Integrand size = 29, antiderivative size = 717 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\frac {2 (c-d) \sqrt {c+d} \left (12 b c+27 d-7 b^2 d\right ) E\left (\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right )|\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{3 (3-b)^2 b^2 (3+b)^{3/2} f}+\frac {2 d^2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {b (c+d)}{(3+b) d},\arcsin \left (\frac {\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}\right ),\frac {(3-b) (c+d)}{(3+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-3 d) (1-\sin (e+f x))}{(c+d) (3+b \sin (e+f x))}} \sqrt {\frac {(b c-3 d) (1+\sin (e+f x))}{(c-d) (3+b \sin (e+f x))}} (3+b \sin (e+f x))}{b^3 \sqrt {3+b} f}+\frac {2 (b c-3 d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b \left (9-b^2\right ) f (3+b \sin (e+f x))^{3/2}}+\frac {2 \left (27 b (c-2 d) d+81 d^2+3 b^2 \left (3 c^2-4 c d-2 d^2\right )+b^3 \left (c^2-7 c d+9 d^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {3+b \sin (e+f x)}}{\sqrt {3+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(3+b) (c-d)}{(3-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-3 d) (1-\sin (e+f x))}{(3+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-3 d) (1+\sin (e+f x))}{(3-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{3 (3-b)^2 b^3 \sqrt {3+b} \sqrt {c+d} f} \]
2/3*(c-d)*(3*a^2*d+4*a*b*c-7*b^2*d)*EllipticE((a+b)^(1/2)*(c+d*sin(f*x+e)) ^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),((a-b)*(c+d)/(a+b)/(c-d))^(1/2)) *sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c)*(1-sin(f*x+e))/(c+d) /(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/(c-d)/(a+b*sin(f*x+e)) )^(1/2)/(a-b)^2/b^2/(a+b)^(3/2)/f+2*d^2*EllipticPi((a+b)^(1/2)*(c+d*sin(f* x+e))^(1/2)/(c+d)^(1/2)/(a+b*sin(f*x+e))^(1/2),b*(c+d)/(a+b)/d,((a-b)*(c+d )/(a+b)/(c-d))^(1/2))*sec(f*x+e)*(a+b*sin(f*x+e))*(c+d)^(1/2)*(-(-a*d+b*c) *(1-sin(f*x+e))/(c+d)/(a+b*sin(f*x+e)))^(1/2)*((-a*d+b*c)*(1+sin(f*x+e))/( c-d)/(a+b*sin(f*x+e)))^(1/2)/b^3/f/(a+b)^(1/2)+2/3*(3*a^2*b*(c-2*d)*d+3*a^ 3*d^2+a*b^2*(3*c^2-4*c*d-2*d^2)+b^3*(c^2-7*c*d+9*d^2))*EllipticF((c+d)^(1/ 2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/ (a-b)/(c+d))^(1/2))*sec(f*x+e)*(c+d*sin(f*x+e))*((-a*d+b*c)*(1-sin(f*x+e)) /(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin( f*x+e)))^(1/2)/(a-b)^2/b^3/f/(a+b)^(1/2)/(c+d)^(1/2)+2/3*(-a*d+b*c)^2*cos( f*x+e)*(c+d*sin(f*x+e))^(1/2)/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^(3/2)
Leaf count is larger than twice the leaf count of optimal. \(2103\) vs. \(2(717)=1434\).
Time = 11.02 (sec) , antiderivative size = 2103, normalized size of antiderivative = 2.93 \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\text {Result too large to show} \]
(Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]*((-2*(b^2*c^2*Cos[e + f *x] - 6*b*c*d*Cos[e + f*x] + 9*d^2*Cos[e + f*x]))/(3*b*(-9 + b^2)*(3 + b*S in[e + f*x])^2) - (2*(-12*b^2*c^2*Cos[e + f*x] + 9*b*c*d*Cos[e + f*x] + 7* b^3*c*d*Cos[e + f*x] + 81*d^2*Cos[e + f*x] - 21*b^2*d^2*Cos[e + f*x]))/(3* b*(-9 + b^2)^2*(3 + b*Sin[e + f*x]))))/f + ((-4*(-(b*c) + 3*d)*(27*b*c^3 + b^3*c^3 - 24*b^2*c^2*d + 18*b*c*d^2 + 2*b^3*c*d^2 - 27*d^3 + 3*b^2*d^3)*S qrt[((c + d)*Cot[(-e + Pi/2 - f*x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[( (-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/ Sqrt[2]], (2*(-(b*c) + 3*d))/((3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi /2 - f*x)/2]^4*Sqrt[((c + d)*Csc[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x ]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[ c + d*Sin[e + f*x]]) - 4*(-(b*c) + 3*d)*(12*b^2*c^3 + 27*b*c^2*d - 7*b^3*c ^2*d - 108*c*d^2 + 9*b*d^3 + 3*b^3*d^3)*((Sqrt[((c + d)*Cot[(-e + Pi/2 - f *x)/2]^2)/(-c + d)]*EllipticF[ArcSin[Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/ 2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)]/Sqrt[2]], (2*(-(b*c) + 3*d))/(( 3 + b)*(-c + d))]*Sec[e + f*x]*Sin[(-e + Pi/2 - f*x)/2]^4*Sqrt[((c + d)*Cs c[(-e + Pi/2 - f*x)/2]^2*(3 + b*Sin[e + f*x]))/(-(b*c) + 3*d)]*Sqrt[((-3 - b)*Csc[(-e + Pi/2 - f*x)/2]^2*(c + d*Sin[e + f*x]))/(-(b*c) + 3*d)])/((3 + b)*(c + d)*Sqrt[3 + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) - (Sqrt...
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.
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {2 \int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{2 (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -\frac {a^2 d^3-3 \left (a^2-b^2\right ) \sin ^2(e+f x) d^3+7 b^2 c^2 d-a b c \left (3 c^2+5 d^2\right )-\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}+\frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {\int -\frac {3 a b c^3-7 b^2 d c^2+5 a b d^2 c-a^2 d^3+3 \left (a^2-b^2\right ) d^3 \sin ^2(e+f x)+\left (-\left (\left (c^3+9 d^2 c\right ) b^2\right )+a d \left (5 c^2+3 d^2\right ) b+2 a^2 c d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}dx}{3 b \left (a^2-b^2\right )}\) |
3.8.95.3.1 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_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* (n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin [e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 67.75 (sec) , antiderivative size = 1793484, normalized size of antiderivative = 2501.37
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
integral((d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)*sqrt(b*sin( f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(3*a*b^2*cos(f*x + e)^2 - a^3 - 3*a *b^2 + (b^3*cos(f*x + e)^2 - 3*a^2*b - b^3)*sin(f*x + e)), x)
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(c+d \sin (e+f x))^{5/2}}{(3+b \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]