Integrand size = 30, antiderivative size = 96 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {c^2 \cos (e+f x) \log (1+\sin (e+f x))}{3 f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f (3+3 \sin (e+f x))^{3/2}} \]
-c^2*cos(f*x+e)*ln(1+sin(f*x+e))/a/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e ))^(1/2)-c*cos(f*x+e)*(c-c*sin(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^(3/2)
Time = 1.23 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.45 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{3/2}} \, dx=-\frac {2 c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)} \left (1+\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)\right )}{3 \sqrt {3} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2}} \]
(-2*c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(1 + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x]))/(3*Sqrt[3]*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2] )*(1 + Sin[e + f*x])^(3/2))
Time = 0.53 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3042, 3218, 3042, 3216, 3042, 3146, 16}
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.
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-c \sin (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{3/2}}dx\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle -\frac {c \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {c \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx}{a}-\frac {c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle -\frac {c^2 \cos (e+f x) \int \frac {\cos (e+f x)}{\sin (e+f x) a+a}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {c^2 \cos (e+f x) \int \frac {\cos (e+f x)}{\sin (e+f x) a+a}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle -\frac {c^2 \cos (e+f x) \int \frac {1}{\sin (e+f x) a+a}d(a \sin (e+f x))}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {c^2 \cos (e+f x) \log (a \sin (e+f x)+a)}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {c \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f (a \sin (e+f x)+a)^{3/2}}\) |
-((c^2*Cos[e + f*x]*Log[a + a*Sin[e + f*x]])/(a*f*Sqrt[a + a*Sin[e + f*x]] *Sqrt[c - c*Sin[e + f*x]])) - (c*Cos[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(f *(a + a*Sin[e + f*x])^(3/2))
3.4.91.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[a*c*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x ]]*Sqrt[c + d*Sin[e + f*x]])) Int[Cos[e + f*x]/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0 ]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*(a + b*Sin[e + f*x])^ (m - 1)*((c + d*Sin[e + f*x])^n/(f*(2*n + 1))), x] - Simp[b*((2*m - 1)/(d*( 2*n + 1))) Int[(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}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b ^2, 0] && IGtQ[m - 1/2, 0] && LtQ[n, -1] && !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])
Time = 2.82 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(-\frac {\sec \left (f x +e \right ) \left (2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )-\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )+2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-2 \sin \left (f x +e \right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c}{f a \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}\) | \(130\) |
-1/f*sec(f*x+e)*(2*ln(-cot(f*x+e)+csc(f*x+e)+1)*sin(f*x+e)-ln(2/(cos(f*x+e )+1))*sin(f*x+e)+2*ln(-cot(f*x+e)+csc(f*x+e)+1)-ln(2/(cos(f*x+e)+1))-2*sin (f*x+e))*(-c*(sin(f*x+e)-1))^(1/2)*c/a/(a*(sin(f*x+e)+1))^(1/2)
\[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
integral(-sqrt(a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(3/2)/(a^2*cos(f* x + e)^2 - 2*a^2*sin(f*x + e) - 2*a^2), x)
\[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.42 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\frac {2 \, c^{\frac {3}{2}} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{\frac {3}{2}}} - \frac {c^{\frac {3}{2}} \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{a^{\frac {3}{2}}} - \frac {4 \, \sqrt {a} c^{\frac {3}{2}} \sin \left (f x + e\right )}{{\left (a^{2} + \frac {2 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}}{f} \]
(2*c^(3/2)*log(sin(f*x + e)/(cos(f*x + e) + 1) + 1)/a^(3/2) - c^(3/2)*log( sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/a^(3/2) - 4*sqrt(a)*c^(3/2)*sin(f *x + e)/((a^2 + 2*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + a^2*sin(f*x + e)^2 /(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1)))/f
Time = 0.51 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.15 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\frac {\sqrt {2} \sqrt {a} c^{\frac {3}{2}} {\left (\frac {\sqrt {2} \log \left (-2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{2 \, f} \]
1/2*sqrt(2)*sqrt(a)*c^(3/2)*(sqrt(2)*log(-2*sin(-1/4*pi + 1/2*f*x + 1/2*e) ^2 + 2)/(a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))) - sqrt(2)/((sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))))*sgn(si n(-1/4*pi + 1/2*f*x + 1/2*e))/f
Timed out. \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]