Optimal. Leaf size=229 \[ -\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right )+\frac{4 a^3 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d} \]
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Rubi [A] time = 0.802126, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3048, 3049, 3033, 3023, 2735, 3770} \[ -\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right )+\frac{4 a^3 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d} \]
Antiderivative was successfully verified.
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Rule 3048
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 ^2(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\int (a+b \cos (c+d x))^3 \left (4 A b+a C \cos (c+d x)-b (4 A-C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac{1}{4} \int (a+b \cos (c+d x))^2 \left (16 a A b+\left (4 A b^2+4 a^2 C+3 b^2 C\right ) \cos (c+d x)-a b (12 A-7 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{a b (12 A-7 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac{1}{12} \int (a+b \cos (c+d x)) \left (48 a^2 A b+a \left (36 A b^2+12 a^2 C+23 b^2 C\right ) \cos (c+d x)-b \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac{1}{24} \int \left (96 a^3 A b+3 \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \cos (c+d x)-4 a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\frac{1}{24} \int \left (96 a^3 A b+3 \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=\frac{1}{8} \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) x-\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}+\left (4 a^3 A b\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{8} \left (8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) x+\frac{4 a^3 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b \left (a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}-\frac{b^2 \left (a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac{a b (12 A-7 C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{A (a+b \cos (c+d x))^4 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 1.34126, size = 274, normalized size = 1.2 \[ \frac{12 (c+d x) \left (24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right )+96 a b \left (4 a^2 C+4 A b^2+3 b^2 C\right ) \sin (c+d x)+24 b^2 \left (C \left (6 a^2+b^2\right )+A b^2\right ) \sin (2 (c+d x))-384 a^3 A b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+384 a^3 A b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{96 a^4 A \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{96 a^4 A \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}+32 a b^3 C \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.061, size = 296, normalized size = 1.3 \begin{align*}{\frac{A{b}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{4}x}{2}}+{\frac{A{b}^{4}c}{2\,d}}+{\frac{C{b}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,C{b}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{b}^{4}Cx}{8}}+{\frac{3\,C{b}^{4}c}{8\,d}}+4\,{\frac{aA{b}^{3}\sin \left ( dx+c \right ) }{d}}+{\frac{4\,C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}a{b}^{3}}{3\,d}}+{\frac{8\,Ca{b}^{3}\sin \left ( dx+c \right ) }{3\,d}}+6\,{a}^{2}A{b}^{2}x+6\,{\frac{A{a}^{2}{b}^{2}c}{d}}+3\,{\frac{{a}^{2}{b}^{2}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+3\,{a}^{2}{b}^{2}Cx+3\,{\frac{{a}^{2}{b}^{2}Cc}{d}}+4\,{\frac{A{a}^{3}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{{a}^{3}bC\sin \left ( dx+c \right ) }{d}}+{\frac{A{a}^{4}\tan \left ( dx+c \right ) }{d}}+{a}^{4}Cx+{\frac{C{a}^{4}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07936, size = 275, normalized size = 1.2 \begin{align*} \frac{96 \,{\left (d x + c\right )} C a^{4} + 576 \,{\left (d x + c\right )} A a^{2} b^{2} + 144 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b^{2} - 128 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{3} + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} + 3 \,{\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 b^{4} + 192 \, A 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 )} + 384 \, C a^{3} b \sin \left (d x + c\right ) + 384 \, A a b^{3} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69908, size = 504, normalized size = 2.2 \begin{align*} \frac{48 \, A a^{3} b \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, A a^{3} b \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (8 \, C a^{4} + 24 \,{\left (2 \, A + C\right )} a^{2} b^{2} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} d x \cos \left (d x + c\right ) +{\left (6 \, C b^{4} \cos \left (d x + c\right )^{4} + 32 \, C a b^{3} \cos \left (d x + c\right )^{3} + 24 \, A a^{4} + 3 \,{\left (24 \, C a^{2} b^{2} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 32 \,{\left (3 \, C a^{3} b +{\left (3 \, A + 2 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, 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 = 1.71155, size = 753, normalized size = 3.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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