Optimal. Leaf size=274 \[ -\frac{b \sin (c+d x) \left (a^2 b (39 A-34 C)+12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{6 d}+\frac{a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{6 d}+\frac{1}{2} b x \left (12 a^2 b B+8 a^3 C+4 a b^2 (2 A+C)+b^3 B\right )-\frac{b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{6 d}+\frac{(a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d} \]
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Rubi [A] time = 0.973756, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {3047, 3049, 3033, 3023, 2735, 3770} \[ -\frac{b \sin (c+d x) \left (a^2 b (39 A-34 C)+12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)\right )}{6 d}+\frac{a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) \cos (c+d x) \left (6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right )}{6 d}+\frac{1}{2} b x \left (12 a^2 b B+8 a^3 C+4 a b^2 (2 A+C)+b^3 B\right )-\frac{b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{6 d}+\frac{(a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d} \]
Antiderivative was successfully verified.
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Rule 3047
Rule 3049
Rule 3033
Rule 3023
Rule 2735
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int (a+b \cos (c+d x))^3 \left (2 (2 A b+a B)+(2 b B+a (A+2 C)) \cos (c+d x)-b (3 A-2 C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \int (a+b \cos (c+d x))^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)-2 b (a A-b B-2 a C) \cos (c+d x)-b (15 A b+6 a B-2 b C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{6} \int (a+b \cos (c+d x)) \left (3 a \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )+b \left (18 a b B-3 a^2 (A-6 C)+2 b^2 (3 A+2 C)\right ) \cos (c+d x)-2 b \left (18 a A b+6 a^2 B-3 b^2 B-8 a b C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac{b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{12} \int \left (6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )+6 b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) \cos (c+d x)-2 b \left (12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)+a^2 (39 A b-34 b C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac{b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{12} \int \left (6 a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )+6 b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) x-\frac{b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac{b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac{1}{2} \left (a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right )\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} b \left (12 a^2 b B+b^3 B+8 a^3 C+4 a b^2 (2 A+C)\right ) x+\frac{a^2 \left (12 A b^2+8 a b B+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b \left (12 a^3 B-24 a b^2 B+a^2 b (39 A-34 C)-2 b^3 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (6 a^2 B-3 b^2 B+2 a b (9 A-4 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}-\frac{b (15 A b+6 a B-2 b C) (a+b \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac{(2 A b+a B) (a+b \cos (c+d x))^3 \tan (c+d x)}{d}+\frac{A (a+b \cos (c+d x))^4 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 4.73348, size = 367, normalized size = 1.34 \[ \frac{6 b (c+d x) \left (12 a^2 b B+8 a^3 C+4 a b^2 (2 A+C)+b^3 B\right )+3 b^2 \sin (c+d x) \left (24 a^2 C+16 a b B+4 A b^2+3 b^2 C\right )-6 a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+6 a^2 \left (a^2 (A+2 C)+8 a b B+12 A b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{12 a^3 (a B+4 A b) \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{12 a^3 (a B+4 A b) \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+\frac{3 a^4 A}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{3 a^4 A}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+3 b^3 (4 a C+b B) \sin (2 (c+d x))+b^4 C \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 374, normalized size = 1.4 \begin{align*}{\frac{A{b}^{4}\sin \left ( dx+c \right ) }{d}}+{\frac{{b}^{4}B\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{b}^{4}Bx}{2}}+{\frac{{b}^{4}Bc}{2\,d}}+{\frac{C{b}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{2\,C{b}^{4}\sin \left ( dx+c \right ) }{3\,d}}+4\,aA{b}^{3}x+4\,{\frac{Aa{b}^{3}c}{d}}+4\,{\frac{a{b}^{3}B\sin \left ( dx+c \right ) }{d}}+2\,{\frac{C\cos \left ( dx+c \right ) a{b}^{3}\sin \left ( dx+c \right ) }{d}}+2\,a{b}^{3}Cx+2\,{\frac{Ca{b}^{3}c}{d}}+6\,{\frac{{a}^{2}A{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+6\,{a}^{2}{b}^{2}Bx+6\,{\frac{B{a}^{2}{b}^{2}c}{d}}+6\,{\frac{{a}^{2}{b}^{2}C\sin \left ( dx+c \right ) }{d}}+4\,{\frac{A{a}^{3}b\tan \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{3}bB\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{a}^{3}bCx+4\,{\frac{C{a}^{3}bc}{d}}+{\frac{A{a}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{a}^{4}B\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01502, size = 420, normalized size = 1.53 \begin{align*} \frac{48 \,{\left (d x + c\right )} C a^{3} b + 72 \,{\left (d x + c\right )} B a^{2} b^{2} + 48 \,{\left (d x + c\right )} A a b^{3} + 12 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b^{4} - 3 \, A a^{4}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B a^{3} b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, A a^{2} b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 72 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 48 \, B a b^{3} \sin \left (d x + c\right ) + 12 \, A b^{4} \sin \left (d x + c\right ) + 12 \, B a^{4} \tan \left (d x + c\right ) + 48 \, A a^{3} b \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07179, size = 628, normalized size = 2.29 \begin{align*} \frac{6 \,{\left (8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 4 \,{\left (2 \, A + C\right )} a b^{3} + B b^{4}\right )} d x \cos \left (d x + c\right )^{2} + 3 \,{\left ({\left (A + 2 \, C\right )} a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left ({\left (A + 2 \, C\right )} a^{4} + 8 \, B a^{3} b + 12 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (2 \, C b^{4} \cos \left (d x + c\right )^{4} + 3 \, A a^{4} + 3 \,{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (18 \, C a^{2} b^{2} + 12 \, B a b^{3} +{\left (3 \, A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \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 = 1.29781, size = 730, normalized size = 2.66 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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