Optimal. Leaf size=485 \[ \frac{\left (-645 a^2 b^2+630 a^4+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac{\sqrt{a^2-b^2} \left (-29 a^2 b^2+42 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^8 d}-\frac{\left (-60 a^2 b^2+63 a^4+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left (-54 a^2 b^2+63 a^4+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left (-187 a^2 b^2+210 a^4+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{30 a^2 b^5 d}-\frac{\left (-79 a^2 b^2+84 a^4+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{8 a b^6 d}+\frac{a x \left (-200 a^2 b^2+168 a^4+45 b^4\right )}{8 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 1.71693, antiderivative size = 485, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2896, 3047, 3049, 3023, 2735, 2660, 618, 204} \[ \frac{\left (-645 a^2 b^2+630 a^4+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac{\sqrt{a^2-b^2} \left (-29 a^2 b^2+42 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^8 d}-\frac{\left (-60 a^2 b^2+63 a^4+5 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{\left (-54 a^2 b^2+63 a^4+4 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left (-187 a^2 b^2+210 a^4+15 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{30 a^2 b^5 d}-\frac{\left (-79 a^2 b^2+84 a^4+8 b^4\right ) \sin (c+d x) \cos (c+d x)}{8 a b^6 d}+\frac{a x \left (-200 a^2 b^2+168 a^4+45 b^4\right )}{8 b^8}-\frac{b \sin ^4(c+d x) \cos (c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\sin ^6(c+d x) \cos (c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{\sin ^3(c+d x) \cos (c+d x)}{3 a d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2896
Rule 3047
Rule 3049
Rule 3023
Rule 2735
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x) \sin ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{\int \frac{\sin ^4(c+d x) \left (20 \left (21 a^4-20 a^2 b^2+2 b^4\right )-12 a b \left (3 a^2-5 b^2\right ) \sin (c+d x)-12 \left (42 a^4-44 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{240 a^2 b^2}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\int \frac{\sin ^3(c+d x) \left (-32 \left (63 a^6-123 a^4 b^2+65 a^2 b^4-5 b^6\right )+8 a b \left (21 a^4-41 a^2 b^2+20 b^4\right ) \sin (c+d x)+24 \left (105 a^6-209 a^4 b^2+114 a^2 b^4-10 b^6\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{480 a^2 b^3 \left (a^2-b^2\right )}\\ &=\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\sin ^2(c+d x) \left (120 \left (a^2-b^2\right )^2 \left (63 a^4-54 a^2 b^2+4 b^4\right )-24 a b \left (21 a^2-10 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-48 \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{480 a^2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{\sin (c+d x) \left (-96 a \left (a^2-b^2\right )^2 \left (210 a^4-187 a^2 b^2+15 b^4\right )+24 a^2 b \left (105 a^2-62 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+360 a \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^2 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac{\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{360 a^2 \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right )-24 a^3 b \left (420 a^2-311 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-96 a^2 \left (a^2-b^2\right )^2 \left (630 a^4-645 a^2 b^2+91 b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{2880 a^2 b^6 \left (a^2-b^2\right )^2}\\ &=\frac{\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac{\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac{\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\int \frac{360 a^2 b \left (a^2-b^2\right )^2 \left (84 a^4-79 a^2 b^2+8 b^4\right )+360 a^3 \left (a^2-b^2\right )^2 \left (168 a^4-200 a^2 b^2+45 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{2880 a^2 b^7 \left (a^2-b^2\right )^2}\\ &=\frac{a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}+\frac{\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac{\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac{\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}-\frac{\left (\left (a^2-b^2\right ) \left (42 a^4-29 a^2 b^2+2 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 b^8}\\ &=\frac{a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}+\frac{\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac{\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac{\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}-\frac{\left (\left (a^2-b^2\right ) \left (42 a^4-29 a^2 b^2+2 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^8 d}\\ &=\frac{a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}+\frac{\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac{\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac{\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}+\frac{\left (2 \left (a^2-b^2\right ) \left (42 a^4-29 a^2 b^2+2 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^8 d}\\ &=\frac{a \left (168 a^4-200 a^2 b^2+45 b^4\right ) x}{8 b^8}-\frac{\sqrt{a^2-b^2} \left (42 a^4-29 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^8 d}+\frac{\left (630 a^4-645 a^2 b^2+91 b^4\right ) \cos (c+d x)}{30 b^7 d}-\frac{\left (84 a^4-79 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 a b^6 d}+\frac{\left (210 a^4-187 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a^2 b^5 d}+\frac{\cos (c+d x) \sin ^3(c+d x)}{3 a d (a+b \sin (c+d x))^2}-\frac{b \cos (c+d x) \sin ^4(c+d x)}{12 a^2 d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-60 a^2 b^2+5 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac{7 a \cos (c+d x) \sin ^5(c+d x)}{20 b^2 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \sin ^6(c+d x)}{5 b d (a+b \sin (c+d x))^2}-\frac{\left (63 a^4-54 a^2 b^2+4 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{12 a^2 b^4 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 12.8154, size = 517, normalized size = 1.07 \[ \frac{\frac{\left (a^2-b^2\right )^2 \left (30240 a^5 b^2 \sin (2 (c+d x))-96000 a^4 b^3 c \sin (c+d x)-96000 a^4 b^3 d x \sin (c+d x)-32640 a^3 b^4 \sin (2 (c+d x))+420 a^3 b^4 \sin (4 (c+d x))+21600 a^2 b^5 c \sin (c+d x)+21600 a^2 b^5 d x \sin (c+d x)-3360 a^4 b^3 \cos (3 (c+d x))+3580 a^2 b^5 \cos (3 (c+d x))+84 a^2 b^5 \cos (5 (c+d x))-120 a b^2 \left (-200 a^2 b^2+168 a^4+45 b^4\right ) (c+d x) \cos (2 (c+d x))+10 b \left (-3792 a^4 b^2+216 a^2 b^4+4032 a^6+59 b^6\right ) \cos (c+d x)-27840 a^5 b^2 c-13200 a^3 b^4 c-27840 a^5 b^2 d x-13200 a^3 b^4 d x+80640 a^6 b c \sin (c+d x)+80640 a^6 b d x \sin (c+d x)+40320 a^7 c+40320 a^7 d x+5675 a b^6 \sin (2 (c+d x))-374 a b^6 \sin (4 (c+d x))-21 a b^6 \sin (6 (c+d x))+5400 a b^6 c+5400 a b^6 d x-526 b^7 \cos (3 (c+d x))-58 b^7 \cos (5 (c+d x))-6 b^7 \cos (7 (c+d x))\right )}{(a+b \sin (c+d x))^2}-1920 \left (a^2-b^2\right )^{5/2} \left (-29 a^2 b^2+42 a^4+2 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{1920 b^8 d (a-b)^2 (a+b)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 1676, normalized size = 3.5 \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.69394, size = 2326, normalized size = 4.8 \begin{align*} \text{result too large to display} \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 [A] time = 1.40237, size = 977, normalized size = 2.01 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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