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 = 0.23, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 12
Rule 1629
Rule 2837
Rule 3872
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*}
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Mathematica [A] time = 0.35, 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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fricas [A] time = 0.64, size = 210, normalized size = 1.93 \[ \frac {2 \, a^{2} b^{2} - 2 \, b^{4} + 4 \, {\left (a^{3} b \cos \left (d x + c\right ) + a^{2} b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3} + {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3} + {\left (a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left ({\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 213, normalized size = 1.95 \[ \frac {\frac {4 \, a b \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {\log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {4 \, {\left (a b + b^{2} + \frac {a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} {\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 106, normalized size = 0.97 \[ \frac {b^{2}}{d \left (a +b \right ) \left (a -b \right ) a \left (b +a \cos \left (d x +c \right )\right )}+\frac {2 a b \ln \left (b +a \cos \left (d x +c \right )\right )}{d \left (a -b \right )^{2} \left (a +b \right )^{2}}+\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{2 d \left (a +b \right )^{2}}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 \left (a -b \right )^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 123, normalized size = 1.13 \[ \frac {\frac {4 \, a b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, b^{2}}{a^{3} b - a b^{3} + {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )} - \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 103, normalized size = 0.94 \[ \frac {\ln \left (\cos \left (c+d\,x\right )-1\right )}{2\,d\,{\left (a+b\right )}^2}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )}{2\,d\,{\left (a-b\right )}^2}+\frac {b^2}{a\,d\,\left (a^2-b^2\right )\,\left (b+a\,\cos \left (c+d\,x\right )\right )}+\frac {2\,a\,b\,\ln \left (b+a\,\cos \left (c+d\,x\right )\right )}{d\,{\left (a^2-b^2\right )}^2} \]
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 \[ \int \frac {\csc {\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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