Integrand size = 25, antiderivative size = 116 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a+a \sin (c+d x))^2}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e \left (a^2+a^2 \sin (c+d x)\right )} \]
2/7*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+ 1/2*c),2^(1/2))*cos(d*x+c)^(1/2)/a^2/d/(e*cos(d*x+c))^(1/2)-2/7*(e*cos(d*x +c))^(1/2)/d/e/(a+a*sin(d*x+c))^2-2/7*(e*cos(d*x+c))^(1/2)/d/e/(a^2+a^2*si n(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx=-\frac {\sqrt {e \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{4},\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{2^{3/4} a^2 d e \sqrt [4]{1+\sin (c+d x)}} \]
-((Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 11/4, 5/4, (1 - Sin[c + d*x ])/2])/(2^(3/4)*a^2*d*e*(1 + Sin[c + d*x])^(1/4)))
Time = 0.52 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3160, 3042, 3162, 3042, 3121, 3042, 3120}
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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^2 \sqrt {e \cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {3 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)}dx}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)}dx}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {e \cos (c+d x)}}dx}{3 a}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 a \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {3 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\) |
(-2*Sqrt[e*Cos[c + d*x]])/(7*d*e*(a + a*Sin[c + d*x])^2) + (3*((2*Sqrt[Cos [c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a*d*Sqrt[e*Cos[c + d*x]]) - (2*Sq rt[e*Cos[c + d*x]])/(3*d*e*(a + a*Sin[c + d*x]))))/(7*a)
3.3.49.3.1 Defintions of rubi rules used
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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. \(371\) vs. \(2(128)=256\).
Time = 5.21 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.21
| method | result | size |
| default | \(-\frac {2 \left (8 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 \left (8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\) | \(372\) |
-2/7/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*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)*(8*(sin(1/ 2*d*x+1/2*c)^2)^(1/2)*EllipticF(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)^6+8*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x +1/2*c)-12*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(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)^4-8*sin(1/2*d*x+1/ 2*c)^4*cos(1/2*d*x+1/2*c)+6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(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 +6*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2 *sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-2*sin (1/2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx=-\frac {{\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} \sin \left (d x + c\right ) - 2 i \, \sqrt {2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} \sin \left (d x + c\right ) + 2 i \, \sqrt {2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, \sqrt {e \cos \left (d x + c\right )} {\left (\sin \left (d x + c\right ) + 2\right )}}{7 \, {\left (a^{2} d e \cos \left (d x + c\right )^{2} - 2 \, a^{2} d e \sin \left (d x + c\right ) - 2 \, a^{2} d e\right )}} \]
-1/7*((I*sqrt(2)*cos(d*x + c)^2 - 2*I*sqrt(2)*sin(d*x + c) - 2*I*sqrt(2))* sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + (-I*sq rt(2)*cos(d*x + c)^2 + 2*I*sqrt(2)*sin(d*x + c) + 2*I*sqrt(2))*sqrt(e)*wei erstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 2*sqrt(e*cos(d*x + c))*(sin(d*x + c) + 2))/(a^2*d*e*cos(d*x + c)^2 - 2*a^2*d*e*sin(d*x + c) - 2*a^2*d*e)
Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx=\int \frac {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]