Integrand size = 25, antiderivative size = 318 \[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=-\frac {2 a \cos (e+f x)}{5 (c+d) f (c+d \sin (e+f x))^{5/2}}-\frac {2 a (3 c-5 d) \cos (e+f x)}{15 (c-d) (c+d)^2 f (c+d \sin (e+f x))^{3/2}}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{15 (c-d)^2 (c+d)^3 f \sqrt {c+d \sin (e+f x)}}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{15 (c-d)^2 d (c+d)^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a (3 c-5 d) \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{15 (c-d) d (c+d)^2 f \sqrt {c+d \sin (e+f x)}} \] Output:
-2/5*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(5/2)-2/15*a*(3*c-5*d)*cos(f*x+ e)/(c-d)/(c+d)^2/f/(c+d*sin(f*x+e))^(3/2)-2/15*a*(3*c^2-20*c*d+9*d^2)*cos( f*x+e)/(c-d)^2/(c+d)^3/f/(c+d*sin(f*x+e))^(1/2)+2/15*a*(3*c^2-20*c*d+9*d^2 )*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f* x+e))^(1/2)/(c-d)^2/d/(c+d)^3/f/((c+d*sin(f*x+e))/(c+d))^(1/2)+2/15*a*(3*c -5*d)*InverseJacobiAM(1/2*e-1/4*Pi+1/2*f*x,2^(1/2)*(d/(c+d))^(1/2))*((c+d* sin(f*x+e))/(c+d))^(1/2)/(c-d)/d/(c+d)^2/f/(c+d*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 7.80 (sec) , antiderivative size = 2815, normalized size of antiderivative = 8.85 \[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2),x]
Output:
a*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*(3*c^2 - 20*c*d + 9*d ^2)*Csc[e]*Sec[e])/(15*(c - d)^2*d*(c + d)^3*f) + (2*Csc[e]*(c*Cos[e] + d* Sin[f*x]))/(5*d*(c + d)*f*(c + d*Sin[e + f*x])^3) - (2*Csc[e]*(5*c*Cos[e] - 3*d*Cos[e] - 3*c*Sin[f*x] + 5*d*Sin[f*x]))/(15*(c - d)*(c + d)^2*f*(c + d*Sin[e + f*x])^2) - (2*Csc[e]*(15*c^2*Cos[e] - 12*c*d*Cos[e] + 5*d^2*Cos[ e] - 3*c^2*Sin[f*x] + 20*c*d*Sin[f*x] - 9*d^2*Sin[f*x]))/(15*(c - d)^2*(c + d)^3*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2]) ^2 - (c^2*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -( (Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sq rt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e ]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[C ot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcT an[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d *Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*S qrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqr t[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]] *Sqrt[1 + Cot[e]^2]*Sin[e]]))/(5*(c - d)^2*(c + d)^3*f*(Cos[e/2 + (f*x)...
Time = 1.67 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.720, Rules used = {3042, 3233, 27, 3042, 3233, 27, 3042, 3233, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c+d \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (e+f x)+a}{(c+d \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {2 \int -\frac {5 a (c-d)+3 a \sin (e+f x) (c-d)}{2 (c+d \sin (e+f x))^{5/2}}dx}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 a (c-d)+3 a \sin (e+f x) (c-d)}{(c+d \sin (e+f x))^{5/2}}dx}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a (c-d)+3 a \sin (e+f x) (c-d)}{(c+d \sin (e+f x))^{5/2}}dx}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {3 a (5 c-3 d) (c-d)+a (3 c-5 d) \sin (e+f x) (c-d)}{2 (c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 a (5 c-3 d) (c-d)+a (3 c-5 d) \sin (e+f x) (c-d)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 a (5 c-3 d) (c-d)+a (3 c-5 d) \sin (e+f x) (c-d)}{(c+d \sin (e+f x))^{3/2}}dx}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {a (c-d) \left (15 c^2-12 d c+5 d^2\right )-a (c-d) \left (3 c^2-20 d c+9 d^2\right ) \sin (e+f x)}{2 \sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a (c-d) \left (15 c^2-12 d c+5 d^2\right )-a (c-d) \left (3 c^2-20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a (c-d) \left (15 c^2-12 d c+5 d^2\right )-a (c-d) \left (3 c^2-20 d c+9 d^2\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}}dx}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (3 c-5 d) (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (3 c-5 d) (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \int \sqrt {c+d \sin (e+f x)}dx}{d}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (3 c-5 d) (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (3 c-5 d) (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}dx}{d \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (3 c-5 d) (c-d)^2 (c+d) \int \frac {1}{\sqrt {c+d \sin (e+f x)}}dx}{d}-\frac {2 a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (3 c-5 d) (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {a (3 c-5 d) (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}dx}{d \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 a (3 c-5 d) (c-d)^2 (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a (c-d) \left (3 c^2-20 c d+9 d^2\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}}{c^2-d^2}-\frac {2 a \left (3 c^2-20 c d+9 d^2\right ) \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}}{3 \left (c^2-d^2\right )}-\frac {2 a (3 c-5 d) \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^{3/2}}}{5 \left (c^2-d^2\right )}-\frac {2 a \cos (e+f x)}{5 f (c+d) (c+d \sin (e+f x))^{5/2}}\) |
Input:
Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(7/2),x]
Output:
(-2*a*Cos[e + f*x])/(5*(c + d)*f*(c + d*Sin[e + f*x])^(5/2)) + ((-2*a*(3*c - 5*d)*Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^(3/2)) + ((-2*a*(3 *c^2 - 20*c*d + 9*d^2)*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) + ((-2*a*(c - d)*(3*c^2 - 20*c*d + 9*d^2)*EllipticE[(e - Pi/2 + f*x)/2, (2 *d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(d*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*(3*c - 5*d)*(c - d)^2*(c + d)*EllipticF[(e - Pi/2 + f*x)/2, ( 2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]]))/(c^2 - d^2))/(3*(c^2 - d^2)))/(5*(c^2 - d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1042\) vs. \(2(299)=598\).
Time = 2.86 (sec) , antiderivative size = 1043, normalized size of antiderivative = 3.28
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1043\) |
| parts | \(\text {Expression too large to display}\) | \(1622\) |
Input:
int((a+sin(f*x+e)*a)/(c+d*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)
Output:
(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)*a*(1/d*(2/3/(c^2-d^2)/d*(-(-c-d*si n(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e)+c/d)^2+8/3*d*cos(f*x+e)^2/(c^2-d ^2)^2*c/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)+2*(3*c^2+d^2)/(3*c^4-6*c^2 *d^2+3*d^4)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d) )^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^ (1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+8/3*c* d/(c^2-d^2)^2*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+ d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2 )^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^( 1/2))+EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))))-(c-d )/d*(2/5/(c^2-d^2)/d^2*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin(f*x+e) +c/d)^3+16/15*c/(c^2-d^2)^2/d*(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^(1/2)/(sin (f*x+e)+c/d)^2+2/15*cos(f*x+e)^2*d/(c^2-d^2)^3*(23*c^2+9*d^2)/(-(-c-d*sin( f*x+e))*cos(f*x+e)^2)^(1/2)+2*(15*c^3+17*c*d^2)/(15*c^6-45*c^4*d^2+45*c^2* d^4-15*d^6)*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-sin(f*x+e))/(c+d) )^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e))*cos(f*x+e)^2)^ (1/2)*EllipticF(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))+2/15*d *(23*c^2+9*d^2)/(c^2-d^2)^3*(c/d-1)*((c+d*sin(f*x+e))/(c-d))^(1/2)*(d*(1-s in(f*x+e))/(c+d))^(1/2)*(-d*(1+sin(f*x+e))/(c-d))^(1/2)/(-(-c-d*sin(f*x+e) )*cos(f*x+e)^2)^(1/2)*((-c/d-1)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 1395, normalized size of antiderivative = 4.39 \[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(7/2),x, algorithm="fricas")
Output:
-2/45*((6*a*c^6 + 5*a*c^5*d + 30*a*c^3*d^3 - 54*a*c^2*d^4 + 45*a*c*d^5 - 3 *(6*a*c^4*d^2 + 5*a*c^3*d^3 - 18*a*c^2*d^4 + 15*a*c*d^5)*cos(f*x + e)^2 + (18*a*c^5*d + 15*a*c^4*d^2 - 48*a*c^3*d^3 + 50*a*c^2*d^4 - 18*a*c*d^5 + 15 *a*d^6 - (6*a*c^3*d^3 + 5*a*c^2*d^4 - 18*a*c*d^5 + 15*a*d^6)*cos(f*x + e)^ 2)*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^ 2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) + (6*a*c^6 + 5*a*c^5*d + 30*a*c^3*d^3 - 54*a*c^2*d^4 + 45 *a*c*d^5 - 3*(6*a*c^4*d^2 + 5*a*c^3*d^3 - 18*a*c^2*d^4 + 15*a*c*d^5)*cos(f *x + e)^2 + (18*a*c^5*d + 15*a*c^4*d^2 - 48*a*c^3*d^3 + 50*a*c^2*d^4 - 18* a*c*d^5 + 15*a*d^6 - (6*a*c^3*d^3 + 5*a*c^2*d^4 - 18*a*c*d^5 + 15*a*d^6)*c os(f*x + e)^2)*sin(f*x + e))*sqrt(-1/2*I*d)*weierstrassPInverse(-4/3*(4*c^ 2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(3*I*a*c^5*d - 20*I*a*c^4*d^2 + 18*I*a* c^3*d^3 - 60*I*a*c^2*d^4 + 27*I*a*c*d^5 + 3*(-3*I*a*c^3*d^3 + 20*I*a*c^2*d ^4 - 9*I*a*c*d^5)*cos(f*x + e)^2 + (9*I*a*c^4*d^2 - 60*I*a*c^3*d^3 + 30*I* a*c^2*d^4 - 20*I*a*c*d^5 + 9*I*a*d^6 + (-3*I*a*c^2*d^4 + 20*I*a*c*d^5 - 9* I*a*d^6)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(1/2*I*d)*weierstrassZeta(-4/3* (4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse( -4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f* x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*(-3*I*a*c^5*d + 20*I*a*c^4...
Timed out. \[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(7/2),x, algorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(7/2), x)
\[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(7/2),x, algorithm="giac")
Output:
integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(7/2), x)
Timed out. \[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=\int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(7/2),x)
Output:
int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(7/2), x)
\[ \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx=a \left (\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x +\int \frac {\sqrt {\sin \left (f x +e \right ) d +c}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4} d^{4}+4 \sin \left (f x +e \right )^{3} c \,d^{3}+6 \sin \left (f x +e \right )^{2} c^{2} d^{2}+4 \sin \left (f x +e \right ) c^{3} d +c^{4}}d x \right ) \] Input:
int((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(7/2),x)
Output:
a*(int(sqrt(sin(e + f*x)*d + c)/(sin(e + f*x)**4*d**4 + 4*sin(e + f*x)**3* c*d**3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c**3*d + c**4),x) + int((sqrt(sin(e + f*x)*d + c)*sin(e + f*x))/(sin(e + f*x)**4*d**4 + 4*sin( e + f*x)**3*c*d**3 + 6*sin(e + f*x)**2*c**2*d**2 + 4*sin(e + f*x)*c**3*d + c**4),x))