Integrand size = 30, antiderivative size = 92 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{20 c f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}} \] Output:
1/5*a*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f/(c-c*sin(f*x+e))^(11/2)-1/20*a^2 *cos(f*x+e)/c/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(9/2)
Time = 5.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.15 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (3+5 \sin (e+f x))}{20 c^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^5 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(11/2),x]
Output:
-1/20*(a*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*Sqrt[a*(1 + Sin[e + f*x])]* (3 + 5*Sin[e + f*x]))/(c^5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + S in[e + f*x])^5*Sqrt[c - c*Sin[e + f*x]])
Time = 0.45 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3042, 3218, 3042, 3217}
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.
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2}}{(c-c \sin (e+f x))^{11/2}}dx\) |
\(\Big \downarrow \) 3218 |
\(\displaystyle \frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{5 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{(c-c \sin (e+f x))^{9/2}}dx}{5 c}\) |
\(\Big \downarrow \) 3217 |
\(\displaystyle \frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{5 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{20 c f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}\) |
Input:
Int[(a + a*Sin[e + f*x])^(3/2)/(c - c*Sin[e + f*x])^(11/2),x]
Output:
(a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(5*f*(c - c*Sin[e + f*x])^(11/2) ) - (a^2*Cos[e + f*x])/(20*c*f*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x ])^(9/2))
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f _.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^ n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e, f, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]
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 = 1.85 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.36
method | result | size |
default | \(\frac {\tan \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{3} \sec \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6} \sqrt {a \sin \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, \left (\cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{6}+2 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{4}+3 \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}+4\right ) a \sqrt {4}}{640 f \sqrt {c \cos \left (\frac {\pi }{4}+\frac {f x}{2}+\frac {e}{2}\right )^{2}}\, c^{5}}\) | \(125\) |
Input:
int((a+sin(f*x+e)*a)^(3/2)/(c-c*sin(f*x+e))^(11/2),x,method=_RETURNVERBOSE )
Output:
1/640/f*tan(1/4*Pi+1/2*f*x+1/2*e)^3*sec(1/4*Pi+1/2*f*x+1/2*e)^6*(a*sin(1/4 *Pi+1/2*f*x+1/2*e)^2)^(1/2)*(cos(1/4*Pi+1/2*f*x+1/2*e)^6+2*cos(1/4*Pi+1/2* f*x+1/2*e)^4+3*cos(1/4*Pi+1/2*f*x+1/2*e)^2+4)*a/(c*cos(1/4*Pi+1/2*f*x+1/2* e)^2)^(1/2)/c^5*4^(1/2)
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.42 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {{\left (5 \, a \sin \left (f x + e\right ) + 3 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{20 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{5} - 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} f \cos \left (f x + e\right ) - {\left (c^{6} f \cos \left (f x + e\right )^{5} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="fri cas")
Output:
1/20*(5*a*sin(f*x + e) + 3*a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e ) + c)/(5*c^6*f*cos(f*x + e)^5 - 20*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f* x + e) - (c^6*f*cos(f*x + e)^5 - 12*c^6*f*cos(f*x + e)^3 + 16*c^6*f*cos(f* x + e))*sin(f*x + e))
Timed out. \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((a+a*sin(f*x+e))**(3/2)/(c-c*sin(f*x+e))**(11/2),x)
Output:
Timed out
\[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="max ima")
Output:
integrate((a*sin(f*x + e) + a)^(3/2)/(-c*sin(f*x + e) + c)^(11/2), x)
Exception generated. \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="gia c")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 24.00 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.45 \[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {\left (\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,48{}\mathrm {i}}{5\,c^6\,f}+\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,16{}\mathrm {i}}{c^6\,f}\right )\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,264{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,220{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )\,20{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,330{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,88{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )\,2{}\mathrm {i}} \] Input:
int((a + a*sin(e + f*x))^(3/2)/(c - c*sin(e + f*x))^(11/2),x)
Output:
(((a*exp(e*6i + f*x*6i)*(a + a*sin(e + f*x))^(1/2)*48i)/(5*c^6*f) + (a*exp (e*6i + f*x*6i)*sin(e + f*x)*(a + a*sin(e + f*x))^(1/2)*16i)/(c^6*f))*(c - c*sin(e + f*x))^(1/2))/(cos(e + f*x)*exp(e*6i + f*x*6i)*264i - exp(e*6i + f*x*6i)*cos(3*e + 3*f*x)*220i + exp(e*6i + f*x*6i)*cos(5*e + 5*f*x)*20i - exp(e*6i + f*x*6i)*sin(2*e + 2*f*x)*330i + exp(e*6i + f*x*6i)*sin(4*e + 4 *f*x)*88i - exp(e*6i + f*x*6i)*sin(6*e + 6*f*x)*2i)
\[ \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{6}-6 \sin \left (f x +e \right )^{5}+15 \sin \left (f x +e \right )^{4}-20 \sin \left (f x +e \right )^{3}+15 \sin \left (f x +e \right )^{2}-6 \sin \left (f x +e \right )+1}d x +\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}}{\sin \left (f x +e \right )^{6}-6 \sin \left (f x +e \right )^{5}+15 \sin \left (f x +e \right )^{4}-20 \sin \left (f x +e \right )^{3}+15 \sin \left (f x +e \right )^{2}-6 \sin \left (f x +e \right )+1}d x \right )}{c^{6}} \] Input:
int((a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(11/2),x)
Output:
(sqrt(c)*sqrt(a)*a*(int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)* sin(e + f*x))/(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1),x) + int((sq rt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1))/(sin(e + f*x)**6 - 6*sin(e + f*x)**5 + 15*sin(e + f*x)**4 - 20*sin(e + f*x)**3 + 15*sin(e + f*x)**2 - 6*sin(e + f*x) + 1),x)))/c**6