Optimal. Leaf size=539 \[ \frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}-\frac {\sqrt {a-b} b \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))} \]
[Out]
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Rubi [A]
time = 1.61, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2975,
3126, 3128, 3102, 2814, 2738, 214} \begin {gather*} \frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{20 a^4 b^2 d (a \cos (c+d x)+b)}-\frac {b \sqrt {a-b} \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{24 a^5 b^2 d}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{60 a^3 b^2 d (a \cos (c+d x)+b)^2}+\frac {x \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right )}{16 a^9}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 2738
Rule 2814
Rule 2975
Rule 3102
Rule 3126
Rule 3128
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^6(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}-\frac {\int \frac {\cos ^5(c+d x) \left (30 \left (6 a^4-35 a^2 b^2+32 b^4\right )+30 a b \left (3 a^2-2 b^2\right ) \cos (c+d x)-20 \left (12 a^4-65 a^2 b^2+56 b^4\right ) \cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^3} \, dx}{600 a^2 b^2}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^4(c+d x) \left (100 \left (9 a^6-69 a^4 b^2+116 a^2 b^4-56 b^6\right )+20 a b \left (15 a^4-31 a^2 b^2+16 b^4\right ) \cos (c+d x)-40 \left (30 a^6-215 a^4 b^2+353 a^2 b^4-168 b^6\right ) \cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^2} \, dx}{1200 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {\cos ^3(c+d x) \left (240 \left (a^2-b^2\right )^2 \left (15 a^4-110 a^2 b^2+112 b^4\right )+40 a b \left (15 a^2-28 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-200 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{1200 a^4 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\int \frac {\cos ^2(c+d x) \left (600 b \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right )+840 a b^2 \left (5 a^2-8 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-480 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{4800 a^5 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (960 b^2 \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right )+120 a b^3 \left (207 a^2-280 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-1800 b^2 \left (a^2-b^2\right )^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{14400 a^6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\int \frac {1800 b^3 \left (a^2-b^2\right )^2 \left (43 a^4-244 a^2 b^2+224 b^4\right )-120 a b^2 \left (a^2-b^2\right )^2 \left (75 a^4-996 a^2 b^2+1120 b^4\right ) \cos (c+d x)-960 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \cos ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{28800 a^7 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {-1800 a b^3 \left (a^2-b^2\right )^2 \left (43 a^4-244 a^2 b^2+224 b^4\right )+1800 b^2 \left (a^2-b^2\right )^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{28800 a^8 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\left (b \left (a^2-b^2\right ) \left (6 a^4-47 a^2 b^2+56 b^4\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{2 a^9}\\ &=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\left (b \left (a^2-b^2\right ) \left (6 a^4-47 a^2 b^2+56 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^9 d}\\ &=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}-\frac {\sqrt {a-b} b \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 7.86, size = 599, normalized size = 1.11 \begin {gather*} \frac {-7680 b \left (-a^2+b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2+2 \left (a^2-b^2\right )^{5/2} \left (600 a^8 c-20400 a^6 b^2 c+28800 a^4 b^4 c+90240 a^2 b^6 c-107520 b^8 c+600 a^8 d x-20400 a^6 b^2 d x+28800 a^4 b^4 d x+90240 a^2 b^6 d x-107520 b^8 d x+480 a b \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (c+d x)+120 a^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (2 (c+d x))+2640 a^7 b \sin (c+d x)+16160 a^5 b^3 \sin (c+d x)-117120 a^3 b^5 \sin (c+d x)+107520 a b^7 \sin (c+d x)-405 a^8 \sin (2 (c+d x))+24600 a^6 b^2 \sin (2 (c+d x))-99040 a^4 b^4 \sin (2 (c+d x))+80640 a^2 b^6 \sin (2 (c+d x))+2436 a^7 b \sin (3 (c+d x))-10880 a^5 b^3 \sin (3 (c+d x))+8960 a^3 b^5 \sin (3 (c+d x))-140 a^8 \sin (4 (c+d x))+1164 a^6 b^2 \sin (4 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))-188 a^7 b \sin (5 (c+d x))+224 a^5 b^3 \sin (5 (c+d x))+35 a^8 \sin (6 (c+d x))-56 a^6 b^2 \sin (6 (c+d x))+16 a^7 b \sin (7 (c+d x))-5 a^8 \sin (8 (c+d x))\right )}{7680 a^9 (a-b)^2 (a+b)^2 \sqrt {a^2-b^2} d (b+a \cos (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.48, size = 582, normalized size = 1.08 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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Fricas [A]
time = 4.38, size = 1057, normalized size = 1.96 \begin {gather*} \left [\frac {15 \, {\left (5 \, a^{8} - 180 \, a^{6} b^{2} + 600 \, a^{4} b^{4} - 448 \, a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 30 \, {\left (5 \, a^{7} b - 180 \, a^{5} b^{3} + 600 \, a^{3} b^{5} - 448 \, a b^{7}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b^{2} - 180 \, a^{4} b^{4} + 600 \, a^{2} b^{6} - 448 \, b^{8}\right )} d x + 60 \, {\left (6 \, a^{4} b^{3} - 47 \, a^{2} b^{5} + 56 \, b^{7} + {\left (6 \, a^{6} b - 47 \, a^{4} b^{3} + 56 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 47 \, a^{3} b^{4} + 56 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left (40 \, a^{8} \cos \left (d x + c\right )^{7} - 64 \, a^{7} b \cos \left (d x + c\right )^{6} - 1704 \, a^{5} b^{3} + 7880 \, a^{3} b^{5} - 6720 \, a b^{7} - 2 \, {\left (65 \, a^{8} - 56 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (67 \, a^{7} b - 56 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (165 \, a^{8} - 694 \, a^{6} b^{2} + 560 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (387 \, a^{7} b - 1444 \, a^{5} b^{3} + 1120 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (2763 \, a^{6} b^{2} - 12100 \, a^{4} b^{4} + 10080 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{11} d \cos \left (d x + c\right )^{2} + 2 \, a^{10} b d \cos \left (d x + c\right ) + a^{9} b^{2} d\right )}}, \frac {15 \, {\left (5 \, a^{8} - 180 \, a^{6} b^{2} + 600 \, a^{4} b^{4} - 448 \, a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 30 \, {\left (5 \, a^{7} b - 180 \, a^{5} b^{3} + 600 \, a^{3} b^{5} - 448 \, a b^{7}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b^{2} - 180 \, a^{4} b^{4} + 600 \, a^{2} b^{6} - 448 \, b^{8}\right )} d x - 120 \, {\left (6 \, a^{4} b^{3} - 47 \, a^{2} b^{5} + 56 \, b^{7} + {\left (6 \, a^{6} b - 47 \, a^{4} b^{3} + 56 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 47 \, a^{3} b^{4} + 56 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (40 \, a^{8} \cos \left (d x + c\right )^{7} - 64 \, a^{7} b \cos \left (d x + c\right )^{6} - 1704 \, a^{5} b^{3} + 7880 \, a^{3} b^{5} - 6720 \, a b^{7} - 2 \, {\left (65 \, a^{8} - 56 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (67 \, a^{7} b - 56 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (165 \, a^{8} - 694 \, a^{6} b^{2} + 560 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (387 \, a^{7} b - 1444 \, a^{5} b^{3} + 1120 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (2763 \, a^{6} b^{2} - 12100 \, a^{4} b^{4} + 10080 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{11} d \cos \left (d x + c\right )^{2} + 2 \, a^{10} b d \cos \left (d x + c\right ) + a^{9} b^{2} d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{6}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1030 vs.
\(2 (508) = 1016\).
time = 0.64, size = 1030, normalized size = 1.91 \begin {gather*} \frac {\frac {15 \, {\left (5 \, a^{6} - 180 \, a^{4} b^{2} + 600 \, a^{2} b^{4} - 448 \, b^{6}\right )} {\left (d x + c\right )}}{a^{9}} - \frac {240 \, {\left (6 \, a^{6} b - 53 \, a^{4} b^{3} + 103 \, a^{2} b^{5} - 56 \, b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{9}} - \frac {240 \, {\left (6 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 19 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 14 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2} a^{8}} + \frac {2 \, {\left (75 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 720 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1260 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4800 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1800 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 5040 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 425 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4560 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5220 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 27200 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5400 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 25200 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12384 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3960 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 57600 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3600 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 50400 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 990 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12384 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3960 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 57600 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3600 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 50400 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 425 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4560 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5220 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27200 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5400 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25200 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 75 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 720 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1260 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4800 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5040 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{8}}}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.71, size = 2500, normalized size = 4.64 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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