Integrand size = 23, antiderivative size = 131 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}-\frac {2 a \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}-\frac {6 a \cos (c+d x)}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {6 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d e^4 \sqrt {\sin (c+d x)}} \]
-2/5*b/d/e/(e*sin(d*x+c))^(5/2)-2/5*a*cos(d*x+c)/d/e/(e*sin(d*x+c))^(5/2)- 6/5*a*cos(d*x+c)/d/e^3/(e*sin(d*x+c))^(1/2)+6/5*a*(sin(1/2*c+1/4*Pi+1/2*d* x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x), 2^(1/2))*(e*sin(d*x+c))^(1/2)/d/e^4/sin(d*x+c)^(1/2)
Time = 0.57 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=\frac {-4 b-7 a \cos (c+d x)+3 a \cos (3 (c+d x))+12 a E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sin ^{\frac {5}{2}}(c+d x)}{10 d e (e \sin (c+d x))^{5/2}} \]
(-4*b - 7*a*Cos[c + d*x] + 3*a*Cos[3*(c + d*x)] + 12*a*EllipticE[(-2*c + P i - 2*d*x)/4, 2]*Sin[c + d*x]^(5/2))/(10*d*e*(e*Sin[c + d*x])^(5/2))
Time = 0.54 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3148, 3042, 3116, 3042, 3116, 3042, 3121, 3042, 3119}
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.
| \(\displaystyle \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a-b \sin \left (c+d x-\frac {\pi }{2}\right )}{\left (e \cos \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{7/2}}dx-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{7/2}}dx-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle a \left (\frac {3 \int \frac {1}{(e \sin (c+d x))^{3/2}}dx}{5 e^2}-\frac {2 \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}\right )-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \int \frac {1}{(e \sin (c+d x))^{3/2}}dx}{5 e^2}-\frac {2 \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}\right )-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle a \left (\frac {3 \left (-\frac {\int \sqrt {e \sin (c+d x)}dx}{e^2}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )}{5 e^2}-\frac {2 \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}\right )-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \left (-\frac {\int \sqrt {e \sin (c+d x)}dx}{e^2}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )}{5 e^2}-\frac {2 \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}\right )-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle a \left (\frac {3 \left (-\frac {\sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{e^2 \sqrt {\sin (c+d x)}}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )}{5 e^2}-\frac {2 \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}\right )-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {3 \left (-\frac {\sqrt {e \sin (c+d x)} \int \sqrt {\sin (c+d x)}dx}{e^2 \sqrt {\sin (c+d x)}}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )}{5 e^2}-\frac {2 \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}\right )-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle a \left (\frac {3 \left (-\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}\right )}{5 e^2}-\frac {2 \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}\right )-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\) |
(-2*b)/(5*d*e*(e*Sin[c + d*x])^(5/2)) + a*((-2*Cos[c + d*x])/(5*d*e*(e*Sin [c + d*x])^(5/2)) + (3*((-2*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2* EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]])))/(5*e^2))
3.1.40.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Time = 2.57 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {-\frac {2 b}{5 e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {a \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+6 \left (\sin ^{5}\left (d x +c \right )\right )-4 \left (\sin ^{3}\left (d x +c \right )\right )-2 \sin \left (d x +c \right )\right )}{5 e^{3} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(187\) |
| parts | \(\frac {a \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) E\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+6 \left (\sin ^{5}\left (d x +c \right )\right )-4 \left (\sin ^{3}\left (d x +c \right )\right )-2 \sin \left (d x +c \right )\right )}{5 e^{3} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b}{5 d e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(189\) |
(-2/5*b/e/(e*sin(d*x+c))^(5/2)+1/5*a/e^3*(6*(1-sin(d*x+c))^(1/2)*(2*sin(d* x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2)) -3*(1-sin(d*x+c))^(1/2)*(2*sin(d*x+c)+2)^(1/2)*sin(d*x+c)^(7/2)*EllipticF( (1-sin(d*x+c))^(1/2),1/2*2^(1/2))+6*sin(d*x+c)^5-4*sin(d*x+c)^3-2*sin(d*x+ c))/sin(d*x+c)^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.37 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=-\frac {3 \, {\left (i \, \sqrt {2} a \cos \left (d x + c\right )^{2} - i \, \sqrt {2} a\right )} \sqrt {-i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, \sqrt {2} a \cos \left (d x + c\right )^{2} + i \, \sqrt {2} a\right )} \sqrt {i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, a \cos \left (d x + c\right )^{3} - 4 \, a \cos \left (d x + c\right ) - b\right )} \sqrt {e \sin \left (d x + c\right )}}{5 \, {\left (d e^{4} \cos \left (d x + c\right )^{2} - d e^{4}\right )} \sin \left (d x + c\right )} \]
-1/5*(3*(I*sqrt(2)*a*cos(d*x + c)^2 - I*sqrt(2)*a)*sqrt(-I*e)*sin(d*x + c) *weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*(-I*sqrt(2)*a*cos(d*x + c)^2 + I*sqrt(2)*a)*sqrt(I*e)*sin(d*x + c)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d *x + c))) + 2*(3*a*cos(d*x + c)^3 - 4*a*cos(d*x + c) - b)*sqrt(e*sin(d*x + c)))/((d*e^4*cos(d*x + c)^2 - d*e^4)*sin(d*x + c))
Timed out. \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=\int { \frac {b \cos \left (d x + c\right ) + a}{\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=\int { \frac {b \cos \left (d x + c\right ) + a}{\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=\int \frac {a+b\,\cos \left (c+d\,x\right )}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}} \,d x \]