Optimal. Leaf size=126 \[ -\frac{\cot ^4(c+d x) \left (a^2+2 a b \sec (c+d x)+b^2\right )}{4 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac{a (4 a-3 b) \log (\sec (c+d x)+1)}{8 d}+\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d} \]
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Rubi [A] time = 0.159482, antiderivative size = 126, 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 = {3885, 1805, 823, 801} \[ -\frac{\cot ^4(c+d x) \left (a^2+2 a b \sec (c+d x)+b^2\right )}{4 d}+\frac{a^2 \log (\cos (c+d x))}{d}+\frac{a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac{a (4 a-3 b) \log (\sec (c+d x)+1)}{8 d}+\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 1805
Rule 823
Rule 801
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+b \sec (c+d x))^2 \, dx &=-\frac{b^6 \operatorname{Subst}\left (\int \frac{(a+x)^2}{x \left (b^2-x^2\right )^3} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac{\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}+\frac{b^4 \operatorname{Subst}\left (\int \frac{-4 a^2-6 a x}{x \left (b^2-x^2\right )^2} \, dx,x,b \sec (c+d x)\right )}{4 d}\\ &=\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}-\frac{\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}+\frac{\operatorname{Subst}\left (\int \frac{-8 a^2 b^2-6 a b^2 x}{x \left (b^2-x^2\right )} \, dx,x,b \sec (c+d x)\right )}{8 d}\\ &=\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}-\frac{\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}+\frac{\operatorname{Subst}\left (\int \left (-\frac{a (4 a+3 b)}{b-x}-\frac{8 a^2}{x}+\frac{a (4 a-3 b)}{b+x}\right ) \, dx,x,b \sec (c+d x)\right )}{8 d}\\ &=\frac{a^2 \log (\cos (c+d x))}{d}+\frac{a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac{a (4 a-3 b) \log (1+\sec (c+d x))}{8 d}+\frac{a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}-\frac{\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 3.22511, size = 148, normalized size = 1.17 \[ \frac{2 \left (7 a^2+10 a b+3 b^2\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )+2 \left (7 a^2-10 a b+3 b^2\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )-(a+b)^2 \csc ^4\left (\frac{1}{2} (c+d x)\right )-(a-b)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right )+16 a \left ((4 a+3 b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+(4 a-3 b) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{64 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 169, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{{a}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+{\frac{{a}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,a\cos \left ( dx+c \right ) b}{4\,d}}+{\frac{3\,ab\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{4\,d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06366, size = 165, normalized size = 1.31 \begin{align*} \frac{{\left (4 \, a^{2} - 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) +{\left (4 \, a^{2} + 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (5 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right ) + 2 \,{\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - b^{2}\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.989525, size = 504, normalized size = 4. \begin{align*} -\frac{10 \, a b \cos \left (d x + c\right )^{3} - 6 \, a b \cos \left (d x + c\right ) + 4 \,{\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 6 \, a^{2} - 2 \, b^{2} -{\left ({\left (4 \, a^{2} - 3 \, a b\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (4 \, a^{2} - 3 \, a b\right )} \cos \left (d x + c\right )^{2} + 4 \, a^{2} - 3 \, a b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left ({\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (d x + c\right )^{2} + 4 \, a^{2} + 3 \, a b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{8 \,{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.45574, size = 486, normalized size = 3.86 \begin{align*} -\frac{64 \, a^{2} \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) + \frac{12 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{16 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{4 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{2 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 8 \,{\left (4 \, a^{2} + 3 \, a b\right )} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac{{\left (a^{2} + 2 \, a b + b^{2} + \frac{12 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{16 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{4 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{48 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{36 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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