Integrand size = 38, antiderivative size = 193 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f (c-c \sin (e+f x))^{7/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 \cos (e+f x) \log (1-\sin (e+f x))}{c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
1/3*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c/f/(c-c*sin(f*x+e))^(7/2)-1/2*a*cos (f*x+e)*(a+a*sin(f*x+e))^(3/2)/c^2/f/(c-c*sin(f*x+e))^(5/2)+a^2*cos(f*x+e) *(a+a*sin(f*x+e))^(1/2)/c^3/f/(c-c*sin(f*x+e))^(3/2)+a^3*cos(f*x+e)*ln(1-s in(f*x+e))/c^4/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Time = 6.01 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.21 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {a (1+\sin (e+f x))} \left (34+30 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-18 \cos (2 (e+f x)) \left (1+\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )-9 \left (4+5 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin (e+f x)+3 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{6 c^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^4 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x]) ^(9/2),x]
Output:
(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(3 4 + 30*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 18*Cos[2*(e + f*x)]*(1 + Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) - 9*(4 + 5*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]])*Sin[e + f*x] + 3*Log[Cos[(e + f*x)/2] - Sin[(e + f*x )/2]]*Sin[3*(e + f*x)]))/(6*c^4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(- 1 + Sin[e + f*x])^4*Sqrt[c - c*Sin[e + f*x]])
Time = 1.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.342, Rules used = {3042, 3320, 3042, 3218, 3042, 3218, 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.
| \(\displaystyle \int \frac {\cos ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (e+f x)^2 (a \sin (e+f x)+a)^{5/2}}{(c-c \sin (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 3320 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{7/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{7/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{5/2}}dx}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{5/2}}dx}{c}}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a^2 \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a^2 \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a^2 \cos (e+f x) \int \frac {1}{c-c \sin (e+f x)}d(-c \sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {a \left (\frac {a^2 \cos (e+f x) \log (c-c \sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}}\right )}{c}\right )}{c}}{a c}\) |
Input:
Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(9/2) ,x]
Output:
((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*f*(c - c*Sin[e + f*x])^(7/ 2)) - (a*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*f*(c - c*Sin[e + f*x])^(5/2)) - (a*((a*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(f*(c - c*Sin [e + f*x])^(3/2)) + (a^2*Cos[e + f*x]*Log[c - c*Sin[e + f*x]])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])))/c))/c)/(a*c)
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])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(a^(p/ 2)*c^(p/2)) Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2]
\[\int \frac {\cos \left (f x +e \right )^{2} \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {9}{2}}}d x\]
Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x)
Output:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x)
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x, al gorithm="fricas")
Output:
integral(-(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2*sin(f*x + e) - 2*a^2* cos(f*x + e)^2)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(5*c^5* cos(f*x + e)^4 - 20*c^5*cos(f*x + e)^2 + 16*c^5 - (c^5*cos(f*x + e)^4 - 12 *c^5*cos(f*x + e)^2 + 16*c^5)*sin(f*x + e)), x)
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(5/2)/(c-c*sin(f*x+e))**(9/2),x)
Output:
Timed out
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x, al gorithm="maxima")
Output:
integrate((a*sin(f*x + e) + a)^(5/2)*cos(f*x + e)^2/(-c*sin(f*x + e) + c)^ (9/2), x)
Exception generated. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x, al gorithm="giac")
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
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \] Input:
int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(9/2) ,x)
Output:
int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(9/2) , x)
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a^{2} \left (-\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{5}-5 \sin \left (f x +e \right )^{4}+10 \sin \left (f x +e \right )^{3}-10 \sin \left (f x +e \right )^{2}+5 \sin \left (f x +e \right )-1}d x \right )-2 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{5}-5 \sin \left (f x +e \right )^{4}+10 \sin \left (f x +e \right )^{3}-10 \sin \left (f x +e \right )^{2}+5 \sin \left (f x +e \right )-1}d x \right )-\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{5}-5 \sin \left (f x +e \right )^{4}+10 \sin \left (f x +e \right )^{3}-10 \sin \left (f x +e \right )^{2}+5 \sin \left (f x +e \right )-1}d x \right )\right )}{c^{5}} \] Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x)
Output:
(sqrt(c)*sqrt(a)*a**2*( - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x)**2)/(sin(e + f*x)**5 - 5*sin(e + f*x)** 4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x) - 2*i nt((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x))/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*si n(e + f*x)**2 + 5*sin(e + f*x) - 1),x) - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2)/(sin(e + f*x)**5 - 5*sin(e + f*x)**4 + 10*sin(e + f*x)**3 - 10*sin(e + f*x)**2 + 5*sin(e + f*x) - 1),x)))/c**5