Integrand size = 35, antiderivative size = 712 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^7 (a-b)^2 (a+b)^{5/2} \sqrt {d} f}-\frac {16 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right ) \tan (e+f x)}{693 a^6 (a-b)^2 (a+b)^{5/2} \sqrt {d} f} \]
2/11*cos(f*x+e)^5*(d*sin(f*x+e))^(1/2)/a/d/f/(a+b*sin(f*x+e))^(11/2)-20/99 *(a^2-b^2)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^2/b^2/d/f/(a+b*sin(f*x+e))^(9 /2)+80/693*(3*a^2+2*b^2)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^3/b^2/d/f/(a+b* sin(f*x+e))^(7/2)-4/231*(5*a^4-17*a^2*b^2+16*b^4)*cos(f*x+e)*(d*sin(f*x+e) )^(1/2)/a^4/b^2/(a^2-b^2)/d/f/(a+b*sin(f*x+e))^(5/2)-8/693*(5*a^6-22*a^4*b ^2+65*a^2*b^4-32*b^6)*cos(f*x+e)*(d*sin(f*x+e))^(1/2)/a^5/b^2/(a^2-b^2)^2/ d/f/(a+b*sin(f*x+e))^(3/2)+16/693*b*(93*a^4-93*a^2*b^2+32*b^4)*cos(f*x+e)/ a^5/(a^2-b^2)^3/f/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e))^(1/2)-16/693*b*(93 *a^4-93*a^2*b^2+32*b^4)*EllipticE(d^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/ 2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1/2))*(a*(1-csc(f*x+e))/(a+b))^(1/ 2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x+e)/a^7/(a-b)^2/(a+b)^(5/2)/f/d^( 1/2)-16/693*(45*a^4-48*a^3*b-69*a^2*b^2+24*a*b^3+32*b^4)*EllipticF(d^(1/2) *(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2),((-a-b)/(a-b))^(1 /2))*(a*(1-csc(f*x+e))/(a+b))^(1/2)*(a*(1+csc(f*x+e))/(a-b))^(1/2)*tan(f*x +e)/a^6/(a-b)^2/(a+b)^(5/2)/f/d^(1/2)
Result contains complex when optimal does not.
Time = 8.57 (sec) , antiderivative size = 1906, normalized size of antiderivative = 2.68 \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx =\text {Too large to display} \]
(Sin[e + f*x]*Sqrt[a + b*Sin[e + f*x]]*((2*(a^4*Cos[e + f*x] - 2*a^2*b^2*C os[e + f*x] + b^4*Cos[e + f*x]))/(11*a*b^4*(a + b*Sin[e + f*x])^6) - (4*(1 8*a^4*Cos[e + f*x] - 13*a^2*b^2*Cos[e + f*x] - 5*b^4*Cos[e + f*x]))/(99*a^ 2*b^4*(a + b*Sin[e + f*x])^5) + (4*(189*a^4*Cos[e + f*x] - 3*a^2*b^2*Cos[e + f*x] + 40*b^4*Cos[e + f*x]))/(693*a^3*b^4*(a + b*Sin[e + f*x])^4) - (4* (42*a^6*Cos[e + f*x] - 37*a^4*b^2*Cos[e + f*x] - 17*a^2*b^4*Cos[e + f*x] + 16*b^6*Cos[e + f*x]))/(231*a^4*b^4*(a^2 - b^2)*(a + b*Sin[e + f*x])^3) + (2*(63*a^8*Cos[e + f*x] - 146*a^6*b^2*Cos[e + f*x] + 151*a^4*b^4*Cos[e + f *x] - 260*a^2*b^6*Cos[e + f*x] + 128*b^8*Cos[e + f*x]))/(693*a^5*b^4*(a^2 - b^2)^2*(a + b*Sin[e + f*x])^2) - (16*(93*a^4*b^2*Cos[e + f*x] - 93*a^2*b ^4*Cos[e + f*x] + 32*b^6*Cos[e + f*x]))/(693*a^6*(a^2 - b^2)^3*(a + b*Sin[ e + f*x]))))/(f*Sqrt[d*Sin[e + f*x]]) + (8*Sqrt[Sin[e + f*x]]*((4*a*(45*a^ 6 - 114*a^4*b^2 + 101*a^2*b^4 - 32*b^6)*Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x )/2]^2)/(-a + b)]*EllipticF[ArcSin[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b *Sin[e + f*x]))/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[-(((a + b)*Csc[(-e + Pi/2 - f*x)/2]^2*Sin[e + f*x])/a)]*S qrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a])/((a + b)*Sqrt[Si n[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + 4*a*(-93*a^5*b + 93*a^3*b^3 - 32*a *b^5)*((Sqrt[((a + b)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-a + b)]*EllipticF[ArcS in[Sqrt[(Csc[(-e + Pi/2 - f*x)/2]^2*(a + b*Sin[e + f*x]))/a]/Sqrt[2]], ...
Time = 3.58 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.486, Rules used = {3042, 3366, 3042, 3370, 27, 3042, 3534, 3042, 3534, 27, 3042, 3472, 3042, 3477, 3042, 3295, 3473}
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.
\(\displaystyle \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (e+f x)^6}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}}dx\) |
\(\Big \downarrow \) 3366 |
\(\displaystyle \frac {10 \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{11/2}}dx}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {10 \int \frac {\cos (e+f x)^4}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{11/2}}dx}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3370 |
\(\displaystyle \frac {10 \left (-\frac {4 \int \frac {5 a^2-14 b \sin (e+f x) a-48 b^2-\left (15 a^2-32 b^2\right ) \sin ^2(e+f x)}{4 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {10 \left (-\frac {\int \frac {5 a^2-14 b \sin (e+f x) a-48 b^2-\left (15 a^2-32 b^2\right ) \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {10 \left (-\frac {\int \frac {5 a^2-14 b \sin (e+f x) a-48 b^2-\left (15 a^2-32 b^2\right ) \sin (e+f x)^2}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}}dx}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \int \frac {-3 \left (5 a^4-17 b^2 a^2+16 b^4\right ) d \sin ^2(e+f x)-10 a b \left (a^2-4 b^2\right ) d \sin (e+f x)+\left (5 a^4-107 b^2 a^2+96 b^4\right ) d}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}dx}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \int \frac {-3 \left (5 a^4-17 b^2 a^2+16 b^4\right ) d \sin (e+f x)^2-10 a b \left (a^2-4 b^2\right ) d \sin (e+f x)+\left (5 a^4-107 b^2 a^2+96 b^4\right ) d}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}dx}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3534 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \int -\frac {3 \left (b^2 \left (45 a^4-69 b^2 a^2+32 b^4\right ) d^2-24 a b^3 \left (2 a^2-b^2\right ) d^2 \sin (e+f x)\right )}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{3 a d \left (a^2-b^2\right )}+\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \int \frac {b^2 \left (45 a^4-69 b^2 a^2+32 b^4\right ) d^2-24 a b^3 \left (2 a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \int \frac {b^2 \left (45 a^4-69 b^2 a^2+32 b^4\right ) d^2-24 a b^3 \left (2 a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}dx}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3472 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \int \frac {\left (93 a^4-93 b^2 a^2+32 b^4\right ) d^2 b^3+a \left (45 a^4-21 b^2 a^2+8 b^4\right ) d^2 \sin (e+f x) b^2}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \int \frac {\left (93 a^4-93 b^2 a^2+32 b^4\right ) d^2 b^3+a \left (45 a^4-21 b^2 a^2+8 b^4\right ) d^2 \sin (e+f x) b^2}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3477 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \left (b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx+b^2 d (a-b) \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx\right )}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \left (b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx+b^2 d (a-b) \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}dx\right )}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3295 |
\(\displaystyle \frac {10 \left (-\frac {\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {d \left (b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \int \frac {\sin (e+f x)+1}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}}dx-\frac {2 b^2 \sqrt {d} (a-b) \sqrt {a+b} \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a f}\right )}{a^2-b^2}+\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}+\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}}{63 a^2 b^2}+\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
\(\Big \downarrow \) 3473 |
\(\displaystyle \frac {10 \left (\frac {8 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{63 a^2 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {2 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{9 a b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac {\frac {6 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{5 a d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}+\frac {2 \left (\frac {2 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{a f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \left (\frac {2 b^3 d^2 \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{f \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {d \left (-\frac {2 b^3 \sqrt {d} (a-b) \sqrt {a+b} \left (93 a^4-93 a^2 b^2+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 f}-\frac {2 b^2 \sqrt {d} (a-b) \sqrt {a+b} \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right ),-\frac {a+b}{a-b}\right )}{a f}\right )}{a^2-b^2}\right )}{a d \left (a^2-b^2\right )}\right )}{5 a d \left (a^2-b^2\right )}}{63 a^2 b^2}\right )}{11 a}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}\) |
(2*Cos[e + f*x]^5*Sqrt[d*Sin[e + f*x]])/(11*a*d*f*(a + b*Sin[e + f*x])^(11 /2)) + (10*((-2*(a^2 - b^2)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(9*a*b^2*d* f*(a + b*Sin[e + f*x])^(9/2)) + (8*(3*a^2 + 2*b^2)*Cos[e + f*x]*Sqrt[d*Sin [e + f*x]])/(63*a^2*b^2*d*f*(a + b*Sin[e + f*x])^(7/2)) - ((6*(5*a^4 - 17* a^2*b^2 + 16*b^4)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(5*a*(a^2 - b^2)*d*f* (a + b*Sin[e + f*x])^(5/2)) + (2*((2*(5*a^6 - 22*a^4*b^2 + 65*a^2*b^4 - 32 *b^6)*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]])/(a*(a^2 - b^2)*f*(a + b*Sin[e + f *x])^(3/2)) - (2*((2*b^3*(93*a^4 - 93*a^2*b^2 + 32*b^4)*d^2*Cos[e + f*x])/ ((a^2 - b^2)*f*Sqrt[d*Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]) + (d*((-2*(a - b)*b^3*Sqrt[a + b]*(93*a^4 - 93*a^2*b^2 + 32*b^4)*Sqrt[d]*Sqrt[(a*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticE[Arc Sin[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])] , -((a + b)/(a - b))]*Tan[e + f*x])/(a^2*f) - (2*(a - b)*b^2*Sqrt[a + b]*( 45*a^4 - 48*a^3*b - 69*a^2*b^2 + 24*a*b^3 + 32*b^4)*Sqrt[d]*Sqrt[(a*(1 - C sc[e + f*x]))/(a + b)]*Sqrt[(a*(1 + Csc[e + f*x]))/(a - b)]*EllipticF[ArcS in[(Sqrt[d]*Sqrt[a + b*Sin[e + f*x]])/(Sqrt[a + b]*Sqrt[d*Sin[e + f*x]])], -((a + b)/(a - b))]*Tan[e + f*x])/(a*f)))/(a^2 - b^2)))/(a*(a^2 - b^2)*d) ))/(5*a*(a^2 - b^2)*d))/(63*a^2*b^2)))/(11*a)
3.13.76.3.1 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[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_))/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[(-g)*(g* Cos[e + f*x])^(p - 1)*Sqrt[d*Sin[e + f*x]]*((a + b*Sin[e + f*x])^(m + 1)/(a *d*f*(m + 1))), x] + Simp[g^2*((2*m + 3)/(2*a*(m + 1))) Int[(g*Cos[e + f* x])^(p - 2)*((a + b*Sin[e + f*x])^(m + 1)/Sqrt[d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && EqQ[m + p + 1/2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 1))), x] + (Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b*S in[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2) )), x] - Simp[1/(a^2*b^2*(m + 1)*(m + 2)) Int[(a + b*Sin[e + f*x])^(m + 2 )*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a ^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] && !LtQ[n, -1] && (LtQ[m , -2] || EqQ[m + n + 4, 0])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(d_.)*sin[(e_.) + (f_.)*( x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)), x_Symbol] :> Simp[2*(A *b - a*B)*(Cos[e + f*x]/(f*(a^2 - b^2)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[d*Sin[ e + f*x]])), x] + Simp[d/(a^2 - b^2) Int[(A*b - a*B + (a*A - b*B)*Sin[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*(d*Sin[e + f*x])^(3/2)), x], x] /; FreeQ[{ a, b, d, e, f, A, B}, x] && NeQ[a^2 - b^2, 0]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int [(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(37794\) vs. \(2(648)=1296\).
Time = 15.30 (sec) , antiderivative size = 37795, normalized size of antiderivative = 53.08
\[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
integral(sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))*cos(f*x + e)^6/(b^7 *d*cos(f*x + e)^8 - (21*a^2*b^5 + 4*b^7)*d*cos(f*x + e)^6 + (35*a^4*b^3 + 63*a^2*b^5 + 6*b^7)*d*cos(f*x + e)^4 - (7*a^6*b + 70*a^4*b^3 + 63*a^2*b^5 + 4*b^7)*d*cos(f*x + e)^2 + (7*a^6*b + 35*a^4*b^3 + 21*a^2*b^5 + b^7)*d - (7*a*b^6*d*cos(f*x + e)^6 - 7*(5*a^3*b^4 + 3*a*b^6)*d*cos(f*x + e)^4 + 7*( 3*a^5*b^2 + 10*a^3*b^4 + 3*a*b^6)*d*cos(f*x + e)^2 - (a^7 + 21*a^5*b^2 + 3 5*a^3*b^4 + 7*a*b^6)*d)*sin(f*x + e)), x)
Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}} \,d x } \]
Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^6}{\sqrt {d\,\sin \left (e+f\,x\right )}\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \]