Optimal. Leaf size=355 \[ \frac{\left (84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+35 b^4 (2 A+3 C)\right ) \tan (c+d x)}{105 d}+\frac{a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{210 d}+\frac{\left (3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{105 d}+\frac{a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{\left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{21 d} \]
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Rubi [A] time = 1.23683, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3048, 3047, 3031, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{\left (84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)+35 b^4 (2 A+3 C)\right ) \tan (c+d x)}{105 d}+\frac{a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{210 d}+\frac{\left (3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{105 d}+\frac{a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{\left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{21 d} \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3047
Rule 3031
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{7} \int (a+b \cos (c+d x))^3 \left (4 A b+a (6 A+7 C) \cos (c+d x)+b (2 A+7 C) \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx\\ &=\frac{2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{42} \int (a+b \cos (c+d x))^2 \left (6 \left (2 A b^2+a^2 (6 A+7 C)\right )+4 a b (17 A+21 C) \cos (c+d x)+2 b^2 (10 A+21 C) \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx\\ &=\frac{\left (2 A b^2+a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{210} \int (a+b \cos (c+d x)) \left (4 b \left (6 A b^2+a^2 (103 A+126 C)\right )+2 a \left (12 a^2 (6 A+7 C)+b^2 (244 A+315 C)\right ) \cos (c+d x)+2 b \left (6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac{a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac{\left (2 A b^2+a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac{1}{840} \int \left (-24 \left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right )-420 a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x)-8 b^2 \left (6 a^2 (6 A+7 C)+b^2 (62 A+105 C)\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{\left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{105 d}+\frac{a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac{\left (2 A b^2+a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}-\frac{\int \left (-1260 a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )-24 \left (35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx}{2520}\\ &=\frac{\left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{105 d}+\frac{a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac{\left (2 A b^2+a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{2} \left (a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int \sec ^3(c+d x) \, dx-\frac{1}{105} \left (-35 b^4 (2 A+3 C)-84 a^2 b^2 (4 A+5 C)-8 a^4 (6 A+7 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{4 d}+\frac{\left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{105 d}+\frac{a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac{\left (2 A b^2+a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}+\frac{1}{4} \left (a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int \sec (c+d x) \, dx-\frac{\left (35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 d}\\ &=\frac{a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{\left (35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \tan (c+d x)}{105 d}+\frac{a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{4 d}+\frac{\left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{105 d}+\frac{a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac{\left (2 A b^2+a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 2.2006, size = 233, normalized size = 0.66 \[ \frac{105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (84 a^2 \left (a^2 (3 A+C)+6 A b^2\right ) \tan ^4(c+d x)+140 \left (6 a^2 b^2 (2 A+C)+a^4 (3 A+2 C)+A b^4\right ) \tan ^2(c+d x)+70 a b \left (a^2 (5 A+6 C)+6 A b^2\right ) \sec ^3(c+d x)+105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \sec (c+d x)+420 \left (6 a^2 b^2+a^4+b^4\right ) (A+C)+280 a^3 A b \sec ^5(c+d x)+60 a^4 A \tan ^6(c+d x)\right )}{420 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 591, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04678, size = 637, normalized size = 1.79 \begin{align*} \frac{24 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a^{4} + 56 \,{\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 a^{4} + 336 \,{\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^{2} b^{2} + 1680 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b^{2} + 280 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{4} - 35 \, A a^{3} b{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, C a^{3} 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 )} - 210 \, A 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 )} - 840 \, C 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 )} + 840 \, C b^{4} \tan \left (d x + c\right )}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66217, size = 783, normalized size = 2.21 \begin{align*} \frac{105 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \,{\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \,{\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (4 \,{\left (8 \,{\left (6 \, A + 7 \, C\right )} a^{4} + 84 \,{\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 35 \,{\left (2 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 280 \, A a^{3} b \cos \left (d x + c\right ) + 105 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \,{\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 60 \, A a^{4} + 4 \,{\left (4 \,{\left (6 \, A + 7 \, C\right )} a^{4} + 42 \,{\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 12 \,{\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \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 = 1.46617, size = 1728, normalized size = 4.87 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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