Optimal. Leaf size=215 \[ -\frac{\sin ^3(c+d x) \left (a^2 (4 A+5 C)+10 a b B+2 A b^2\right )}{15 d}+\frac{\sin (c+d x) \left (a^2 (4 A+5 C)+10 a b B+b^2 (4 A+5 C)\right )}{5 d}+\frac{\sin (c+d x) \cos (c+d x) \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )}{8 d}+\frac{1}{8} x \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )+\frac{a (5 a B+2 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d} \]
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Rubi [A] time = 0.513512, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {4094, 4074, 4047, 2635, 8, 4044, 3013} \[ -\frac{\sin ^3(c+d x) \left (a^2 (4 A+5 C)+10 a b B+2 A b^2\right )}{15 d}+\frac{\sin (c+d x) \left (a^2 (4 A+5 C)+10 a b B+b^2 (4 A+5 C)\right )}{5 d}+\frac{\sin (c+d x) \cos (c+d x) \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )}{8 d}+\frac{1}{8} x \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )+\frac{a (5 a B+2 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4074
Rule 4047
Rule 2635
Rule 8
Rule 4044
Rule 3013
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (2 A b+5 a B+(4 a A+5 b B+5 a C) \sec (c+d x)+b (2 A+5 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (2 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac{1}{20} \int \cos ^3(c+d x) \left (-4 \left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right )-5 \left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \sec (c+d x)-4 b^2 (2 A+5 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a (2 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac{1}{20} \int \cos ^3(c+d x) \left (-4 \left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right )-4 b^2 (2 A+5 C) \sec ^2(c+d x)\right ) \, dx-\frac{1}{4} \left (-6 a A b-3 a^2 B-4 b^2 B-8 a b C\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{\left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a (2 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac{1}{20} \int \cos (c+d x) \left (-4 b^2 (2 A+5 C)-4 \left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx-\frac{1}{8} \left (-6 a A b-3 a^2 B-4 b^2 B-8 a b C\right ) \int 1 \, dx\\ &=\frac{1}{8} \left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) x+\frac{\left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a (2 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac{\operatorname{Subst}\left (\int \left (-4 b^2 (2 A+5 C)-4 \left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right )+4 \left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{20 d}\\ &=\frac{1}{8} \left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) x+\frac{\left (10 a b B+a^2 (4 A+5 C)+b^2 (4 A+5 C)\right ) \sin (c+d x)}{5 d}+\frac{\left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac{a (2 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac{A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac{\left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.802154, size = 169, normalized size = 0.79 \[ \frac{60 (c+d x) \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )+60 \sin (c+d x) \left (a^2 (5 A+6 C)+12 a b B+2 b^2 (3 A+4 C)\right )+120 \sin (2 (c+d x)) \left (a^2 B+2 a b (A+C)+b^2 B\right )+10 \sin (3 (c+d x)) \left (a^2 (5 A+4 C)+8 a b B+4 A b^2\right )+6 a^2 A \sin (5 (c+d x))+15 a (a B+2 A b) \sin (4 (c+d x))}{480 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 244, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2}A\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+B{a}^{2} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{{a}^{2}C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,Aab \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{2\,Bab \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,abC \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{\frac{A{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+B{b}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{b}^{2}C\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0328, size = 315, normalized size = 1.47 \begin{align*} \frac{32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 30 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b - 320 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b + 240 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b - 160 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{2} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{2} + 480 \, C b^{2} \sin \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.535981, size = 414, normalized size = 1.93 \begin{align*} \frac{15 \,{\left (3 \, B a^{2} + 2 \,{\left (3 \, A + 4 \, C\right )} a b + 4 \, B b^{2}\right )} d x +{\left (24 \, A a^{2} \cos \left (d x + c\right )^{4} + 30 \,{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )^{3} + 16 \,{\left (4 \, A + 5 \, C\right )} a^{2} + 160 \, B a b + 40 \,{\left (2 \, A + 3 \, C\right )} b^{2} + 8 \,{\left ({\left (4 \, A + 5 \, C\right )} a^{2} + 10 \, B a b + 5 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (3 \, B a^{2} + 2 \,{\left (3 \, A + 4 \, C\right )} a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \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 = 1.20855, size = 972, normalized size = 4.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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