Optimal. Leaf size=109 \[ \frac{b^2}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}+\frac{2 a b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^2}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^2} \]
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Rubi [A] time = 0.226432, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2837, 12, 1629} \[ \frac{b^2}{a d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}+\frac{2 a b \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^2}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2837
Rule 12
Rule 1629
Rubi steps
\begin{align*} \int \frac{\csc (c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos (c+d x) \cot (c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{x^2}{a^2 (-b+x)^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(-b+x)^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{2 (a-b)^2 (a-x)}+\frac{b^2}{(a-b) (a+b) (b-x)^2}-\frac{2 a^2 b}{(a-b)^2 (a+b)^2 (b-x)}+\frac{a}{2 (a+b)^2 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{b^2}{a \left (a^2-b^2\right ) d (b+a \cos (c+d x))}+\frac{\log (1-\cos (c+d x))}{2 (a+b)^2 d}-\frac{\log (1+\cos (c+d x))}{2 (a-b)^2 d}+\frac{2 a b \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^2 d}\\ \end{align*}
Mathematica [A] time = 0.281333, size = 165, normalized size = 1.51 \[ \frac{b \left (2 a^2 b \log (a \cos (c+d x)+b)+(a-b) \left (a (a-b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+b (a+b)\right )-a (a+b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-a^2 \cos (c+d x) \left ((a-b)^2 \left (-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )+(a+b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-2 a b \log (a \cos (c+d x)+b)\right )}{a d (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 106, normalized size = 1. \begin{align*}{\frac{{b}^{2}}{d \left ( a+b \right ) \left ( a-b \right ) a \left ( b+a\cos \left ( dx+c \right ) \right ) }}+2\,{\frac{ab\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{2\, \left ( a-b \right ) ^{2}d}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,d \left ( a+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0816, size = 166, normalized size = 1.52 \begin{align*} \frac{\frac{4 \, a b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \, b^{2}}{a^{3} b - a b^{3} +{\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )} - \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19723, size = 486, normalized size = 4.46 \begin{align*} \frac{2 \, a^{2} b^{2} - 2 \, b^{4} + 4 \,{\left (a^{3} b \cos \left (d x + c\right ) + a^{2} b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) -{\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3} +{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3} +{\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left ({\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d \cos \left (d x + c\right ) +{\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32882, size = 288, normalized size = 2.64 \begin{align*} \frac{\frac{4 \, a b \log \left ({\left | -a - b - \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{\log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{4 \,{\left (a b + b^{2} + \frac{a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )}{\left (a + b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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