Integrand size = 25, antiderivative size = 181 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx=-\frac {42 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{65 a^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {42 \sin (c+d x)}{65 a^2 d e^3 \sqrt {e \cos (c+d x)}}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2}-\frac {2}{13 d e (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )} \]
14/65*sin(d*x+c)/a^2/d/e/(e*cos(d*x+c))^(5/2)-2/13/d/e/(e*cos(d*x+c))^(5/2 )/(a+a*sin(d*x+c))^2-2/13/d/e/(e*cos(d*x+c))^(5/2)/(a^2+a^2*sin(d*x+c))+42 /65*sin(d*x+c)/a^2/d/e^3/(e*cos(d*x+c))^(1/2)-42/65*(cos(1/2*d*x+1/2*c)^2) ^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x +c))^(1/2)/a^2/d/e^4/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {17}{4},-\frac {1}{4},\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{5/4}}{20 \sqrt [4]{2} a^2 d e (e \cos (c+d x))^{5/2}} \]
(Hypergeometric2F1[-5/4, 17/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d* x])^(5/4))/(20*2^(1/4)*a^2*d*e*(e*Cos[c + d*x])^(5/2))
Time = 0.82 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 3160, 3042, 3162, 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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {9 \int \frac {1}{(e \cos (c+d x))^{7/2} (\sin (c+d x) a+a)}dx}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \int \frac {1}{(e \cos (c+d x))^{7/2} (\sin (c+d x) a+a)}dx}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{(e \cos (c+d x))^{7/2}}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {3 \int \frac {1}{(e \cos (c+d x))^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {3 \int \frac {1}{\left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \cos (c+d x)}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\int \sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\cos (c+d x)}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {\sqrt {e \cos (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {9 \left (\frac {7 \left (\frac {3 \left (\frac {2 \sin (c+d x)}{d e \sqrt {e \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}\right )}{5 e^2}+\frac {2 \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}\right )}{9 a}-\frac {2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}\right )}{13 a}-\frac {2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}\) |
-2/(13*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2) + (9*(-2/(9*d*e* (e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])) + (7*((2*Sin[c + d*x])/(5*d*e *(e*Cos[c + d*x])^(5/2)) + (3*((-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x )/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])))/(5*e^2)))/(9*a)))/(13*a)
3.3.52.3.1 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 _)])^(m_), x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x ])^m/(a*f*g*(2*m + p + 1))), x] + Simp[(m + p + 1)/(a*(2*m + p + 1)) Int[ (g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] & & IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[b*((g*Cos[e + f*x])^(p + 1)/(a*f*g*(p - 1)*(a + b*S in[e + f*x]))), x] + Simp[p/(a*(p - 1)) Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && !GeQ[p, 1] && Intege rQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(669\) vs. \(2(185)=370\).
Time = 10.38 (sec) , antiderivative size = 670, normalized size of antiderivative = 3.70
2/65/(64*sin(1/2*d*x+1/2*c)^12-192*sin(1/2*d*x+1/2*c)^10+240*sin(1/2*d*x+1 /2*c)^8-160*sin(1/2*d*x+1/2*c)^6+60*sin(1/2*d*x+1/2*c)^4-12*sin(1/2*d*x+1/ 2*c)^2+1)/a^2/sin(1/2*d*x+1/2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)/e^3*( 2688*sin(1/2*d*x+1/2*c)^14*cos(1/2*d*x+1/2*c)-1344*(2*sin(1/2*d*x+1/2*c)^2 -1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*sin(1/2*d*x+1/2*c)^12-8064*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+403 2*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*( sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^10+10304*sin(1/2*d*x+1/2*c) ^10*cos(1/2*d*x+1/2*c)-5040*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(2*sin(1 /2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^8 -7168*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+3360*EllipticE(cos(1/2*d*x+1 /2*c),2^(1/2))*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*sin(1/2*d*x+1/2*c)^6+2896*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-1260* (2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(si n(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-656*sin(1/2*d*x+1/2*c)^4*co s(1/2*d*x+1/2*c)+252*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d* x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+86*sin (1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-10*sin(1 /2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx=\frac {21 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{5} + 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{5} - 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 i \, \sqrt {2} \cos \left (d x + c\right )^{3}\right )} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (42 \, \cos \left (d x + c\right )^{4} - 28 \, \cos \left (d x + c\right )^{2} + {\left (21 \, \cos \left (d x + c\right )^{4} - 35 \, \cos \left (d x + c\right )^{2} - 9\right )} \sin \left (d x + c\right ) - 4\right )} \sqrt {e \cos \left (d x + c\right )}}{65 \, {\left (a^{2} d e^{4} \cos \left (d x + c\right )^{5} - 2 \, a^{2} d e^{4} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d e^{4} \cos \left (d x + c\right )^{3}\right )}} \]
1/65*(21*(-I*sqrt(2)*cos(d*x + c)^5 + 2*I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) + 2*I*sqrt(2)*cos(d*x + c)^3)*sqrt(e)*weierstrassZeta(-4, 0, weierstra ssPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(I*sqrt(2)*cos(d*x + c)^5 - 2*I*sqrt(2)*cos(d*x + c)^3*sin(d*x + c) - 2*I*sqrt(2)*cos(d*x + c )^3)*sqrt(e)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c ) - I*sin(d*x + c))) + 2*(42*cos(d*x + c)^4 - 28*cos(d*x + c)^2 + (21*cos( d*x + c)^4 - 35*cos(d*x + c)^2 - 9)*sin(d*x + c) - 4)*sqrt(e*cos(d*x + c)) )/(a^2*d*e^4*cos(d*x + c)^5 - 2*a^2*d*e^4*cos(d*x + c)^3*sin(d*x + c) - 2* a^2*d*e^4*cos(d*x + c)^3)
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]