Optimal. Leaf size=157 \[ -\frac{b^4 \log (a+b \sec (c+d x))}{a d \left (a^2-b^2\right )^2}+\frac{1}{4 d (a+b) (1-\sec (c+d x))}+\frac{1}{4 d (a-b) (\sec (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sec (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sec (c+d x)+1)}{4 d (a-b)^2}-\frac{\log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.181046, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3885, 894} \[ -\frac{b^4 \log (a+b \sec (c+d x))}{a d \left (a^2-b^2\right )^2}+\frac{1}{4 d (a+b) (1-\sec (c+d x))}+\frac{1}{4 d (a-b) (\sec (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sec (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sec (c+d x)+1)}{4 d (a-b)^2}-\frac{\log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x)}{a+b \sec (c+d x)} \, dx &=\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{x (a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{b^4 \operatorname{Subst}\left (\int \left (\frac{1}{4 b^3 (a+b) (b-x)^2}+\frac{2 a+3 b}{4 b^4 (a+b)^2 (b-x)}+\frac{1}{a b^4 x}-\frac{1}{a (a-b)^2 (a+b)^2 (a+x)}-\frac{1}{4 (a-b) b^3 (b+x)^2}+\frac{-2 a+3 b}{4 (a-b)^2 b^4 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac{\log (\cos (c+d x))}{a d}-\frac{(2 a+3 b) \log (1-\sec (c+d x))}{4 (a+b)^2 d}-\frac{(2 a-3 b) \log (1+\sec (c+d x))}{4 (a-b)^2 d}-\frac{b^4 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right )^2 d}+\frac{1}{4 (a+b) d (1-\sec (c+d x))}+\frac{1}{4 (a-b) d (1+\sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.0088, size = 141, normalized size = 0.9 \[ -\frac{8 b^4 \log (a \cos (c+d x)+b)+a (a-b)^2 (a+b) \csc ^2\left (\frac{1}{2} (c+d x)\right )+a (a-b) (a+b)^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )+4 a (a-b)^2 (2 a+3 b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+4 a (2 a-3 b) (a+b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{8 a d (a-b)^2 (a+b)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 167, normalized size = 1.1 \begin{align*} -{\frac{{b}^{4}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}a}}-{\frac{1}{d \left ( 4\,a-4\,b \right ) \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) a}{2\,d \left ( a-b \right ) ^{2}}}+{\frac{3\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ) b}{4\,d \left ( a-b \right ) ^{2}}}+{\frac{1}{d \left ( 4\,a+4\,b \right ) \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) a}{2\,d \left ( a+b \right ) ^{2}}}-{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{4\,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 = 1.0286, size = 194, normalized size = 1.24 \begin{align*} -\frac{\frac{4 \, b^{4} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac{{\left (2 \, a - 3 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{{\left (2 \, a + 3 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} + \frac{2 \,{\left (b \cos \left (d x + c\right ) - a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + b^{2}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07828, size = 587, normalized size = 3.74 \begin{align*} \frac{2 \, a^{4} - 2 \, a^{2} b^{2} - 2 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) - 4 \,{\left (b^{4} \cos \left (d x + c\right )^{2} - b^{4}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) +{\left (2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a b^{3} -{\left (2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left (2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3} -{\left (2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{4 \,{\left ({\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \cos \left (d x + c\right )^{2} -{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\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{\cot ^{3}{\left (c + d x \right )}}{a + b \sec{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.45078, size = 346, normalized size = 2.2 \begin{align*} -\frac{\frac{8 \, b^{4} \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^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac{2 \,{\left (2 \, a + 3 \, b\right )} \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{{\left (a + b + \frac{4 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{6 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{{\left (a^{2} + 2 \, a b + b^{2}\right )}{\left (\cos \left (d x + c\right ) - 1\right )}} - \frac{8 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac{\cos \left (d x + c\right ) - 1}{{\left (a - b\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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