\frac {a+b}{2 (a-b) b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=-\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {2 a b \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^2 d}+\frac {b}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 102, normalized size = 0.98 \[ \frac {b \left (\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 188, normalized size = 1.81 \[ \frac {2 \, a^{2} b - 2 \, b^{3} - 4 \, {\left (a b^{2} \sin \left (d x + c\right ) + a^{2} b\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (a^{3} + 2 \, a^{2} b + a b^{2} + {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a^{3} - 2 \, a^{2} b + a b^{2} + {\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, {\left ({\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\right ) + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 147, normalized size = 1.41 \[ -\frac {\frac {4 \, a b^{2} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {2 \, {\left (2 \, a b^{2} \sin \left (d x + c\right ) + 3 \, a^{2} b - b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 101, normalized size = 0.97 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 d \left (a +b \right )^{2}}+\frac {b}{d \left (a +b \right ) \left (a -b \right ) \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 a b \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 118, normalized size = 1.13 \[ -\frac {\frac {4 \, a b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, b}{a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 98, normalized size = 0.94 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,d\,{\left (a-b\right )}^2}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,d\,{\left (a+b\right )}^2}+\frac {b}{d\,\left (a^2-b^2\right )\,\left (a+b\,\sin \left (c+d\,x\right )\right )}-\frac {2\,a\,b\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,{\left (a^2-b^2\right )}^2} \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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