Optimal. Leaf size=124 \[ \frac{\left (2 a^2-b^2\right ) \cos ^3(c+d x)}{3 d}-\frac{\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a b \cos ^4(c+d x)}{2 d}+\frac{2 a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d} \]
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Rubi [A] time = 0.195642, antiderivative size = 124, 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, 2837, 12, 948} \[ \frac{\left (2 a^2-b^2\right ) \cos ^3(c+d x)}{3 d}-\frac{\left (a^2-2 b^2\right ) \cos (c+d x)}{d}-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a b \cos ^4(c+d x)}{2 d}+\frac{2 a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2837
Rule 12
Rule 948
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^2 \sin ^5(c+d x) \, dx &=\int (-b-a \cos (c+d x))^2 \sin ^3(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a^2 (-b+x)^2 \left (a^2-x^2\right )^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-b+x)^2 \left (a^2-x^2\right )^2}{x^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^4 \left (1-\frac{2 b^2}{a^2}\right )+\frac{a^4 b^2}{x^2}-\frac{2 a^4 b}{x}+4 a^2 b x-\left (2 a^2-b^2\right ) x^2-2 b x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac{\left (a^2-2 b^2\right ) \cos (c+d x)}{d}+\frac{2 a b \cos ^2(c+d x)}{d}+\frac{\left (2 a^2-b^2\right ) \cos ^3(c+d x)}{3 d}-\frac{a b \cos ^4(c+d x)}{2 d}-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.36251, size = 112, normalized size = 0.9 \[ -\frac{30 \left (5 a^2-14 b^2\right ) \cos (c+d x)-25 a^2 \cos (3 (c+d x))+3 a^2 \cos (5 (c+d x))-180 a b \cos (2 (c+d x))+15 a b \cos (4 (c+d x))+480 a b \log (\cos (c+d x))+20 b^2 \cos (3 (c+d x))-240 b^2 \sec (c+d x)}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 184, normalized size = 1.5 \begin{align*} -{\frac{8\,{a}^{2}\cos \left ( dx+c \right ) }{15\,d}}-{\frac{{a}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5\,d}}-{\frac{4\,{a}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{15\,d}}-{\frac{ab \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2\,d}}-{\frac{ab \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-2\,{\frac{ab\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{d\cos \left ( dx+c \right ) }}+{\frac{8\,{b}^{2}\cos \left ( dx+c \right ) }{3\,d}}+{\frac{{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) }{d}}+{\frac{4\,{b}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968799, size = 142, normalized size = 1.15 \begin{align*} -\frac{6 \, a^{2} \cos \left (d x + c\right )^{5} + 15 \, a b \cos \left (d x + c\right )^{4} - 60 \, a b \cos \left (d x + c\right )^{2} - 10 \,{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{3} + 60 \, a b \log \left (\cos \left (d x + c\right )\right ) + 30 \,{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right ) - \frac{30 \, b^{2}}{\cos \left (d x + c\right )}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80296, size = 328, normalized size = 2.65 \begin{align*} -\frac{48 \, a^{2} \cos \left (d x + c\right )^{6} + 120 \, a b \cos \left (d x + c\right )^{5} - 480 \, a b \cos \left (d x + c\right )^{3} - 80 \,{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 480 \, a b \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 195 \, a b \cos \left (d x + c\right ) + 240 \,{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 240 \, b^{2}}{240 \, d \cos \left (d x + c\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 = 1.39652, size = 564, normalized size = 4.55 \begin{align*} \frac{60 \, a b \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a b \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{60 \,{\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 )}}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1} + \frac{32 \, a^{2} + 137 \, a b - 100 \, b^{2} - \frac{160 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{805 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{440 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{320 \, a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{1970 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{640 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1970 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{360 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{805 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{60 \, b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{137 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{30 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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