Integrand size = 38, antiderivative size = 48 \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{3/2}}{4 a c f (c-c \sin (e+f x))^{5/2}} \] Output:
1/4*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/a/c/f/(c-c*sin(f*x+e))^(5/2)
Time = 3.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.88 \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sin (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}}{c^4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \] Input:
Integrate[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^( 7/2),x]
Output:
(Sin[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(c^4*f* (Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/ 2]))
Time = 0.57 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3042, 3320, 3042, 3221}
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.
\(\displaystyle \int \frac {\cos ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (e+f x)^2 \sqrt {a \sin (e+f x)+a}}{(c-c \sin (e+f x))^{7/2}}dx\) |
\(\Big \downarrow \) 3320 |
\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{5/2}}dx}{a c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{5/2}}dx}{a c}\) |
\(\Big \downarrow \) 3221 |
\(\displaystyle \frac {\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{4 a c f (c-c \sin (e+f x))^{5/2}}\) |
Input:
Int[(Cos[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]])/(c - c*Sin[e + f*x])^(7/2),x ]
Output:
(Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(4*a*c*f*(c - c*Sin[e + f*x])^(5 /2))
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*( (c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Ne Q[m, -2^(-1)]
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]
\[\int \frac {\cos \left (f x +e \right )^{2} \sqrt {a +a \sin \left (f x +e \right )}}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]
Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x)
Output:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x)
Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.65 \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{c^{4} f \cos \left (f x + e\right )^{3} + 2 \, c^{4} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, c^{4} f \cos \left (f x + e\right )} \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x, al gorithm="fricas")
Output:
-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*sin(f*x + e)/(c^4*f*co s(f*x + e)^3 + 2*c^4*f*cos(f*x + e)*sin(f*x + e) - 2*c^4*f*cos(f*x + e))
Timed out. \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(1/2)/(c-c*sin(f*x+e))**(7/2),x)
Output:
Timed out
\[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {\sqrt {a \sin \left (f x + e\right ) + a} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x, al gorithm="maxima")
Output:
integrate(sqrt(a*sin(f*x + e) + a)*cos(f*x + e)^2/(-c*sin(f*x + e) + c)^(7 /2), x)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (42) = 84\).
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.90 \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {{\left (2 \, \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a}}{4 \, c^{4} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}} \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x, al gorithm="giac")
Output:
1/4*(2*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/(c^4*f*sg n(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^4)
Timed out. \[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(1/2))/(c - c*sin(e + f*x))^(7/2) ,x)
Output:
int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(1/2))/(c - c*sin(e + f*x))^(7/2) , x)
\[ \int \frac {\cos ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right )}{c^{4}} \] Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(c)*sqrt(a)*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos (e + f*x)**2)/(sin(e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4 *sin(e + f*x) + 1),x))/c**4