Integrand size = 21, antiderivative size = 411 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {3 a \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{11/2} d}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6} \]
3/8*a*(2*a^2+b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^ 2)^(11/2)/d-1/7*cos(d*x+c)^3/b/d/(a+b*sin(d*x+c))^7-1/140*(a^2-3*b^2)*cos( d*x+c)/b^3/(a^2-b^2)/d/(a+b*sin(d*x+c))^5-1/280*a*(2*a^2-11*b^2)*cos(d*x+c )/b^3/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^4-1/280*(2*a^4-15*a^2*b^2-8*b^4)*cos( d*x+c)/b^3/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^3-1/560*a*(4*a^4-36*a^2*b^2-73*b ^4)*cos(d*x+c)/b^3/(a^2-b^2)^4/d/(a+b*sin(d*x+c))^2-1/560*(4*a^6-40*a^4*b^ 2-247*a^2*b^4-32*b^6)*cos(d*x+c)/b^3/(a^2-b^2)^5/d/(a+b*sin(d*x+c))+1/28*c os(d*x+c)*(a+3*b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^6
Leaf count is larger than twice the leaf count of optimal. \(1167\) vs. \(2(411)=822\).
Time = 6.10 (sec) , antiderivative size = 1167, normalized size of antiderivative = 2.84 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\cos ^5(c+d x)}{5 (a-b) d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x) \left (-\frac {b (1-\sin (c+d x))^{5/2} (1+\sin (c+d x))^{7/2}}{7 (-a+b) (a+b) (a+b \sin (c+d x))^7}-\frac {-\frac {(a b+(7 a-b) b) (1-\sin (c+d x))^{5/2} (1+\sin (c+d x))^{7/2}}{6 (-a+b) (a+b) (a+b \sin (c+d x))^6}-\frac {7 \left (6 a^2-2 a b+b^2\right ) \left (-\frac {(1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{7/2}}{5 (-a+b) (a+b \sin (c+d x))^5}-\frac {3 \left (-\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 {-\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)}}{4 (-a+b)}\right )}{5 (-a+b)}\right )}{6 (-a+b) (a+b)}}{7 (-a+b) (a+b)}\right )}{(a-b) d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}+\frac {2 b \left (\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)}}\right )}{5 (a-b)} \]
Cos[c + d*x]^5/(5*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/ 7*(b*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)* (a + b*Sin[c + d*x])^7) - (-1/6*((a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(5 /2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (7*(6*a^2 - 2*a*b + b^2)*(-1/5*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x] )^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (3*(-1/4*(Sqrt[1 - Sin[c + d* x]]*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^4) - (-1/3*(S qrt[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))))/(6*( -a + b)*(a + b)))/(7*(-a + b)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]] *Sqrt[1 + Sin[c + d*x]]) + (2*b*(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 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))...
Time = 2.06 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.17, 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, 3233, 27, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{(a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3172 |
| \(\displaystyle -\frac {3 \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \int \frac {\cos (c+d x)^2 \sin (c+d x)}{(a+b \sin (c+d x))^7}dx}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3342 |
| \(\displaystyle -\frac {3 \left (\frac {\int -\frac {2 (3 b+a \sin (c+d x))}{(a+b \sin (c+d x))^6}dx}{24 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {3 b+a \sin (c+d x)}{(a+b \sin (c+d x))^6}dx}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {\int \frac {3 b+a \sin (c+d x)}{(a+b \sin (c+d x))^6}dx}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {3 \left (-\frac {-\frac {\int -\frac {2 \left (5 a b+2 \left (a^2-3 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5}dx}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \int \frac {5 a b+2 \left (a^2-3 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \int \frac {5 a b+2 \left (a^2-3 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5}dx}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (-\frac {\int -\frac {3 \left (4 b \left (a^2+2 b^2\right )+a \left (2 a^2-11 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \int \frac {4 b \left (a^2+2 b^2\right )+a \left (2 a^2-11 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \int \frac {4 b \left (a^2+2 b^2\right )+a \left (2 a^2-11 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (-\frac {\int -\frac {3 a b \left (2 a^2+19 b^2\right )+2 \left (a^2-8 b^2\right ) \left (2 a^2+b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\int \frac {3 a b \left (2 a^2+19 b^2\right )+2 \left (a^2-8 b^2\right ) \left (2 a^2+b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\int \frac {3 a b \left (2 a^2+19 b^2\right )+2 \left (a^2-8 b^2\right ) \left (2 a^2+b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {-\frac {\int -\frac {2 b \left (2 a^4+87 b^2 a^2+16 b^4\right )+a \left (4 a^4-36 b^2 a^2-73 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {\int \frac {2 b \left (2 a^4+87 b^2 a^2+16 b^4\right )+a \left (4 a^4-36 b^2 a^2-73 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {\int \frac {2 b \left (2 a^4+87 b^2 a^2+16 b^4\right )+a \left (4 a^4-36 b^2 a^2-73 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {-\frac {\int -\frac {105 a b^3 \left (2 a^2+b^2\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {\frac {105 a b^3 \left (2 a^2+b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {\frac {105 a b^3 \left (2 a^2+b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {\frac {210 a b^3 \left (2 a^2+b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {-\frac {420 a b^3 \left (2 a^2+b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2-b^2\right )}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {3 \left (-\frac {\frac {2 \left (\frac {3 \left (\frac {\frac {\frac {210 a b^3 \left (2 a^2+b^2\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{3/2}}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{3 \left (a^2-b^2\right )}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}\right )}{4 \left (a^2-b^2\right )}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}\right )}{5 \left (a^2-b^2\right )}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{12 b^2}-\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{12 b^2 d (a+b \sin (c+d x))^6}\right )}{7 b}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}\) |
-1/7*Cos[c + d*x]^3/(b*d*(a + b*Sin[c + d*x])^7) - (3*(-1/12*(Cos[c + d*x] *(a + 3*b*Sin[c + d*x]))/(b^2*d*(a + b*Sin[c + d*x])^6) - (-1/5*((a^2 - 3* b^2)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) + (2*(-1/4*(a*(2 *a^2 - 11*b^2)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])^4) + (3*( -1/3*((2*a^4 - 15*a^2*b^2 - 8*b^4)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin [c + d*x])^3) + (-1/2*(a*(4*a^4 - 36*a^2*b^2 - 73*b^4)*Cos[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((210*a*b^3*(2*a^2 + b^2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*d) - ((4* a^6 - 40*a^4*b^2 - 247*a^2*b^4 - 32*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))))/(5 *(a^2 - b^2)))/(12*b^2)))/(7*b)
3.5.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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]
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]
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]
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]
Result contains complex when optimal does not.
Time = 14.07 (sec) , antiderivative size = 1533, normalized size of antiderivative = 3.73
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1533\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1660\) |
| default | \(\text {Expression too large to display}\) | \(1660\) |
1/280*I*(-2240*I*b^13*exp(6*I*(d*x+c))-448*I*b^13*exp(4*I*(d*x+c))+185472* a^5*b^8*exp(5*I*(d*x+c))+55930*a^3*b^10*exp(5*I*(d*x+c))+3395*a*b^12*exp(5 *I*(d*x+c))-1344*a^11*b^2*exp(5*I*(d*x+c))+15680*a^9*b^4*exp(5*I*(d*x+c))+ 146272*a^7*b^6*exp(5*I*(d*x+c))+336*I*a^8*b^5*exp(2*I*(d*x+c))+256*a^13*ex p(7*I*(d*x+c))+896*I*b*a^12*exp(6*I*(d*x+c))-245448*I*b^7*a^6*exp(6*I*(d*x +c))+60368*I*a^8*b^5*exp(8*I*(d*x+c))-896*I*b*a^12*exp(8*I*(d*x+c))-5600*I *a^10*b^3*exp(6*I*(d*x+c))-114128*I*a^8*b^5*exp(6*I*(d*x+c))+1365*I*b^11*a ^2*exp(12*I*(d*x+c))-224*I*b^13*exp(2*I*(d*x+c))-1120*I*b^13*exp(10*I*(d*x +c))-1120*I*b^13*exp(8*I*(d*x+c))+1120*I*b^3*a^10*exp(10*I*(d*x+c))-5600*I *b^5*a^8*exp(10*I*(d*x+c))-48930*I*b^9*a^4*exp(10*I*(d*x+c))-2100*I*b^11*a ^2*exp(10*I*(d*x+c))-162820*I*a^4*b^9*exp(6*I*(d*x+c))-700*b^12*a*exp(11*I *(d*x+c))-1344*b^2*a^11*exp(9*I*(d*x+c))+128*b^2*a^11*exp(7*I*(d*x+c))-4*I *a^6*b^7+40*I*a^4*b^9+247*I*a^2*b^11-61460*a^3*b^10*exp(7*I*(d*x+c))-4480* a*b^12*exp(7*I*(d*x+c))-56*exp(I*(d*x+c))*b^6*a^7+560*b^8*exp(I*(d*x+c))*a ^5+3248*exp(I*(d*x+c))*b^10*a^3+343*exp(I*(d*x+c))*b^12*a+6720*b^4*a^9*exp (9*I*(d*x+c))+63672*b^6*a^7*exp(9*I*(d*x+c))+121072*b^8*a^5*exp(9*I*(d*x+c ))+31780*b^10*a^3*exp(9*I*(d*x+c))+3325*b^12*a*exp(9*I*(d*x+c))+121576*I*a ^6*b^7*exp(4*I*(d*x+c))-3332*I*a^6*b^7*exp(2*I*(d*x+c))+5600*I*b^3*a^10*ex p(8*I*(d*x+c))+217308*I*a^6*b^7*exp(8*I*(d*x+c))+2730*I*b^9*a^4*exp(12*I*( d*x+c))-1120*I*b^3*a^10*exp(4*I*(d*x+c))+32*I*b^13-14980*b^8*a^5*exp(11...
Leaf count of result is larger than twice the leaf count of optimal. 1286 vs. \(2 (390) = 780\).
Time = 0.48 (sec) , antiderivative size = 2657, normalized size of antiderivative = 6.46 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
[-1/1120*(2*(4*a^8*b^3 - 44*a^6*b^5 - 207*a^4*b^7 + 215*a^2*b^9 + 32*b^11) *cos(d*x + c)^7 - 28*(6*a^10*b - 65*a^8*b^3 - 224*a^6*b^5 + 222*a^4*b^7 + 53*a^2*b^9 + 8*b^11)*cos(d*x + c)^5 - 70*(14*a^10*b + 173*a^8*b^3 - 3*a^6* b^5 - 137*a^4*b^7 - 47*a^2*b^9)*cos(d*x + c)^3 + 105*(2*a^10 + 43*a^8*b^2 + 91*a^6*b^4 + 49*a^4*b^6 + 7*a^2*b^8 - 7*(2*a^4*b^6 + a^2*b^8)*cos(d*x + c)^6 + 7*(10*a^6*b^4 + 11*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^4 - 7*(6*a^8*b ^2 + 23*a^6*b^4 + 16*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^2 + (14*a^9*b + 77* a^7*b^3 + 77*a^5*b^5 + 23*a^3*b^7 + a*b^9 - (2*a^3*b^7 + a*b^9)*cos(d*x + c)^6 + 3*(14*a^5*b^5 + 9*a^3*b^7 + a*b^9)*cos(d*x + c)^4 - (70*a^7*b^3 + 1 19*a^5*b^5 + 48*a^3*b^7 + 3*a*b^9)*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*(6*a^10*b + 17*a ^8*b^3 - 7*a^6*b^5 - 13*a^4*b^7 - 3*a^2*b^9)*cos(d*x + c) - 14*((4*a^9*b^2 - 44*a^7*b^4 - 177*a^5*b^6 + 200*a^3*b^8 + 17*a*b^10)*cos(d*x + c)^5 - 10 *(2*a^11 - 21*a^9*b^2 - 61*a^7*b^4 + 37*a^5*b^6 + 39*a^3*b^8 + 4*a*b^10)*c os(d*x + c)^3 - 15*(2*a^11 + 29*a^9*b^2 + 14*a^7*b^4 - 28*a^5*b^6 - 16*a^3 *b^8 - a*b^10)*cos(d*x + c))*sin(d*x + c))/(7*(a^13*b^6 - 6*a^11*b^8 + 15* a^9*b^10 - 20*a^7*b^12 + 15*a^5*b^14 - 6*a^3*b^16 + a*b^18)*d*cos(d*x + c) ^6 - 7*(5*a^15*b^4 - 27*a^13*b^6 + 57*a^11*b^8 - 55*a^9*b^10 + 15*a^7*b...
Timed out. \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 1932 vs. \(2 (390) = 780\).
Time = 0.52 (sec) , antiderivative size = 1932, normalized size of antiderivative = 4.70 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
1/280*(105*(2*a^3 + a*b^2)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arct an((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^10 - 5*a^8*b^2 + 10* a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*sqrt(a^2 - b^2)) - (350*a^16*tan( 1/2*d*x + 1/2*c)^13 - 2905*a^14*b^2*tan(1/2*d*x + 1/2*c)^13 + 5600*a^12*b^ 4*tan(1/2*d*x + 1/2*c)^13 - 5600*a^10*b^6*tan(1/2*d*x + 1/2*c)^13 + 2800*a ^8*b^8*tan(1/2*d*x + 1/2*c)^13 - 560*a^6*b^10*tan(1/2*d*x + 1/2*c)^13 + 63 0*a^15*b*tan(1/2*d*x + 1/2*c)^12 - 18165*a^13*b^3*tan(1/2*d*x + 1/2*c)^12 + 33600*a^11*b^5*tan(1/2*d*x + 1/2*c)^12 - 33600*a^9*b^7*tan(1/2*d*x + 1/2 *c)^12 + 16800*a^7*b^9*tan(1/2*d*x + 1/2*c)^12 - 3360*a^5*b^11*tan(1/2*d*x + 1/2*c)^12 + 840*a^16*tan(1/2*d*x + 1/2*c)^11 - 15680*a^14*b^2*tan(1/2*d *x + 1/2*c)^11 - 41090*a^12*b^4*tan(1/2*d*x + 1/2*c)^11 + 89600*a^10*b^6*t an(1/2*d*x + 1/2*c)^11 - 100800*a^8*b^8*tan(1/2*d*x + 1/2*c)^11 + 53760*a^ 6*b^10*tan(1/2*d*x + 1/2*c)^11 - 11200*a^4*b^12*tan(1/2*d*x + 1/2*c)^11 - 840*a^15*b*tan(1/2*d*x + 1/2*c)^10 - 102760*a^13*b^3*tan(1/2*d*x + 1/2*c)^ 10 + 11270*a^11*b^5*tan(1/2*d*x + 1/2*c)^10 + 78400*a^9*b^7*tan(1/2*d*x + 1/2*c)^10 - 151200*a^7*b^9*tan(1/2*d*x + 1/2*c)^10 + 97440*a^5*b^11*tan(1/ 2*d*x + 1/2*c)^10 - 22400*a^3*b^13*tan(1/2*d*x + 1/2*c)^10 + 630*a^16*tan( 1/2*d*x + 1/2*c)^9 - 51905*a^14*b^2*tan(1/2*d*x + 1/2*c)^9 - 249410*a^12*b ^4*tan(1/2*d*x + 1/2*c)^9 + 202244*a^10*b^6*tan(1/2*d*x + 1/2*c)^9 - 12936 0*a^8*b^8*tan(1/2*d*x + 1/2*c)^9 - 62832*a^6*b^10*tan(1/2*d*x + 1/2*c)^...
Time = 10.51 (sec) , antiderivative size = 2184, normalized size of antiderivative = 5.31 \[ \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
((686*a^8*b + 80*b^9 - 408*a^2*b^7 + 842*a^4*b^5 - 885*a^6*b^3)/(280*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + ( d*x)/2)*(50*a^10 + 80*b^10 - 416*a^2*b^8 + 884*a^4*b^6 - 970*a^6*b^4 + 957 *a^8*b^2))/(40*a*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^ 8*b^2)) + (tan(c/2 + (d*x)/2)^9*(3840*b^14 - 90*a^14 - 13184*a^2*b^12 + 89 76*a^4*b^10 + 18480*a^6*b^8 - 28892*a^8*b^6 + 35630*a^10*b^4 + 7415*a^12*b ^2))/(40*a^5*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^ 2)) + (tan(c/2 + (d*x)/2)^5*(90*a^14 + 3840*b^14 - 13184*a^2*b^12 + 8976*a ^4*b^10 + 19040*a^6*b^8 - 21592*a^8*b^6 + 47580*a^10*b^4 + 13165*a^12*b^2) )/(40*a^5*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^11*(160*b^12 - 12*a^12 - 768*a^2*b^10 + 1440*a^4*b^ 8 - 1280*a^6*b^6 + 587*a^8*b^4 + 224*a^10*b^2))/(4*a^3*(a^10 - b^10 + 5*a^ 2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^3*(60* a^12 + 800*b^12 - 3840*a^2*b^10 + 7192*a^4*b^8 - 6248*a^6*b^6 + 5475*a^8*b ^4 + 2996*a^10*b^2))/(20*a^3*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^ 6*b^4 - 5*a^8*b^2)) - (tan(c/2 + (d*x)/2)^13*(10*a^10 - 16*b^10 + 80*a^2*b ^8 - 160*a^4*b^6 + 160*a^6*b^4 - 83*a^8*b^2))/(8*a*(a^10 - b^10 + 5*a^2*b^ 8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^6*(560*a^1 4*b + 640*b^15 - 864*a^2*b^13 - 4304*a^4*b^11 + 12140*a^6*b^9 - 8312*a^8*b ^7 + 8855*a^10*b^5 + 10590*a^12*b^3))/(10*a^6*(a^10 - b^10 + 5*a^2*b^8 ...