Optimal. Leaf size=307 \[ \frac{4 a b \left (2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{15 d}+\frac{\left (12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 b^4 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b \left (a^2 (39 A+50 C)+4 A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{\left (10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right ) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\left (5 a^2 (5 A+6 C)+12 A b^2\right ) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{120 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}+\frac{2 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d} \]
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Rubi [A] time = 1.12453, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{4 a b \left (2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right ) \tan (c+d x)}{15 d}+\frac{\left (12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 b^4 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b \left (a^2 (39 A+50 C)+4 A b^2\right ) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{\left (10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right ) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\left (5 a^2 (5 A+6 C)+12 A b^2\right ) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{120 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}+\frac{2 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d} \]
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))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^7(c+d x) \, dx &=\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{6} \int (a+b \cos (c+d x))^3 \left (4 A b+a (5 A+6 C) \cos (c+d x)+b (A+6 C) \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx\\ &=\frac{2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{30} \int (a+b \cos (c+d x))^2 \left (12 A b^2+5 a^2 (5 A+6 C)+2 a b (23 A+30 C) \cos (c+d x)+3 b^2 (3 A+10 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{120} \int (a+b \cos (c+d x)) \left (6 b \left (4 A b^2+a^2 (39 A+50 C)\right )+a \left (15 a^2 (5 A+6 C)+8 b^2 (32 A+45 C)\right ) \cos (c+d x)+b \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{1}{360} \int \left (-3 \left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right )-96 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \cos (c+d x)-3 b^2 \left (24 b^2 (2 A+5 C)+5 a^2 (5 A+6 C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{1}{720} \int \left (-192 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )-45 \left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac{1}{15} \left (4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right )\right ) \int \sec ^2(c+d x) \, dx-\frac{1}{16} \left (-8 b^4 (A+2 C)-12 a^2 b^2 (3 A+4 C)-a^4 (5 A+6 C)\right ) \int \sec (c+d x) \, dx\\ &=\frac{\left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac{\left (4 a b \left (5 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{\left (8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{4 a b \left (5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac{\left (24 A b^4+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{a b \left (4 A b^2+a^2 (39 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac{\left (12 A b^2+5 a^2 (5 A+6 C)\right ) (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{120 d}+\frac{2 A b (a+b \cos (c+d x))^3 \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac{A (a+b \cos (c+d x))^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 4.90567, size = 204, normalized size = 0.66 \[ \frac{15 \left (12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 b^4 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (64 a b \left (5 \left (a^2 (2 A+C)+A b^2\right ) \tan ^2(c+d x)+15 \left (a^2+b^2\right ) (A+C)+3 a^2 A \tan ^4(c+d x)\right )+10 a^2 \left (a^2 (5 A+6 C)+36 A b^2\right ) \sec ^3(c+d x)+15 \left (12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+8 A b^4\right ) \sec (c+d x)+40 a^4 A \sec ^5(c+d x)\right )}{240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 511, 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 = 1.0382, size = 629, normalized size = 2.05 \begin{align*} \frac{128 \,{\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} b + 640 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} b + 640 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b^{3} - 5 \, A a^{4}{\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 )} - 30 \, C a^{4}{\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 \, A a^{2} b^{2}{\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 )} - 720 \, C a^{2} b^{2}{\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 \, A b^{4}{\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 \, C b^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1920 \, C a b^{3} \tan \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72725, size = 716, normalized size = 2.33 \begin{align*} \frac{15 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \,{\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \,{\left (A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \,{\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \,{\left (A + 2 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (192 \, A a^{3} b \cos \left (d x + c\right ) + 64 \,{\left (2 \,{\left (4 \, A + 5 \, C\right )} a^{3} b + 5 \,{\left (2 \, A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 40 \, A a^{4} + 15 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 12 \,{\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 64 \,{\left ({\left (4 \, A + 5 \, C\right )} a^{3} b + 5 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \,{\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \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 = 1.55314, size = 1485, normalized size = 4.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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