Integrand size = 28, antiderivative size = 148 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{108 \sqrt {2} \sqrt {c} f}-\frac {\sec (e+f x) \sqrt {c-c \sin (e+f x)}}{108 c f}-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{162 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{135 c^3 f} \]
-1/6*sec(f*x+e)^3*(c-c*sin(f*x+e))^(3/2)/a^3/c^2/f-1/5*sec(f*x+e)^5*(c-c*s in(f*x+e))^(5/2)/a^3/c^3/f+1/8*arctanh(1/2*cos(f*x+e)*c^(1/2)*2^(1/2)/(c-c *sin(f*x+e))^(1/2))/a^3/f*2^(1/2)/c^(1/2)-1/4*sec(f*x+e)*(c-c*sin(f*x+e))^ (1/2)/a^3/c/f
Result contains complex when optimal does not.
Time = 1.62 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-12-10 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-15 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-(15+15 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5\right )}{1620 f (1+\sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2 ])*(-12 - 10*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 - 15*(Cos[(e + f*x)/2 ] + Sin[(e + f*x)/2])^4 - (15 + 15*I)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^( 1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5))/(16 20*f*(1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])
Time = 0.74 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {3042, 3215, 3042, 3154, 3042, 3154, 3042, 3154, 3042, 3128, 219}
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.
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 \sqrt {c-c \sin (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a \sin (e+f x)+a)^3 \sqrt {c-c \sin (e+f x)}}dx\) |
| \(\Big \downarrow \) 3215 |
| \(\displaystyle \frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{5/2}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(c-c \sin (e+f x))^{5/2}}{\cos (e+f x)^6}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {\frac {1}{2} c \int \sec ^4(e+f x) (c-c \sin (e+f x))^{3/2}dx-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} c \int \frac {(c-c \sin (e+f x))^{3/2}}{\cos (e+f x)^4}dx-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {\frac {1}{2} c \left (\frac {1}{2} c \int \sec ^2(e+f x) \sqrt {c-c \sin (e+f x)}dx-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} c \left (\frac {1}{2} c \int \frac {\sqrt {c-c \sin (e+f x)}}{\cos (e+f x)^2}dx-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3154 |
| \(\displaystyle \frac {\frac {1}{2} c \left (\frac {1}{2} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx-\frac {\sec (e+f x) \sqrt {c-c \sin (e+f x)}}{f}\right )-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} c \left (\frac {1}{2} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c-c \sin (e+f x)}}dx-\frac {\sec (e+f x) \sqrt {c-c \sin (e+f x)}}{f}\right )-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 3128 |
| \(\displaystyle \frac {\frac {1}{2} c \left (\frac {1}{2} c \left (-\frac {c \int \frac {1}{2 c-\frac {c^2 \cos ^2(e+f x)}{c-c \sin (e+f x)}}d\left (-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{f}-\frac {\sec (e+f x) \sqrt {c-c \sin (e+f x)}}{f}\right )-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{2} c \left (\frac {1}{2} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} f}-\frac {\sec (e+f x) \sqrt {c-c \sin (e+f x)}}{f}\right )-\frac {\sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 f}\right )-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{5/2}}{5 f}}{a^3 c^3}\) |
(-1/5*(Sec[e + f*x]^5*(c - c*Sin[e + f*x])^(5/2))/f + (c*(-1/3*(Sec[e + f* x]^3*(c - c*Sin[e + f*x])^(3/2))/f + (c*((Sqrt[c]*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(Sqrt[2]*f) - (Sec[e + f*x]*Sq rt[c - c*Sin[e + f*x]])/f))/2))/2)/(a^3*c^3)
3.4.37.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
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*(p + 1))), x] + Simp[a*((m + p + 1)/(g^2*(p + 1))) Int[(g* Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[p, -2*m] && IntegersQ [m + 1/2, 2*p]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((Lt Q[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Time = 0.91 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(\frac {\left (\sin \left (f x +e \right )-1\right ) \left (30 \left (\sin ^{2}\left (f x +e \right )\right ) c^{\frac {11}{2}}+80 \sin \left (f x +e \right ) c^{\frac {11}{2}}+74 c^{\frac {11}{2}}-15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3} \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}}\right )}{120 a^{3} c^{\frac {11}{2}} \left (\sin \left (f x +e \right )+1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(122\) |
1/120*(sin(f*x+e)-1)*(30*sin(f*x+e)^2*c^(11/2)+80*sin(f*x+e)*c^(11/2)+74*c ^(11/2)-15*2^(1/2)*arctanh(1/2*(c*(sin(f*x+e)+1))^(1/2)*2^(1/2)/c^(1/2))*c ^3*(c*(sin(f*x+e)+1))^(5/2))/a^3/c^(11/2)/(sin(f*x+e)+1)^2/cos(f*x+e)/(c-c *sin(f*x+e))^(1/2)/f
Time = 0.28 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {15 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (15 \, \cos \left (f x + e\right )^{2} - 40 \, \sin \left (f x + e\right ) - 52\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{240 \, {\left (a^{3} c f \cos \left (f x + e\right )^{3} - 2 \, a^{3} c f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} c f \cos \left (f x + e\right )\right )}} \]
1/240*(15*sqrt(2)*(cos(f*x + e)^3 - 2*cos(f*x + e)*sin(f*x + e) - 2*cos(f* x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 + 2*sqrt(2)*sqrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f *x + e) - 2*c)*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*si n(f*x + e) - cos(f*x + e) - 2)) - 4*(15*cos(f*x + e)^2 - 40*sin(f*x + e) - 52)*sqrt(-c*sin(f*x + e) + c))/(a^3*c*f*cos(f*x + e)^3 - 2*a^3*c*f*cos(f* x + e)*sin(f*x + e) - 2*a^3*c*f*cos(f*x + e))
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\int \frac {1}{\sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{3}{\left (e + f x \right )} + 3 \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )} + 3 \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx}{a^{3}} \]
Integral(1/(sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**3 + 3*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x)**2 + 3*sqrt(-c*sin(e + f*x) + c)*sin(e + f*x) + sqr t(-c*sin(e + f*x) + c)), x)/a**3
\[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{3} \sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (137) = 274\).
Time = 0.40 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\frac {15 \, \sqrt {2} \log \left (-\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )}{a^{3} \sqrt {c} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {4 \, \sqrt {2} {\left (23 \, \sqrt {c} + \frac {70 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {140 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \frac {90 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}} + \frac {45 \, \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{4}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{4}}\right )}}{a^{3} c {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{5} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{240 \, f} \]
1/240*(15*sqrt(2)*log(-(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1))/(a^3*sqrt(c)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + 4*sqrt(2)*(23*sqrt(c) + 70*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/ (cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 140*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^2/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^2 + 90*sqrt(c)*(cos(- 1/4*pi + 1/2*f*x + 1/2*e) - 1)^3/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^3 + 45*sqrt(c)*(cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)^4/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1)^4)/(a^3*c*((cos(-1/4*pi + 1/2*f*x + 1/2*e) - 1)/(cos(-1/4*pi + 1/2*f*x + 1/2*e) + 1) + 1)^5*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))))/f
Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]