Optimal. Leaf size=227 \[ \frac{\left (a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac{a b \left (6 a^2 C+40 A b^2+29 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{1}{2} a b x \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]
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Rubi [A] time = 0.792801, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3050, 3049, 3033, 3023, 2735, 3770} \[ \frac{\left (a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right ) \sin (c+d x)}{15 d}+\frac{a b \left (6 a^2 C+40 A b^2+29 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{1}{2} a b x \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 3050
Rule 3049
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} \int (a+b \cos (c+d x))^3 \left (5 a A+b (5 A+4 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{a C (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{20} \int (a+b \cos (c+d x))^2 \left (20 a^2 A+4 a b (10 A+7 C) \cos (c+d x)+4 \left (3 a^2 C+b^2 (5 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac{a C (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{60} \int (a+b \cos (c+d x)) \left (60 a^3 A+4 b \left (9 a^2 (5 A+3 C)+2 b^2 (5 A+4 C)\right ) \cos (c+d x)+4 a \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac{a C (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{120} \int \left (120 a^4 A+60 a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \cos (c+d x)+8 \left (6 a^4 C+2 b^4 (5 A+4 C)+a^2 b^2 (85 A+56 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{\left (6 a^4 C+2 b^4 (5 A+4 C)+a^2 b^2 (85 A+56 C)\right ) \sin (c+d x)}{15 d}+\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac{a C (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{120} \int \left (120 a^4 A+60 a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) x+\frac{\left (6 a^4 C+2 b^4 (5 A+4 C)+a^2 b^2 (85 A+56 C)\right ) \sin (c+d x)}{15 d}+\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac{a C (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) x+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\left (6 a^4 C+2 b^4 (5 A+4 C)+a^2 b^2 (85 A+56 C)\right ) \sin (c+d x)}{15 d}+\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{15 d}+\frac{a C (a+b \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+b \cos (c+d x))^4 \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 1.03833, size = 226, normalized size = 1. \[ \frac{120 a b (c+d x) \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )+240 a b \left (C \left (a^2+b^2\right )+A b^2\right ) \sin (2 (c+d x))+30 \left (12 a^2 b^2 (4 A+3 C)+8 a^4 C+b^4 (6 A+5 C)\right ) \sin (c+d x)+5 b^2 \left (24 a^2 C+4 A b^2+5 b^2 C\right ) \sin (3 (c+d x))-240 a^4 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+240 a^4 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+30 a b^3 C \sin (4 (c+d x))+3 b^4 C \sin (5 (c+d x))}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 364, normalized size = 1.6 \begin{align*}{\frac{A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{4}}{3\,d}}+{\frac{2\,A{b}^{4}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{8\,C{b}^{4}\sin \left ( dx+c \right ) }{15\,d}}+{\frac{C{b}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{4\,C{b}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+2\,{\frac{aA{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+2\,aA{b}^{3}x+2\,{\frac{aA{b}^{3}c}{d}}+{\frac{Ca{b}^{3}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{3\,Ca{b}^{3}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{3\,a{b}^{3}Cx}{2}}+{\frac{3\,Ca{b}^{3}c}{2\,d}}+6\,{\frac{{a}^{2}A{b}^{2}\sin \left ( dx+c \right ) }{d}}+2\,{\frac{C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{2}{b}^{2}}{d}}+4\,{\frac{{a}^{2}{b}^{2}C\sin \left ( dx+c \right ) }{d}}+4\,A{a}^{3}bx+4\,{\frac{A{a}^{3}bc}{d}}+2\,{\frac{{a}^{3}bC\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+2\,{a}^{3}bCx+2\,{\frac{{a}^{3}bCc}{d}}+{\frac{A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{4}C\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00588, size = 313, normalized size = 1.38 \begin{align*} \frac{480 \,{\left (d x + c\right )} A a^{3} b + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 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 )} C a b^{3} - 40 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} + 8 \,{\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 )} C b^{4} + 120 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 120 \, C a^{4} \sin \left (d x + c\right ) + 720 \, A a^{2} b^{2} \sin \left (d x + c\right )}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62882, size = 473, normalized size = 2.08 \begin{align*} \frac{15 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (4 \,{\left (2 \, A + C\right )} a^{3} b +{\left (4 \, A + 3 \, C\right )} a b^{3}\right )} d x +{\left (6 \, C b^{4} \cos \left (d x + c\right )^{4} + 30 \, C a b^{3} \cos \left (d x + c\right )^{3} + 30 \, C a^{4} + 60 \,{\left (3 \, A + 2 \, C\right )} a^{2} b^{2} + 4 \,{\left (5 \, A + 4 \, C\right )} b^{4} + 2 \,{\left (30 \, C a^{2} b^{2} +{\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \,{\left (4 \, C a^{3} b +{\left (4 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, 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 = 1.78789, size = 1017, normalized size = 4.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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