Integrand size = 25, antiderivative size = 153 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{77 a^3 d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a+a \sin (c+d x))^3}-\frac {10 \sqrt {e \cos (c+d x)}}{77 a d e (a+a \sin (c+d x))^2}-\frac {10 \sqrt {e \cos (c+d x)}}{77 d e \left (a^3+a^3 \sin (c+d x)\right )} \] Output:
10/77*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(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*c os(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 = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=-\frac {\sqrt {e \cos (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {15}{4},\frac {5}{4},\frac {1}{2} (1-\sin (c+d x))\right )}{2\ 2^{3/4} a^3 d e \sqrt [4]{1+\sin (c+d x)}} \] Input:
Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3),x]
Output:
-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 = 0.70 (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 |
| \(\displaystyle \frac {5 \left (\frac {3 \left (\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 a d \sqrt {e \cos (c+d x)}}-\frac {2 \sqrt {e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}\right )}{7 a}-\frac {2 \sqrt {e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}\right )}{11 a}-\frac {2 \sqrt {e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}\) |
Input:
Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3),x]
Output:
(-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)
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. \(2(136)=272\).
Time = 3.77 (sec) , antiderivative size = 580, normalized size of antiderivative = 3.79
| method | result | size |
| default | \(-\frac {2 \left (160 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+160 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-400 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-320 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+400 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+264 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-200 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-104 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+50 \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+44 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+72 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \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 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-44 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-17 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 \left (32 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-80 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+80 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-40 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+10 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\) | \(580\) |
Input:
int(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-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*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(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*(2*s in(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/ 2*d*x+1/2*c),2^(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*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(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*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(si n(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(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*(2*sin(1/2*d*x+1 /2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c) ,2^(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 complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.32 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=-\frac {2 \, {\left (5 \, \sqrt {\frac {1}{2}} {\left (3 i \, \cos \left (d x + c\right )^{2} + {\left (i \, \cos \left (d x + c\right )^{2} - 4 i\right )} \sin \left (d x + c\right ) - 4 i\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {\frac {1}{2}} {\left (-3 i \, \cos \left (d x + c\right )^{2} + {\left (-i \, \cos \left (d x + c\right )^{2} + 4 i\right )} \sin \left (d x + c\right ) + 4 i\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + \sqrt {e \cos \left (d x + c\right )} {\left (5 \, \cos \left (d x + c\right )^{2} - 15 \, \sin \left (d x + c\right ) - 22\right )}\right )}}{77 \, {\left (3 \, a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e + {\left (a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^3,x, algorithm="fricas")
Output:
-2/77*(5*sqrt(1/2)*(3*I*cos(d*x + c)^2 + (I*cos(d*x + c)^2 - 4*I)*sin(d*x + c) - 4*I)*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(1/2)*(-3*I*cos(d*x + c)^2 + (-I*cos(d*x + c)^2 + 4*I)*sin(d*x + c) + 4*I)*sqrt(e)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 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. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(e*cos(d*x+c))**(1/2)/(a+a*sin(d*x+c))**3,x)
Output:
Timed out
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^3,x, algorithm="maxima")
Output:
integrate(1/(sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + a)^3), x)
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:
integrate(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
integrate(1/(sqrt(e*cos(d*x + c))*(a*sin(d*x + c) + a)^3), x)
Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\int \frac {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \] Input:
int(1/((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^3),x)
Output:
int(1/((e*cos(c + d*x))^(1/2)*(a + a*sin(c + d*x))^3), x)
\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos \left (d x +c \right )}d x \right )}{a^{3} e} \] Input:
int(1/(e*cos(d*x+c))^(1/2)/(a+a*sin(d*x+c))^3,x)
Output:
(sqrt(e)*int(sqrt(cos(c + d*x))/(cos(c + d*x)*sin(c + d*x)**3 + 3*cos(c + d*x)*sin(c + d*x)**2 + 3*cos(c + d*x)*sin(c + d*x) + cos(c + d*x)),x))/(a* *3*e)