Integrand size = 25, antiderivative size = 153 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 a^3 d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a+a \sin (c+d x))^3}-\frac {10 \sqrt {e \cos (c+d x)}}{77 a d e (a+a \sin (c+d x))^2}-\frac {10 \sqrt {e \cos (c+d x)}}{77 d e \left (a^3+a^3 \sin (c+d x)\right )} \]
10/77*(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^3/d/(e*cos(d*x+c))^(1/2)-2/11*(e*cos( d*x+c))^(1/2)/d/e/(a+a*sin(d*x+c))^3-10/77*(e*cos(d*x+c))^(1/2)/a/d/e/(a+a *sin(d*x+c))^2-10/77*(e*cos(d*x+c))^(1/2)/d/e/(a^3+a^3*sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=-\frac {\sqrt {e \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {15}{4},\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{2\ 2^{3/4} a^3 d e \sqrt [4]{1+\sin (c+d x)}} \]
-1/2*(Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 15/4, 5/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^3*d*e*(1 + Sin[c + d*x])^(1/4))
Time = 0.67 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3160, 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.
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {5 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^2}dx}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)^2}dx}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3160 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)}dx}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \int \frac {1}{\sqrt {e \cos (c+d x)} (\sin (c+d x) a+a)}dx}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3162 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {e \cos (c+d x)}}dx}{3 a}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 a \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {\sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
(-2*Sqrt[e*Cos[c + d*x]])/(11*d*e*(a + a*Sin[c + d*x])^3) + (5*((-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*Sqrt[e*Cos[c + d*x]])/(3*d*e*(a + a*Sin[c + d*x]))))/(7*a)))/(11*a)
3.3.61.3.1 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. \(579\) vs. \(2(161)=322\).
Time = 8.72 (sec) , antiderivative size = 580, normalized size of antiderivative = 3.79
| method | result | size |
| default | \(-\frac {2 \left (160 \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 ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-400 \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 ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-320 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+400 \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 )+264 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-200 \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 )-104 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+50 \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 )+44 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+72 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-5 \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 )-44 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-17 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 \left (32 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-80 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+80 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) a^{3} \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}\) | \(580\) |
-2/77/(32*sin(1/2*d*x+1/2*c)^10-80*sin(1/2*d*x+1/2*c)^8+80*sin(1/2*d*x+1/2 *c)^6-40*sin(1/2*d*x+1/2*c)^4+10*sin(1/2*d*x+1/2*c)^2-1)/a^3/sin(1/2*d*x+1 /2*c)/(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*(160*(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)*s in(1/2*d*x+1/2*c)^10+160*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)-400*(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)^8-320*cos(1/2*d*x+1/2*c)*sin(1/2* d*x+1/2*c)^8+400*(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+264*sin(1/ 2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-200*(sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipti cF(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)^4-104*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+50*(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+44*sin(1/2*d*x+1/2*c)^5+72*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)-5*(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))-44*sin(1/2*d*x+1/2*c)^ 3-17*sin(1/2*d*x+1/2*c))/d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=-\frac {5 \, {\left (3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 4 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) - 4 i \, \sqrt {2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (-3 i \, \sqrt {2} \cos \left (d x + c\right )^{2} + {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 4 i \, \sqrt {2}\right )} \sin \left (d x + c\right ) + 4 i \, \sqrt {2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, \sqrt {e \cos \left (d x + c\right )} {\left (5 \, \cos \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) - 22\right )}}{77 \, {\left (3 \, a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e + {\left (a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e\right )} \sin \left (d x + c\right )\right )}} \]
-1/77*(5*(3*I*sqrt(2)*cos(d*x + c)^2 + (I*sqrt(2)*cos(d*x + c)^2 - 4*I*sqr t(2))*sin(d*x + c) - 4*I*sqrt(2))*sqrt(e)*weierstrassPInverse(-4, 0, cos(d *x + c) + I*sin(d*x + c)) + 5*(-3*I*sqrt(2)*cos(d*x + c)^2 + (-I*sqrt(2)*c os(d*x + c)^2 + 4*I*sqrt(2))*sin(d*x + c) + 4*I*sqrt(2))*sqrt(e)*weierstra ssPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 2*sqrt(e*cos(d*x + c))* (5*cos(d*x + c)^2 - 15*sin(d*x + c) - 22))/(3*a^3*d*e*cos(d*x + c)^2 - 4*a ^3*d*e + (a^3*d*e*cos(d*x + c)^2 - 4*a^3*d*e)*sin(d*x + c))
Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\int \frac {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]