\(\int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [465]

Optimal result

Integrand size = 21, antiderivative size = 407 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {5 a \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{9/2} d}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5} \] Output:

5/8*a*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^(9/2)/d-1 
/7*cos(d*x+c)^5/b/d/(a+b*sin(d*x+c))^7+1/168*a*(4*a^2-b^2)*cos(d*x+c)/b^5/ 
(a^2-b^2)/d/(a+b*sin(d*x+c))^4+1/168*(4*a^4-9*a^2*b^2+12*b^4)*cos(d*x+c)/b 
^5/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^3+1/336*a*(8*a^4-30*a^2*b^2+57*b^4)*cos( 
d*x+c)/b^5/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^2+1/336*(8*a^6-38*a^4*b^2+87*a^2 
*b^4+48*b^6)*cos(d*x+c)/b^5/(a^2-b^2)^4/d/(a+b*sin(d*x+c))+5/42*cos(d*x+c) 
^3*(2*a+3*b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^6-1/42*cos(d*x+c)*(4*a^2+9* 
b^2+10*a*b*sin(d*x+c))/b^5/d/(a+b*sin(d*x+c))^5
 

Mathematica [A] (verified)

Time = 6.07 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.36 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\cos ^7(c+d x)}{7 (a-b) d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x) \left (-\frac {(1-\sin (c+d x))^{5/2} (1+\sin (c+d x))^{9/2}}{7 (-a+b) (a+b \sin (c+d x))^7}-\frac {5 \left (-\frac {(1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{9/2}}{6 (-a+b) (a+b \sin (c+d x))^6}-\frac {-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{9/2}}{5 (-a+b) (a+b \sin (c+d x))^5}-\frac {-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{7/2}}{4 (a+b) (a+b \sin (c+d x))^4}+\frac {7 \left (-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}+\frac {5 \left (-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}+\frac {3 \left (-\frac {2 \text {arctanh}\left (\frac {\sqrt {a-b} \sqrt {1-\sin (c+d x)}}{\sqrt {-a-b} \sqrt {1+\sin (c+d x)}}\right )}{(-a-b)^{3/2} \sqrt {a-b}}+\frac {\sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}{(-a-b) (a+b \sin (c+d x))}\right )}{2 (a+b)}\right )}{3 (a+b)}\right )}{4 (a+b)}}{5 (-a+b)}}{2 (-a+b)}\right )}{7 (-a+b)}\right )}{(a-b) d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}} \] Input:

Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^8,x]
 

Output:

Cos[c + d*x]^7/(7*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/ 
7*((1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin 
[c + d*x])^7) - (5*(-1/6*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(9/2 
))/((-a + b)*(a + b*Sin[c + d*x])^6) - (-1/5*(Sqrt[1 - Sin[c + d*x]]*(1 + 
Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (-1/4*(Sqrt[1 - S 
in[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/((a + b)*(a + b*Sin[c + d*x])^4) + 
(7*(-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b 
*Sin[c + d*x])^3) + (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3 
/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[ 
1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2) 
*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)* 
(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b))))/(4*(a + b)))/(5*(-a + 
b)))/(2*(-a + b))))/(7*(-a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[ 
1 + Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 2.14 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.16, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.143, Rules used = {3042, 3172, 3042, 3342, 27, 3042, 3342, 25, 3042, 3233, 27, 3042, 3233, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3139, 1083, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^6}{(a+b \sin (c+d x))^8}dx\)

\(\Big \downarrow \) 3172

\(\displaystyle -\frac {5 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \int \frac {\cos (c+d x)^4 \sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3342

\(\displaystyle -\frac {5 \left (\frac {\int -\frac {2 \cos ^2(c+d x) (3 b+2 a \sin (c+d x))}{(a+b \sin (c+d x))^6}dx}{4 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5 \left (-\frac {\int \frac {\cos ^2(c+d x) (3 b+2 a \sin (c+d x))}{(a+b \sin (c+d x))^6}dx}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \left (-\frac {\int \frac {\cos (c+d x)^2 (3 b+2 a \sin (c+d x))}{(a+b \sin (c+d x))^6}dx}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3342

\(\displaystyle -\frac {5 \left (-\frac {\frac {\int -\frac {10 a b+\left (4 a^2+9 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\int \frac {10 a b+\left (4 a^2+9 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\int \frac {10 a b+\left (4 a^2+9 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3233

\(\displaystyle -\frac {5 \left (-\frac {-\frac {-\frac {\int -\frac {3 \left (4 b \left (2 a^2-3 b^2\right )+a \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \int \frac {4 b \left (2 a^2-3 b^2\right )+a \left (4 a^2-b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \int \frac {4 b \left (2 a^2-3 b^2\right )+a \left (4 a^2-b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3233

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (-\frac {\int -\frac {3 a b \left (4 a^2-11 b^2\right )+2 \left (4 a^4-9 b^2 a^2+12 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\int \frac {3 a b \left (4 a^2-11 b^2\right )+2 \left (4 a^4-9 b^2 a^2+12 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\int \frac {3 a b \left (4 a^2-11 b^2\right )+2 \left (4 a^4-9 b^2 a^2+12 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3233

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {-\frac {\int -\frac {2 b \left (4 a^4-15 b^2 a^2-24 b^4\right )+a \left (8 a^4-30 b^2 a^2+57 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\frac {\int \frac {2 b \left (4 a^4-15 b^2 a^2-24 b^4\right )+a \left (8 a^4-30 b^2 a^2+57 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\frac {\int \frac {2 b \left (4 a^4-15 b^2 a^2-24 b^4\right )+a \left (8 a^4-30 b^2 a^2+57 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3233

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\frac {-\frac {\int \frac {105 a b^5}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\frac {-\frac {105 a b^5 \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\frac {-\frac {105 a b^5 \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 3139

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\frac {-\frac {210 a b^5 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\frac {3 \left (\frac {\frac {\frac {420 a b^5 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2-b^2\right )}-\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

\(\Big \downarrow \) 217

\(\displaystyle -\frac {5 \left (-\frac {-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{15 b^2 d (a+b \sin (c+d x))^5}-\frac {\frac {3 \left (\frac {\frac {-\frac {210 a b^5 \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{3/2}}-\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{15 b^2}}{2 b^2}-\frac {\cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{6 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}\)

Input:

Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^8,x]
 

Output:

-1/7*Cos[c + d*x]^5/(b*d*(a + b*Sin[c + d*x])^7) - (5*(-1/6*(Cos[c + d*x]^ 
3*(2*a + 3*b*Sin[c + d*x]))/(b^2*d*(a + b*Sin[c + d*x])^6) - (-1/15*(Cos[c 
 + d*x]*(4*a^2 + 9*b^2 + 10*a*b*Sin[c + d*x]))/(b^2*d*(a + b*Sin[c + d*x]) 
^5) - (-1/4*(a*(4*a^2 - b^2)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d 
*x])^4) + (3*(-1/3*((4*a^4 - 9*a^2*b^2 + 12*b^4)*Cos[c + d*x])/((a^2 - b^2 
)*d*(a + b*Sin[c + d*x])^3) + (-1/2*(a*(8*a^4 - 30*a^2*b^2 + 57*b^4)*Cos[c 
 + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((-210*a*b^5*ArcTan[(2*b 
 + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - ((8 
*a^6 - 38*a^4*b^2 + 87*a^2*b^4 + 48*b^6)*Cos[c + d*x])/((a^2 - b^2)*d*(a + 
 b*Sin[c + d*x])))/(2*(a^2 - b^2)))/(3*(a^2 - b^2))))/(4*(a^2 - b^2)))/(15 
*b^2))/(2*b^2)))/(7*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 3172

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x 
])^(m + 1)/(b*f*(m + 1))), x] + Simp[g^2*((p - 1)/(b*(m + 1)))   Int[(g*Cos 
[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; Fre 
eQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && I 
ntegersQ[2*m, 2*p]
 

rule 3233

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + 
 f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( 
m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
 

rule 3342

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g*C 
os[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d*p 
 + b*d*(m + 1)*Sin[e + f*x])/(b^2*f*(m + 1)*(m + p + 1))), x] + Simp[g^2*(( 
p - 1)/(b^2*(m + 1)*(m + p + 1)))   Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin 
[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Sin[e + f*x 
], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && Lt 
Q[m, -1] && GtQ[p, 1] && NeQ[m + p + 1, 0] && IntegerQ[2*m]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 7.96 (sec) , antiderivative size = 1388, normalized size of antiderivative = 3.41

method	result	size
risch	\(\text {Expression too large to display}\)	\(1388\)
derivativedivides	\(\text {Expression too large to display}\)	\(1389\)
default	\(\text {Expression too large to display}\)	\(1389\)

Input:

int(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
 

Output:

-1/168*I*(-105*b^12*a*exp(13*I*(d*x+c))+1120*b^4*a^9*exp(11*I*(d*x+c))-448 
0*b^6*a^7*exp(11*I*(d*x+c))+6720*b^8*a^5*exp(11*I*(d*x+c))+3010*b^10*a^3*e 
xp(11*I*(d*x+c))+1820*b^12*a*exp(11*I*(d*x+c))-336*I*b^13*exp(12*I*(d*x+c) 
)-2016*I*b^9*a^4*exp(12*I*(d*x+c))-21*I*b^11*a^2*exp(12*I*(d*x+c))+1792*I* 
b*a^12*exp(6*I*(d*x+c))-8960*I*b^5*a^8*exp(10*I*(d*x+c))+13370*I*b^9*a^4*e 
xp(10*I*(d*x+c))+9940*I*b^11*a^2*exp(10*I*(d*x+c))+2240*I*b^3*a^10*exp(10* 
I*(d*x+c))-12047*I*a^2*b^11*exp(4*I*(d*x+c))+336*I*a^8*b^5*exp(2*I*(d*x+c) 
)+5026*I*a^4*b^9*exp(2*I*(d*x+c))+4620*I*a^2*b^11*exp(2*I*(d*x+c))-1792*I* 
a^10*b^3*exp(6*I*(d*x+c))-9072*I*a^8*b^5*exp(6*I*(d*x+c))+80192*I*a^6*b^7* 
exp(6*I*(d*x+c))-48*I*b^13-42588*b^8*a^5*exp(9*I*(d*x+c))-27370*b^10*a^3*e 
xp(9*I*(d*x+c))-4445*b^12*a*exp(9*I*(d*x+c))+2944*b^2*a^11*exp(7*I*(d*x+c) 
)-2688*a^11*b^2*exp(5*I*(d*x+c))+8288*a^9*b^4*exp(5*I*(d*x+c))-11312*a^7*b 
^6*exp(5*I*(d*x+c))-79128*a^5*b^8*exp(5*I*(d*x+c))-44660*a^3*b^10*exp(5*I* 
(d*x+c))+512*a^13*exp(7*I*(d*x+c))+38*I*a^4*b^9+82180*I*a^4*b^9*exp(6*I*(d 
*x+c))+26880*I*a^2*b^11*exp(6*I*(d*x+c))+1344*I*b^7*a^6*exp(12*I*(d*x+c))- 
2688*b^2*a^11*exp(9*I*(d*x+c))+8288*b^4*a^9*exp(9*I*(d*x+c))-6272*b^6*a^7* 
exp(9*I*(d*x+c))-8*I*a^6*b^7-87*I*a^2*b^11-1792*I*b*a^12*exp(8*I*(d*x+c))+ 
1792*I*b^3*a^10*exp(8*I*(d*x+c))+9072*I*b^5*a^8*exp(8*I*(d*x+c))-70210*I*b 
^9*a^4*exp(8*I*(d*x+c))-1792*I*a^6*b^7*exp(2*I*(d*x+c))-2240*I*a^10*b^3*ex 
p(4*I*(d*x+c))+2212*a*b^12*exp(3*I*(d*x+c))-13248*a^9*b^4*exp(7*I*(d*x+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (386) = 772\).

Time = 0.28 (sec) , antiderivative size = 2250, normalized size of antiderivative = 5.53 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x, algorithm="fricas")
 

Output:

[1/672*(2*(8*a^8*b - 46*a^6*b^3 + 125*a^4*b^5 - 39*a^2*b^7 - 48*b^9)*cos(d 
*x + c)^7 + 28*(7*a^8*b - 56*a^6*b^3 - 44*a^4*b^5 + 93*a^2*b^7)*cos(d*x + 
c)^5 + 70*(7*a^8*b + 83*a^6*b^3 - 43*a^4*b^5 - 47*a^2*b^7)*cos(d*x + c)^3 
- 105*(7*a^2*b^6*cos(d*x + c)^6 - a^8 - 21*a^6*b^2 - 35*a^4*b^4 - 7*a^2*b^ 
6 - 7*(5*a^4*b^4 + 3*a^2*b^6)*cos(d*x + c)^4 + 7*(3*a^6*b^2 + 10*a^4*b^4 + 
 3*a^2*b^6)*cos(d*x + c)^2 + (a*b^7*cos(d*x + c)^6 - 7*a^7*b - 35*a^5*b^3 
- 21*a^3*b^5 - a*b^7 - 3*(7*a^3*b^5 + a*b^7)*cos(d*x + c)^4 + (35*a^5*b^3 
+ 42*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log 
(((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos 
(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + 
c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 420*(3*a^8*b + 7*a^6*b^3 - 7*a^4 
*b^5 - 3*a^2*b^7)*cos(d*x + c) - 14*((8*a^9 - 46*a^7*b^2 + 125*a^5*b^4 - 5 
4*a^3*b^6 - 33*a*b^8)*cos(d*x + c)^5 + 10*(a^9 - 11*a^7*b^2 - 25*a^5*b^4 + 
 31*a^3*b^6 + 4*a*b^8)*cos(d*x + c)^3 + 15*(a^9 + 14*a^7*b^2 - 14*a^3*b^6 
- a*b^8)*cos(d*x + c))*sin(d*x + c))/(7*(a^11*b^6 - 5*a^9*b^8 + 10*a^7*b^1 
0 - 10*a^5*b^12 + 5*a^3*b^14 - a*b^16)*d*cos(d*x + c)^6 - 7*(5*a^13*b^4 - 
22*a^11*b^6 + 35*a^9*b^8 - 20*a^7*b^10 - 5*a^5*b^12 + 10*a^3*b^14 - 3*a*b^ 
16)*d*cos(d*x + c)^4 + 7*(3*a^15*b^2 - 5*a^13*b^4 - 17*a^11*b^6 + 55*a^9*b 
^8 - 55*a^7*b^10 + 17*a^5*b^12 + 5*a^3*b^14 - 3*a*b^16)*d*cos(d*x + c)^2 - 
 (a^17 + 16*a^15*b^2 - 60*a^13*b^4 + 32*a^11*b^6 + 110*a^9*b^8 - 176*a^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**6/(a+b*sin(d*x+c))**8,x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1650 vs. \(2 (386) = 772\).

Time = 0.32 (sec) , antiderivative size = 1650, normalized size of antiderivative = 4.05 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x, algorithm="giac")
 

Output:

1/168*(105*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d* 
x + 1/2*c) + b)/sqrt(a^2 - b^2)))*a/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2* 
b^6 + b^8)*sqrt(a^2 - b^2)) - (231*a^14*tan(1/2*d*x + 1/2*c)^13 - 1344*a^1 
2*b^2*tan(1/2*d*x + 1/2*c)^13 + 2016*a^10*b^4*tan(1/2*d*x + 1/2*c)^13 - 13 
44*a^8*b^6*tan(1/2*d*x + 1/2*c)^13 + 336*a^6*b^8*tan(1/2*d*x + 1/2*c)^13 + 
 651*a^13*b*tan(1/2*d*x + 1/2*c)^12 - 8064*a^11*b^3*tan(1/2*d*x + 1/2*c)^1 
2 + 12096*a^9*b^5*tan(1/2*d*x + 1/2*c)^12 - 8064*a^7*b^7*tan(1/2*d*x + 1/2 
*c)^12 + 2016*a^5*b^9*tan(1/2*d*x + 1/2*c)^12 + 196*a^14*tan(1/2*d*x + 1/2 
*c)^11 - 4354*a^12*b^2*tan(1/2*d*x + 1/2*c)^11 - 21504*a^10*b^4*tan(1/2*d* 
x + 1/2*c)^11 + 36736*a^8*b^6*tan(1/2*d*x + 1/2*c)^11 - 25984*a^6*b^8*tan( 
1/2*d*x + 1/2*c)^11 + 6720*a^4*b^10*tan(1/2*d*x + 1/2*c)^11 + 140*a^13*b*t 
an(1/2*d*x + 1/2*c)^10 - 40250*a^11*b^3*tan(1/2*d*x + 1/2*c)^10 - 6720*a^9 
*b^5*tan(1/2*d*x + 1/2*c)^10 + 49280*a^7*b^7*tan(1/2*d*x + 1/2*c)^10 - 459 
20*a^5*b^9*tan(1/2*d*x + 1/2*c)^10 + 13440*a^3*b^11*tan(1/2*d*x + 1/2*c)^1 
0 + 595*a^14*tan(1/2*d*x + 1/2*c)^9 - 20650*a^12*b^2*tan(1/2*d*x + 1/2*c)^ 
9 - 103740*a^10*b^4*tan(1/2*d*x + 1/2*c)^9 + 70336*a^8*b^6*tan(1/2*d*x + 1 
/2*c)^9 + 2576*a^6*b^8*tan(1/2*d*x + 1/2*c)^9 - 40320*a^4*b^10*tan(1/2*d*x 
 + 1/2*c)^9 + 16128*a^2*b^12*tan(1/2*d*x + 1/2*c)^9 - 3045*a^13*b*tan(1/2* 
d*x + 1/2*c)^8 - 100450*a^11*b^3*tan(1/2*d*x + 1/2*c)^8 - 92120*a^9*b^5*ta 
n(1/2*d*x + 1/2*c)^8 + 129024*a^7*b^7*tan(1/2*d*x + 1/2*c)^8 - 74816*a^...
 

Mupad [B] (verification not implemented)

Time = 21.09 (sec) , antiderivative size = 1868, normalized size of antiderivative = 4.59 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

int(cos(c + d*x)^6/(a + b*sin(c + d*x))^8,x)
 

Output:

((279*a^6*b - 48*b^7 + 200*a^2*b^5 - 326*a^4*b^3)/(168*(a^8 + b^8 - 4*a^2* 
b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2)*(33*a^8 - 48*b^8 + 208 
*a^2*b^6 - 364*a^4*b^4 + 366*a^6*b^2))/(24*a*(a^8 + b^8 - 4*a^2*b^6 + 6*a^ 
4*b^4 - 4*a^6*b^2)) - (tan(c/2 + (d*x)/2)^9*(85*a^12 + 2304*b^12 - 5760*a^ 
2*b^10 + 368*a^4*b^8 + 10048*a^6*b^6 - 14820*a^8*b^4 - 2950*a^10*b^2))/(24 
*a^5*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2 
)^5*(85*a^12 - 2304*b^12 + 5760*a^2*b^10 - 368*a^4*b^8 - 9328*a^6*b^6 + 20 
040*a^8*b^4 + 5420*a^10*b^2))/(24*a^5*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 
 4*a^6*b^2)) - (tan(c/2 + (d*x)/2)^11*(14*a^10 + 480*b^10 - 1856*a^2*b^8 + 
 2624*a^4*b^6 - 1536*a^6*b^4 - 311*a^8*b^2))/(12*a^3*(a^8 + b^8 - 4*a^2*b^ 
6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2)^3*(14*a^10 - 480*b^10 + 
1856*a^2*b^8 - 2696*a^4*b^6 + 2088*a^6*b^4 + 1363*a^8*b^2))/(12*a^3*(a^8 + 
 b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (tan(c/2 + (d*x)/2)^13*(11*a^ 
8 + 16*b^8 - 64*a^2*b^6 + 96*a^4*b^4 - 64*a^6*b^2))/(8*a*(a^8 + b^8 - 4*a^ 
2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2)^6*(240*a^12*b - 384* 
b^13 + 160*a^2*b^11 + 2672*a^4*b^9 - 4548*a^6*b^7 + 3920*a^8*b^5 + 4375*a^ 
10*b^3))/(6*a^6*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/ 
2 + (d*x)/2)^8*(435*a^12*b - 1536*b^13 + 640*a^2*b^11 + 10688*a^4*b^9 - 18 
432*a^6*b^7 + 13160*a^8*b^5 + 14350*a^10*b^3))/(24*a^6*(a^8 + b^8 - 4*a^2* 
b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (5*tan(c/2 + (d*x)/2)^10*(2*a^10*b + 19...
 

Reduce [B] (verification not implemented)

Time = 2.30 (sec) , antiderivative size = 2394, normalized size of antiderivative = 5.88 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx =\text {Too large to display} \] Input:

int(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x)
 

Output:

(210*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*si 
n(c + d*x)**7*a**6*b**8 + 1470*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a 
+ b)/sqrt(a**2 - b**2))*sin(c + d*x)**6*a**7*b**7 + 4410*sqrt(a**2 - b**2) 
*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**5*a**8*b** 
6 + 7350*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2) 
)*sin(c + d*x)**4*a**9*b**5 + 7350*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2 
)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**3*a**10*b**4 + 4410*sqrt(a**2 - 
b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**2*a** 
11*b**3 + 1470*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - 
 b**2))*sin(c + d*x)*a**12*b**2 + 210*sqrt(a**2 - b**2)*atan((tan((c + d*x 
)/2)*a + b)/sqrt(a**2 - b**2))*a**13*b + 8*cos(c + d*x)*sin(c + d*x)**6*a* 
*13*b**2 - 46*cos(c + d*x)*sin(c + d*x)**6*a**11*b**4 + 125*cos(c + d*x)*s 
in(c + d*x)**6*a**9*b**6 - 39*cos(c + d*x)*sin(c + d*x)**6*a**7*b**8 - 48* 
cos(c + d*x)*sin(c + d*x)**6*a**5*b**10 + 56*cos(c + d*x)*sin(c + d*x)**5* 
a**14*b - 322*cos(c + d*x)*sin(c + d*x)**5*a**12*b**3 + 875*cos(c + d*x)*s 
in(c + d*x)**5*a**10*b**5 - 378*cos(c + d*x)*sin(c + d*x)**5*a**8*b**7 - 2 
31*cos(c + d*x)*sin(c + d*x)**5*a**6*b**9 - 122*cos(c + d*x)*sin(c + d*x)* 
*4*a**13*b**2 + 922*cos(c + d*x)*sin(c + d*x)**4*a**11*b**4 + 241*cos(c + 
d*x)*sin(c + d*x)**4*a**9*b**6 - 1185*cos(c + d*x)*sin(c + d*x)**4*a**7*b* 
*8 + 144*cos(c + d*x)*sin(c + d*x)**4*a**5*b**10 - 182*cos(c + d*x)*sin...