Optimal. Leaf size=163 \[ -\frac{b^3}{2 a^2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac{b^2 \left (3 a^2-b^2\right )}{a^2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}+\frac{b \left (3 a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^3}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^3}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^3} \]
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Rubi [A] time = 0.31935, antiderivative size = 163, 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} \[ -\frac{b^3}{2 a^2 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac{b^2 \left (3 a^2-b^2\right )}{a^2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}+\frac{b \left (3 a^2+b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^3}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^3}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^3} \]
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))^3} \, dx &=-\int \frac{\cos ^2(c+d x) \cot (c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{x^3}{a^3 (-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{2 (a-b)^3 (a-x)}-\frac{b^3}{(a-b) (a+b) (b-x)^3}+\frac{3 a^2 b^2-b^4}{(a-b)^2 (a+b)^2 (b-x)^2}-\frac{a^2 b \left (3 a^2+b^2\right )}{(a-b)^3 (a+b)^3 (b-x)}+\frac{a^2}{2 (a+b)^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^2 d}\\ &=-\frac{b^3}{2 a^2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac{b^2 \left (3 a^2-b^2\right )}{a^2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))}+\frac{\log (1-\cos (c+d x))}{2 (a+b)^3 d}-\frac{\log (1+\cos (c+d x))}{2 (a-b)^3 d}+\frac{b \left (3 a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^3 d}\\ \end{align*}
Mathematica [A] time = 0.554559, size = 203, normalized size = 1.25 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left (-\frac{2 b^2 \left (b^2-3 a^2\right ) (a \cos (c+d x)+b)}{a^2 (a-b)^2 (a+b)^2}+\frac{2 b \left (3 a^2+b^2\right ) (a \cos (c+d x)+b)^2 \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^3}+\frac{b^3}{a^2 \left (b^2-a^2\right )}+\frac{2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^2}{(b-a)^3}+\frac{2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^2}{(a+b)^3}\right )}{2 d (a+b \sec (c+d x))^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 206, normalized size = 1.3 \begin{align*} -{\frac{{b}^{3}}{2\,d{a}^{2} \left ( a+b \right ) \left ( a-b \right ) \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ){a}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}}+{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}}+3\,{\frac{{b}^{2}}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-{\frac{{b}^{4}}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}{a}^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{2\, \left ( a-b \right ) ^{3}d}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,d \left ( a+b \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979614, size = 325, normalized size = 1.99 \begin{align*} \frac{\frac{2 \,{\left (3 \, a^{2} b + b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{5 \, a^{2} b^{3} - b^{5} + 2 \,{\left (3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )}{a^{6} b^{2} - 2 \, a^{4} b^{4} + a^{2} b^{6} +{\left (a^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b - 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )} - \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac{\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.60537, size = 1021, normalized size = 6.26 \begin{align*} \frac{5 \, a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7} + 2 \,{\left (3 \, a^{5} b^{2} - 4 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) + 2 \,{\left (3 \, a^{4} b^{3} + a^{2} b^{5} +{\left (3 \, a^{6} b + a^{4} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) -{\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5} +{\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\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^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - a^{2} b^{5} +{\left (a^{7} - 3 \, a^{6} b + 3 \, a^{5} b^{2} - a^{4} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} - a^{3} b^{4}\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^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} d \cos \left (d x + c\right ) +{\left (a^{8} b^{2} - 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8}\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 )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.50738, size = 610, normalized size = 3.74 \begin{align*} \frac{\frac{2 \,{\left (3 \, a^{2} b + b^{3}\right )} \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^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{\log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac{9 \, a^{3} b + 15 \, a^{2} b^{2} + 3 \, a b^{3} - 3 \, b^{4} + \frac{18 \, a^{3} b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{6 \, a^{2} b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{10 \, a b^{3}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{2 \, b^{4}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{9 \, a^{3} b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{9 \, a^{2} b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a b^{3}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{3 \, b^{4}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\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}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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