Optimal. Leaf size=229 \[ -\frac{b^3}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{b \left (8 a^2 b^2+3 a^4+b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^2(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
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Rubi [A] time = 0.512592, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2721, 1647, 1629} \[ -\frac{b^3}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{b \left (8 a^2 b^2+3 a^4+b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^2(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2721
Rule 1647
Rule 1629
Rubi steps
\begin{align*} \int \frac{\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac{\cot ^3(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(-b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{a^4 b^3 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}-\frac{a^2 b^2 \left (3 a^4+3 a^2 b^2-2 b^4\right ) x}{\left (a^2-b^2\right )^3}+\frac{a^4 b \left (3 a^2-7 b^2\right ) x^2}{\left (a^2-b^2\right )^3}+\frac{a^4 \left (a^2+3 b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{2 a^2 d}\\ &=\frac{\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \left (\frac{a^2 (a+2 b)}{2 (a-b)^4 (a-x)}-\frac{2 a^2 b^3}{\left (a^2-b^2\right )^2 (b-x)^3}+\frac{2 a^2 b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 (b-x)^2}-\frac{2 a^2 b \left (3 a^4+8 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4 (b-x)}+\frac{a^2 (a-2 b)}{2 (a+b)^4 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{2 a^2 d}\\ &=-\frac{b^3}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac{b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac{\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac{(a+2 b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac{b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end{align*}
Mathematica [C] time = 6.3066, size = 332, normalized size = 1.45 \[ -\frac{2 i \left (8 a^2 b^3+3 a^4 b+b^5\right ) (c+d x)}{d (a-b)^4 (a+b)^4}-\frac{b^2 \left (3 a^2+b^2\right )}{d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}+\frac{\left (8 a^2 b^3+3 a^4 b+b^5\right ) \log (a \cos (c+d x)+b)}{d \left (b^2-a^2\right )^4}-\frac{b^3}{2 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}-\frac{i (-a-2 b) \tan ^{-1}(\tan (c+d x))}{2 d (b-a)^4}-\frac{i (a-2 b) \tan ^{-1}(\tan (c+d x))}{2 d (a+b)^4}-\frac{\csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d (a+b)^3}-\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d (b-a)^3}+\frac{(a-2 b) \log \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )\right )}{4 d (a+b)^4}+\frac{(-a-2 b) \log \left (\cos ^2\left (\frac{1}{2} (c+d x)\right )\right )}{4 d (b-a)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.087, size = 322, normalized size = 1.4 \begin{align*} -{\frac{{b}^{3}}{2\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ){a}^{4}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+8\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ){a}^{2}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+{\frac{{b}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+3\,{\frac{{a}^{2}{b}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+{\frac{{b}^{4}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+{\frac{1}{4\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) a}{4\,d \left ( a-b \right ) ^{4}}}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) b}{2\,d \left ( a-b \right ) ^{4}}}+{\frac{1}{4\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) a}{4\,d \left ( a+b \right ) ^{4}}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{2\,d \left ( a+b \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04154, size = 587, normalized size = 2.56 \begin{align*} \frac{\frac{4 \,{\left (3 \, a^{4} b + 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac{{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac{{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{2 \,{\left (8 \, a^{2} b^{3} + 4 \, b^{5} -{\left (a^{5} + 9 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} +{\left (a^{4} b - 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2} +{\left (11 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8} -{\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )^{3} +{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.01581, size = 2344, normalized size = 10.24 \begin{align*} \text{result too large to display} \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 ^{3}{\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 = 1.45744, size = 1080, normalized size = 4.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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