Optimal. Leaf size=227 \[ \frac{\left (a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right ) \tan (c+d x)}{15 d}+\frac{a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a b \left (6 a^2 C+40 A b^2+29 b^2 C\right ) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+a^4 A x+\frac{a C \tan (c+d x) (a+b \sec (c+d x))^3}{5 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d} \]
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Rubi [A] time = 0.478201, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {4057, 4056, 4048, 3770, 3767, 8} \[ \frac{\left (a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right ) \tan (c+d x)}{15 d}+\frac{a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a b \left (6 a^2 C+40 A b^2+29 b^2 C\right ) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+a^4 A x+\frac{a C \tan (c+d x) (a+b \sec (c+d x))^3}{5 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d} \]
Antiderivative was successfully verified.
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Rule 4057
Rule 4056
Rule 4048
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{5} \int (a+b \sec (c+d x))^3 \left (5 a A+b (5 A+4 C) \sec (c+d x)+4 a C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a C (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac{C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{20} \int (a+b \sec (c+d x))^2 \left (20 a^2 A+4 a b (10 A+7 C) \sec (c+d x)+4 \left (3 a^2 C+b^2 (5 A+4 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac{a C (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac{C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{60} \int (a+b \sec (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 ) \sec (c+d x)+4 a \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac{a C (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac{C (a+b \sec (c+d x))^4 \tan (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 ) \sec (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 ) \sec ^2(c+d x)\right ) \, dx\\ &=a^4 A x+\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac{a C (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac{C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac{1}{2} \left (a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right )\right ) \int \sec (c+d x) \, dx+\frac{1}{15} \left (6 a^4 C+2 b^4 (5 A+4 C)+a^2 b^2 (85 A+56 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=a^4 A x+\frac{a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac{a C (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac{C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}-\frac{\left (6 a^4 C+2 b^4 (5 A+4 C)+a^2 b^2 (85 A+56 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=a^4 A x+\frac{a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\left (6 a^4 C+2 b^4 (5 A+4 C)+a^2 b^2 (85 A+56 C)\right ) \tan (c+d x)}{15 d}+\frac{a b \left (40 A b^2+6 a^2 C+29 b^2 C\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac{\left (3 a^2 C+b^2 (5 A+4 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac{a C (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac{C (a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}\\ \end{align*}
Mathematica [B] time = 2.5729, size = 503, normalized size = 2.22 \[ \frac{\sec ^5(c+d x) \left (A \cos ^2(c+d x)+C\right ) \left (-120 a b \left (4 a^2 (2 A+C)+b^2 (4 A+3 C)\right ) \cos ^5(c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+180 a^2 A b^2 \sin (c+d x)+270 a^2 A b^2 \sin (3 (c+d x))+90 a^2 A b^2 \sin (5 (c+d x))+150 a^4 A (c+d x) \cos (c+d x)+75 a^4 A (c+d x) \cos (3 (c+d x))+15 a^4 A c \cos (5 (c+d x))+15 a^4 A d x \cos (5 (c+d x))+240 a^2 b^2 C \sin (c+d x)+300 a^2 b^2 C \sin (3 (c+d x))+60 a^2 b^2 C \sin (5 (c+d x))+120 a^3 b C \sin (2 (c+d x))+60 a^3 b C \sin (4 (c+d x))+30 a^4 C \sin (c+d x)+45 a^4 C \sin (3 (c+d x))+15 a^4 C \sin (5 (c+d x))+120 a A b^3 \sin (2 (c+d x))+60 a A b^3 \sin (4 (c+d x))+210 a b^3 C \sin (2 (c+d x))+45 a b^3 C \sin (4 (c+d x))+40 A b^4 \sin (c+d x)+50 A b^4 \sin (3 (c+d x))+10 A b^4 \sin (5 (c+d x))+80 b^4 C \sin (c+d x)+40 b^4 C \sin (3 (c+d x))+8 b^4 C \sin (5 (c+d x))\right )}{120 d (A \cos (2 (c+d x))+A+2 C)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 377, normalized size = 1.7 \begin{align*}{a}^{4}Ax+{\frac{A{a}^{4}c}{d}}+{\frac{{a}^{4}C\tan \left ( dx+c \right ) }{d}}+4\,{\frac{A{a}^{3}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{{a}^{3}bC\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+2\,{\frac{{a}^{3}bC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{A{a}^{2}{b}^{2}\tan \left ( dx+c \right ) }{d}}+4\,{\frac{C{a}^{2}{b}^{2}\tan \left ( dx+c \right ) }{d}}+2\,{\frac{C{a}^{2}{b}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+2\,{\frac{Aa{b}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+2\,{\frac{Aa{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{Ca{b}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{3\,Ca{b}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{3\,Ca{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,A{b}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{A{b}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{8\,C{b}^{4}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{C{b}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{4\,C{b}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979258, size = 429, normalized size = 1.89 \begin{align*} \frac{60 \,{\left (d x + c\right )} A a^{4} + 120 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b^{2} + 20 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{4} + 4 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C b^{4} - 15 \, C a b^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, C a^{3} b{\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 )} - 60 \, A a b^{3}{\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 )} + 240 \, A a^{3} b \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 60 \, C a^{4} \tan \left (d x + c\right ) + 360 \, A a^{2} b^{2} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.582037, size = 610, normalized size = 2.69 \begin{align*} \frac{60 \, A a^{4} d x \cos \left (d x + c\right )^{5} + 15 \,{\left (4 \,{\left (2 \, A + C\right )} a^{3} b +{\left (4 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} \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 )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (30 \, C a b^{3} \cos \left (d x + c\right ) + 6 \, C b^{4} + 2 \,{\left (15 \, C a^{4} + 30 \,{\left (3 \, A + 2 \, C\right )} a^{2} b^{2} + 2 \,{\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \,{\left (4 \, C a^{3} b +{\left (4 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (30 \, C a^{2} b^{2} +{\left (5 \, A + 4 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec{\left (c + d x \right )}\right )^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22534, size = 1050, normalized size = 4.63 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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