Optimal. Leaf size=227 \[ \frac{a \left (2 a^2 (4 A+5 C)+15 b^2 (2 A+3 C)\right ) \tan (c+d x)}{15 d}+\frac{b \left (3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left (2 a^2 (4 A+5 C)+3 A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{3 b \left (5 a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{3 A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d} \]
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Rubi [A] time = 0.71411, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3048, 3047, 3031, 3021, 2748, 3767, 8, 3770} \[ \frac{a \left (2 a^2 (4 A+5 C)+15 b^2 (2 A+3 C)\right ) \tan (c+d x)}{15 d}+\frac{b \left (3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left (2 a^2 (4 A+5 C)+3 A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{3 b \left (5 a^2 (3 A+4 C)+2 A b^2\right ) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{3 A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d} \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3047
Rule 3031
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} \int (a+b \cos (c+d x))^2 \left (3 A b+a (4 A+5 C) \cos (c+d x)+b (A+5 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac{3 A b (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{20} \int (a+b \cos (c+d x)) \left (2 \left (3 A b^2+2 a^2 (4 A+5 C)\right )+a b (29 A+40 C) \cos (c+d x)+b^2 (7 A+20 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a \left (3 A b^2+2 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{3 A b (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{1}{60} \int \left (-9 b \left (2 A b^2+5 a^2 (3 A+4 C)\right )-4 a \left (15 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \cos (c+d x)-3 b^3 (7 A+20 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{3 b \left (2 A b^2+5 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{a \left (3 A b^2+2 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{3 A b (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{1}{120} \int \left (-8 a \left (15 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )-15 b \left (4 b^2 (A+2 C)+3 a^2 (3 A+4 C)\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{3 b \left (2 A b^2+5 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{a \left (3 A b^2+2 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{3 A b (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{8} \left (b \left (4 b^2 (A+2 C)+3 a^2 (3 A+4 C)\right )\right ) \int \sec (c+d x) \, dx+\frac{1}{15} \left (a \left (15 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{b \left (4 b^2 (A+2 C)+3 a^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 b \left (2 A b^2+5 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{a \left (3 A b^2+2 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{3 A b (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{\left (a \left (15 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac{b \left (4 b^2 (A+2 C)+3 a^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left (15 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac{3 b \left (2 A b^2+5 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{40 d}+\frac{a \left (3 A b^2+2 a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{30 d}+\frac{3 A b (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac{A (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 2.24835, size = 150, normalized size = 0.66 \[ \frac{15 b \left (3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (8 a \left (5 \left (a^2 (2 A+C)+3 A b^2\right ) \tan ^2(c+d x)+15 \left (a^2+3 b^2\right ) (A+C)+3 a^2 A \tan ^4(c+d x)\right )+15 b \left (3 a^2 (3 A+4 C)+4 A b^2\right ) \sec (c+d x)+90 a^2 A b \sec ^3(c+d x)\right )}{120 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 338, normalized size = 1.5 \begin{align*}{\frac{A{b}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{C{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{aA{b}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{aA{b}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{d}}+3\,{\frac{Ca{b}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{3\,A{a}^{2}b\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{9\,A{a}^{2}b\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{9\,A{a}^{2}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{3\,{a}^{2}bC\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{3\,{a}^{2}bC\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{8\,A{a}^{3}\tan \left ( dx+c \right ) }{15\,d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{4\,A{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+{\frac{2\,{a}^{3}C\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01956, size = 400, normalized size = 1.76 \begin{align*} \frac{16 \,{\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 )} A a^{3} + 80 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} + 240 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{2} - 45 \, A a^{2} b{\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 )} - 180 \, C a^{2} 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 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 )} + 120 \, C b^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 720 \, C a b^{2} \tan \left (d x + c\right )}{240 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59986, size = 552, normalized size = 2.43 \begin{align*} \frac{15 \,{\left (3 \,{\left (3 \, A + 4 \, C\right )} a^{2} b + 4 \,{\left (A + 2 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (3 \,{\left (3 \, A + 4 \, C\right )} a^{2} b + 4 \,{\left (A + 2 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (90 \, A a^{2} b \cos \left (d x + c\right ) + 8 \,{\left (2 \,{\left (4 \, A + 5 \, C\right )} a^{3} + 15 \,{\left (2 \, A + 3 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} + 24 \, A a^{3} + 15 \,{\left (3 \,{\left (3 \, A + 4 \, C\right )} a^{2} b + 4 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 8 \,{\left ({\left (4 \, A + 5 \, C\right )} a^{3} + 15 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, 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(-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 = 1.5271, size = 886, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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