Integrand size = 38, antiderivative size = 239 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {24 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {12 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {3 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f \sqrt {c-c \sin (e+f x)}} \] Output:
1/2*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(5/2)-2*a*cos(f *x+e)*(a+a*sin(f*x+e))^(5/2)/c^2/f/(c-c*sin(f*x+e))^(3/2)-24*a^4*cos(f*x+e )*ln(1-sin(f*x+e))/c^3/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-12* a^3*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/c^3/f/(c-c*sin(f*x+e))^(1/2)-3*a^2*c os(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c^3/f/(c-c*sin(f*x+e))^(1/2)
Time = 11.96 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {a (1+\sin (e+f x))} \left (273+\cos (4 (e+f x))+\cos (2 (e+f x)) \left (106-384 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+1152 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-320 \sin (e+f x)-1536 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-24 \sin (3 (e+f x))\right )}{16 c^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x]) ^(7/2),x]
Output:
(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(2 73 + Cos[4*(e + f*x)] + Cos[2*(e + f*x)]*(106 - 384*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]) + 1152*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 320* Sin[e + f*x] - 1536*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 24*Sin[3*(e + f*x)]))/(16*c^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(- 1 + Sin[e + f*x])^3*Sqrt[c - c*Sin[e + f*x]])
Time = 1.59 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 3320, 3042, 3218, 3042, 3218, 3042, 3219, 3042, 3219, 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.
| \(\displaystyle \int \frac {\cos ^2(e+f x) (a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (e+f x)^2 (a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3320 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{5/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{5/2}}dx}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{3/2}}dx}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \int \frac {(\sin (e+f x) a+a)^{7/2}}{(c-c \sin (e+f x))^{3/2}}dx}{c}}{a c}\) |
| \(\Big \downarrow \) 3218 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \int \frac {(\sin (e+f x) a+a)^{5/2}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \int \frac {(\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3219 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \left (2 a \int \frac {\sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3216 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \left (\frac {2 a^2 c \cos (e+f x) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \left (-\frac {2 a^2 \cos (e+f x) \int \frac {1}{c-c \sin (e+f x)}d(-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \left (\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{f (c-c \sin (e+f x))^{3/2}}-\frac {3 a \left (2 a \left (-\frac {2 a^2 \cos (e+f x) \log (c-c \sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}}\right )-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{c}}{a c}\) |
Input:
Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2) ,x]
Output:
((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2*f*(c - c*Sin[e + f*x])^(5/ 2)) - (2*a*((a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(f*(c - c*Sin[e + f*x])^(3/2)) - (3*a*(-1/2*(a*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(f*S qrt[c - c*Sin[e + f*x]]) + 2*a*((-2*a^2*Cos[e + f*x]*Log[c - c*Sin[e + f*x ]])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (a*Cos[e + f*x ]*Sqrt[a + a*Sin[e + f*x]])/(f*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[((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 - 1)*((c + d*Sin[e + f*x])^n/(f*(m + n))), x] + Simp[a*((2*m - 1)/(m + n )) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; Fre eQ[{a, b, c, d, e, f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I GtQ[m - 1/2, 0] && !LtQ[n, -1] && !(IGtQ[n - 1/2, 0] && LtQ[n, m]) && !( 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]
\[\int \frac {\cos \left (f x +e \right )^{2} \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {7}{2}}}d x\]
Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x)
Output:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x)
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x, al gorithm="fricas")
Output:
integral(-(3*a^3*cos(f*x + e)^4 - 4*a^3*cos(f*x + e)^2 + (a^3*cos(f*x + e) ^4 - 4*a^3*cos(f*x + e)^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c* sin(f*x + e) + c)/(c^4*cos(f*x + e)^4 - 8*c^4*cos(f*x + e)^2 + 8*c^4 + 4*( c^4*cos(f*x + e)^2 - 2*c^4)*sin(f*x + e)), x)
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(7/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1396 vs. \(2 (217) = 434\).
Time = 0.27 (sec) , antiderivative size = 1396, normalized size of antiderivative = 5.84 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x, al gorithm="maxima")
Output:
1/30*(1440*a^(7/2)*log(sin(f*x + e)/(cos(f*x + e) + 1) - 1)/c^(7/2) - 720* a^(7/2)*log(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)/c^(7/2) + (334*a^(7/2 ) - 1449*a^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 2693*a^(7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 3278*a^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1) ^3 + 3199*a^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 2014*a^(7/2)*sin(f *x + e)^5/(cos(f*x + e) + 1)^5 + 315*a^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 10*a^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 525*a^(7/2)*sin( f*x + e)^8/(cos(f*x + e) + 1)^8 + 75*a^(7/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)/(c^(7/2) - 6*c^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 17*c^(7/2)* sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 32*c^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 46*c^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 52*c^(7/2)*s in(f*x + e)^5/(cos(f*x + e) + 1)^5 + 46*c^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 32*c^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 17*c^(7/2)*si n(f*x + e)^8/(cos(f*x + e) + 1)^8 - 6*c^(7/2)*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + c^(7/2)*sin(f*x + e)^10/(cos(f*x + e) + 1)^10) - (334*a^(7/2) - 2079*a^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 6203*a^(7/2)*sin(f*x + e)^2 /(cos(f*x + e) + 1)^2 - 10698*a^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 15049*a^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 15354*a^(7/2)*sin(f* x + e)^5/(cos(f*x + e) + 1)^5 + 12165*a^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 7410*a^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 2985*a^(7/...
Time = 0.42 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {2 \, a^{\frac {7}{2}} \sqrt {c} {\left (\frac {12 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {8 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {c^{4} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6 \, c^{4} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c^{8}}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \] Input:
integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x, al gorithm="giac")
Output:
2*a^(7/2)*sqrt(c)*(12*log(-cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 + 1)/(c^4*sgn( sin(-1/4*pi + 1/2*f*x + 1/2*e))) - (8*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 7 )/((cos(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1)^2*c^4*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))) + (c^4*cos(-1/4*pi + 1/2*f*x + 1/2*e)^4*sgn(sin(-1/4*pi + 1/2*f *x + 1/2*e)) + 6*c^4*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2*sgn(sin(-1/4*pi + 1/ 2*f*x + 1/2*e)))/c^8)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))/f
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(7/2) ,x)
Output:
int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(7/2) , x)
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x +3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right )+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right )+\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{4}-4 \sin \left (f x +e \right )^{3}+6 \sin \left (f x +e \right )^{2}-4 \sin \left (f x +e \right )+1}d x \right )}{c^{4}} \] Input:
int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(c)*sqrt(a)*a**3*(int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x)**3)/(sin(e + f*x)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1),x) + 3*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) **4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1),x) + 3*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)**4 - 4*sin(e + f*x)**3 + 6*sin(e + f*x)**2 - 4*sin( e + f*x) + 1),x) + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*c os(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