Integrand size = 21, antiderivative size = 539 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}-\frac {\sqrt {a-b} b \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))} \]
[Out]
Time = 2.75 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2975, 3126, 3128, 3102, 2814, 2738, 214} \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{20 a^4 b^2 d (a \cos (c+d x)+b)}-\frac {b \sqrt {a-b} \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{24 a^5 b^2 d}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{60 a^3 b^2 d (a \cos (c+d x)+b)^2}+\frac {x \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right )}{16 a^9}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2} \]
[In]
[Out]
Rule 214
Rule 2738
Rule 2814
Rule 2975
Rule 3102
Rule 3126
Rule 3128
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\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 ) \text {arctanh}\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*}
Time = 9.15 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {-7680 b \left (-a^2+b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2+2 \left (a^2-b^2\right )^{5/2} \left (600 a^8 c-20400 a^6 b^2 c+28800 a^4 b^4 c+90240 a^2 b^6 c-107520 b^8 c+600 a^8 d x-20400 a^6 b^2 d x+28800 a^4 b^4 d x+90240 a^2 b^6 d x-107520 b^8 d x+480 a b \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (c+d x)+120 a^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (2 (c+d x))+2640 a^7 b \sin (c+d x)+16160 a^5 b^3 \sin (c+d x)-117120 a^3 b^5 \sin (c+d x)+107520 a b^7 \sin (c+d x)-405 a^8 \sin (2 (c+d x))+24600 a^6 b^2 \sin (2 (c+d x))-99040 a^4 b^4 \sin (2 (c+d x))+80640 a^2 b^6 \sin (2 (c+d x))+2436 a^7 b \sin (3 (c+d x))-10880 a^5 b^3 \sin (3 (c+d x))+8960 a^3 b^5 \sin (3 (c+d x))-140 a^8 \sin (4 (c+d x))+1164 a^6 b^2 \sin (4 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))-188 a^7 b \sin (5 (c+d x))+224 a^5 b^3 \sin (5 (c+d x))+35 a^8 \sin (6 (c+d x))-56 a^6 b^2 \sin (6 (c+d x))+16 a^7 b \sin (7 (c+d x))-5 a^8 \sin (8 (c+d x))\right )}{7680 a^9 (a-b)^2 (a+b)^2 \sqrt {a^2-b^2} d (b+a \cos (c+d x))^2} \]
[In]
[Out]
Time = 4.42 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+3 a^{5} b -\frac {21}{4} a^{4} b^{2}-20 a^{3} b^{3}+\frac {15}{2} a^{2} b^{4}+21 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (19 a^{5} b -\frac {87}{4} a^{4} b^{2}+\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}+\frac {85}{48} a^{6}-\frac {340}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {258}{5} a^{5} b -\frac {33}{2} a^{4} b^{2}-240 a^{3} b^{3}+15 a^{2} b^{4}+210 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {33}{2} a^{4} b^{2}-15 a^{2} b^{4}+\frac {258}{5} a^{5} b -240 a^{3} b^{3}+210 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (19 a^{5} b +\frac {87}{4} a^{4} b^{2}-\frac {340}{3} a^{3} b^{3}-\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{5} b -20 a^{3} b^{3}+21 a \,b^{5}-\frac {5}{16} a^{6}+\frac {21}{4} a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-180 a^{4} b^{2}+600 a^{2} b^{4}-448 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{9}}+\frac {2 b \left (a +b \right ) \left (a -b \right ) \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}-3 a^{4} b +\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}+3 a^{4} b -\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 a^{4}-47 a^{2} b^{2}+56 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{9}}}{d}\) | \(582\) |
default | \(\frac {\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+3 a^{5} b -\frac {21}{4} a^{4} b^{2}-20 a^{3} b^{3}+\frac {15}{2} a^{2} b^{4}+21 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (19 a^{5} b -\frac {87}{4} a^{4} b^{2}+\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}+\frac {85}{48} a^{6}-\frac {340}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {258}{5} a^{5} b -\frac {33}{2} a^{4} b^{2}-240 a^{3} b^{3}+15 a^{2} b^{4}+210 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {33}{2} a^{4} b^{2}-15 a^{2} b^{4}+\frac {258}{5} a^{5} b -240 a^{3} b^{3}+210 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (19 a^{5} b +\frac {87}{4} a^{4} b^{2}-\frac {340}{3} a^{3} b^{3}-\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{5} b -20 a^{3} b^{3}+21 a \,b^{5}-\frac {5}{16} a^{6}+\frac {21}{4} a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-180 a^{4} b^{2}+600 a^{2} b^{4}-448 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{9}}+\frac {2 b \left (a +b \right ) \left (a -b \right ) \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}-3 a^{4} b +\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}+3 a^{4} b -\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 a^{4}-47 a^{2} b^{2}+56 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{9}}}{d}\) | \(582\) |
risch | \(\frac {5 x}{16 a^{3}}-\frac {45 x \,b^{2}}{4 a^{5}}+\frac {75 x \,b^{4}}{2 a^{7}}-\frac {28 x \,b^{6}}{a^{9}}-\frac {\sin \left (6 d x +6 c \right )}{192 a^{3} d}+\frac {3 \sin \left (4 d x +4 c \right )}{64 d \,a^{3}}+\frac {3 b \sin \left (5 d x +5 c \right )}{80 a^{4} d}-\frac {3 \sin \left (4 d x +4 c \right ) b^{2}}{16 d \,a^{5}}+\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )}}{128 a^{3} d}-\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )}}{128 a^{3} d}+\frac {i b^{2} \left (7 a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-23 a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+16 a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+6 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-27 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+30 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+17 a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-61 a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+44 a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+6 a^{6}-21 a^{4} b^{2}+15 a^{2} b^{4}\right )}{a^{9} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {5 i b^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{12 a^{6} d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{2 a^{5} d}-\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{4}}{8 a^{7} d}+\frac {33 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a^{4} d}-\frac {45 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{4 a^{6} d}+\frac {21 i b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{8} d}+\frac {7 i b \,{\mathrm e}^{3 i \left (d x +c \right )}}{32 a^{4} d}-\frac {5 i b^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{12 a^{6} d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{2 a^{5} d}+\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )} b^{4}}{8 a^{7} d}-\frac {33 i b \,{\mathrm e}^{i \left (d x +c \right )}}{16 a^{4} d}+\frac {45 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{4 a^{6} d}-\frac {21 i b^{5} {\mathrm e}^{i \left (d x +c \right )}}{2 a^{8} d}-\frac {3 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{5}}+\frac {47 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{2 d \,a^{7}}-\frac {28 \sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{9}}+\frac {3 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{5}}-\frac {47 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{2 d \,a^{7}}+\frac {28 \sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{9}}-\frac {7 i b \,{\mathrm e}^{-3 i \left (d x +c \right )}}{32 a^{4} d}\) | \(969\) |
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Time = 0.38 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.96 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\sin ^{6}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (508) = 1016\).
Time = 0.49 (sec) , antiderivative size = 1030, normalized size of antiderivative = 1.91 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
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Time = 18.19 (sec) , antiderivative size = 3975, normalized size of antiderivative = 7.37 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
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