Integrand size = 42, antiderivative size = 123 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
-4*c*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(1/2 )-6*c*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*EllipticE(sin(1/2*f*x+1/2*e) ,2^(1/2))/a/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.33 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.73 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {2 g \sqrt {e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right ) g} \left (-\left (\left (i+5 e^{i (e+f x)}\right ) \sqrt {1+e^{2 i (e+f x)}}\right )+2 e^{2 i (e+f x)} \left (i+e^{i (e+f x)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {c-c \sin (e+f x)}}{a \left (-i+e^{i (e+f x)}\right ) \sqrt {-i a e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2} \sqrt {1+e^{2 i (e+f x)}} f} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]
Output:
(2*g*Sqrt[((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x))]*(-((I + 5*E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]) + 2*E^((2*I)*(e + f*x))*(I + E^(I *(e + f*x)))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[ c - c*Sin[e + f*x]])/(a*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*Sqrt[1 + E^((2*I)*(e + f*x))]*f)
Time = 0.98 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3329, 3042, 3321, 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.
| \(\displaystyle \int \frac {\sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle -\frac {3 c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{a}-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{a}-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle -\frac {3 c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{a \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{a \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle -\frac {3 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{a \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{a \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x]) ^(3/2),x]
Output:
(-4*c*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*S in[e + f*x]]) - (6*c*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[( e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])
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_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* (Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) Int[(g *Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2 *b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* x])^n/(f*g*(2*n + p + 1))), x] - Simp[b*((2*m + p - 1)/(d*(2*n + p + 1))) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^( n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] & & EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && In tegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(111)=222\).
Time = 14.14 (sec) , antiderivative size = 593, normalized size of antiderivative = 4.82
| method | result | size |
| default | \(-\frac {g \left (\left (\left (4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-6\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+4\right ) \sqrt {2}+\left (-4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+6\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-4+\left (3 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+6 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+3\right ) \sqrt {2}\, \operatorname {EllipticE}\left (\left (1+\sqrt {2}\right ) \left (\csc \left (\frac {f x}{2}+\frac {e}{2}\right )-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right ), -2 \sqrt {2}+3\right ) \sqrt {-\frac {2 \left (\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}\, \sqrt {\frac {\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}+\left (\left (6 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+12 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+6\right ) \sqrt {2}-12 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-24 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-12\right ) \operatorname {EllipticF}\left (\left (1+\sqrt {2}\right ) \left (\csc \left (\frac {f x}{2}+\frac {e}{2}\right )-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right ), -2 \sqrt {2}+3\right ) \sqrt {-\frac {2 \left (\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}\, \sqrt {\frac {\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}\right ) \sqrt {g \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}\, \sqrt {-\left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}}{f \left (2 \sqrt {2}-3\right ) \left (1+\sqrt {2}\right ) \left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-\cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) \sqrt {\left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, a}\) | \(593\) |
| risch | \(\text {Expression too large to display}\) | \(1190\) |
Input:
int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x,m ethod=_RETURNVERBOSE)
Output:
-1/f*g/(2*2^(1/2)-3)/(1+2^(1/2))*(((4*cos(1/2*f*x+1/2*e)^2-8*cos(1/2*f*x+1 /2*e)-6)*sin(1/2*f*x+1/2*e)+4*cos(1/2*f*x+1/2*e)+4)*2^(1/2)+(-4*cos(1/2*f* x+1/2*e)^2+8*cos(1/2*f*x+1/2*e)+6)*sin(1/2*f*x+1/2*e)-4*cos(1/2*f*x+1/2*e) -4+(3*cos(1/2*f*x+1/2*e)^2+6*cos(1/2*f*x+1/2*e)+3)*2^(1/2)*EllipticE((1+2^ (1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3)*(-2*(2^(1/2)* cos(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1)) ^(1/2)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos(1 /2*f*x+1/2*e)+1))^(1/2)+((6*cos(1/2*f*x+1/2*e)^2+12*cos(1/2*f*x+1/2*e)+6)* 2^(1/2)-12*cos(1/2*f*x+1/2*e)^2-24*cos(1/2*f*x+1/2*e)-12)*EllipticF((1+2^( 1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3)*(-2*(2^(1/2)*c os(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1))^ (1/2)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos(1/ 2*f*x+1/2*e)+1))^(1/2))*(g*(-1+2*cos(1/2*f*x+1/2*e)^2))^(1/2)*(-(2*cos(1/2 *f*x+1/2*e)*sin(1/2*f*x+1/2*e)-1)*c)^(1/2)/(2*cos(1/2*f*x+1/2*e)*(cos(1/2* f*x+1/2*e)+1)*sin(1/2*f*x+1/2*e)-cos(1/2*f*x+1/2*e)-1)/((2*cos(1/2*f*x+1/2 *e)*sin(1/2*f*x+1/2*e)+1)*a)^(1/2)/a
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.22 \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {\frac {1}{2}} \sqrt {a c g} {\left (-i \, g \sin \left (f x + e\right ) - i \, g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {\frac {1}{2}} \sqrt {a c g} {\left (i \, g \sin \left (f x + e\right ) + i \, g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g\right )}}{a^{2} f \sin \left (f x + e\right ) + a^{2} f} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/ 2),x, algorithm="fricas")
Output:
-2*(3*sqrt(1/2)*sqrt(a*c*g)*(-I*g*sin(f*x + e) - I*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 3*sqrt(1/2 )*sqrt(a*c*g)*(I*g*sin(f*x + e) + I*g)*weierstrassZeta(-4, 0, weierstrassP Inverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) + 2*sqrt(g*cos(f*x + e))*sq rt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*g)/(a^2*f*sin(f*x + e) + a^2*f)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(1/2)/(a+a*sin(f*x+e))** (3/2),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/ 2),x, algorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*sqrt(-c*sin(f*x + e) + c)/(a*sin(f*x + e) + a)^(3/2), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/ 2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(1/2))/(a + a*sin(e + f*x ))^(3/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(1/2))/(a + a*sin(e + f*x ))^(3/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )}{\sin \left (f x +e \right )^{2}+2 \sin \left (f x +e \right )+1}d x \right ) g}{a^{2}} \] Input:
int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1),x)*g)/a**2