Optimal. Leaf size=246 \[ \frac{a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \sin (c+d x)}{12 d}-\frac{b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{24 d}+\frac{\left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{8 d}+\frac{1}{8} x \left (24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right )+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac{4 a b^3 C \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.846502, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4095, 4094, 4076, 4047, 8, 4045, 3770} \[ \frac{a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \sin (c+d x)}{12 d}-\frac{b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \tan (c+d x)}{24 d}+\frac{\left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{8 d}+\frac{1}{8} x \left (24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right )+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac{4 a b^3 C \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 4095
Rule 4094
Rule 4076
Rule 4047
Rule 8
Rule 4045
Rule 3770
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac{1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+a (3 A+4 C) \sec (c+d x)-b (A-4 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A b \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac{1}{12} \int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left (3 \left (4 A b^2+a^2 (3 A+4 C)\right )+2 a b (7 A+12 C) \sec (c+d x)-b^2 (7 A-12 C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac{A b \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac{1}{24} \int \cos (c+d x) (a+b \sec (c+d x)) \left (2 \left (12 A b^3+\frac{1}{2} a^2 (46 A b+72 b C)\right )+a \left (3 a^2 (3 A+4 C)+2 b^2 (13 A+36 C)\right ) \sec (c+d x)-b \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac{A b \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac{b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}+\frac{1}{24} \int \cos (c+d x) \left (2 a \left (12 A b^3+\frac{1}{2} a^2 (46 A b+72 b C)\right )+3 \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \sec (c+d x)+96 a b^3 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac{A b \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac{b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}+\frac{1}{24} \int \cos (c+d x) \left (2 a \left (12 A b^3+\frac{1}{2} a^2 (46 A b+72 b C)\right )+96 a b^3 C \sec ^2(c+d x)\right ) \, dx+\frac{1}{8} \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac{1}{8} \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x+\frac{a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \sin (c+d x)}{12 d}+\frac{\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac{A b \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac{b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}+\left (4 a b^3 C\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{8} \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) x+\frac{4 a b^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \sin (c+d x)}{12 d}+\frac{\left (4 A b^2+a^2 (3 A+4 C)\right ) \cos (c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{8 d}+\frac{A b \cos ^2(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac{A \cos ^3(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{4 d}-\frac{b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \tan (c+d x)}{24 d}\\ \end{align*}
Mathematica [A] time = 1.39823, size = 270, normalized size = 1.1 \[ \frac{12 (c+d x) \left (24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right )+24 a^2 \left (a^2 (A+C)+6 A b^2\right ) \sin (2 (c+d x))+96 a b \left (a^2 (3 A+4 C)+4 A b^2\right ) \sin (c+d x)+32 a^3 A b \sin (3 (c+d x))+3 a^4 A \sin (4 (c+d x))-384 a b^3 C \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+384 a b^3 C \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{96 b^4 C \sin \left (\frac{1}{2} (c+d x)\right )}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{96 b^4 C \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 296, normalized size = 1.2 \begin{align*}{\frac{A{a}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,A{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{4}Ax}{8}}+{\frac{3\,A{a}^{4}c}{8\,d}}+{\frac{{a}^{4}C\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{4}Cx}{2}}+{\frac{{a}^{4}Cc}{2\,d}}+{\frac{4\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ){a}^{3}b}{3\,d}}+{\frac{8\,A{a}^{3}b\sin \left ( dx+c \right ) }{3\,d}}+4\,{\frac{{a}^{3}bC\sin \left ( dx+c \right ) }{d}}+3\,{\frac{A{a}^{2}{b}^{2}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }{d}}+3\,A{a}^{2}{b}^{2}x+3\,{\frac{A{a}^{2}{b}^{2}c}{d}}+6\,C{a}^{2}{b}^{2}x+6\,{\frac{C{a}^{2}{b}^{2}c}{d}}+4\,{\frac{Aa{b}^{3}\sin \left ( dx+c \right ) }{d}}+4\,{\frac{Ca{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+A{b}^{4}x+{\frac{A{b}^{4}c}{d}}+{\frac{C{b}^{4}\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01347, size = 275, normalized size = 1.12 \begin{align*} \frac{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 )} A a^{4} + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 128 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} b + 144 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 576 \,{\left (d x + c\right )} C a^{2} b^{2} + 96 \,{\left (d x + c\right )} A b^{4} + 192 \, C a b^{3}{\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 \, C b^{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 = 0.584909, size = 504, normalized size = 2.05 \begin{align*} \frac{48 \, C a b^{3} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, C a b^{3} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \,{\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} d x \cos \left (d x + c\right ) +{\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, A a^{3} b \cos \left (d x + c\right )^{3} + 24 \, C b^{4} + 3 \,{\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 32 \,{\left ({\left (2 \, A + 3 \, C\right )} a^{3} b + 3 \, A 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 = 1.28809, size = 753, normalized size = 3.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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