\(\int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\) [40]

Optimal result

Integrand size = 38, antiderivative size = 243 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}-\frac {a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

1/4*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(9/2)-1/3*a*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/c^2/f
/(c-c*sin(f*x+e))^(7/2)+1/2*a^2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c^3/f/(c-c*sin(f*x+e))^(5/2)-a^3*cos(f*x+e)*
(a+a*sin(f*x+e))^(1/2)/c^4/f/(c-c*sin(f*x+e))^(3/2)-a^4*cos(f*x+e)*ln(1-sin(f*x+e))/c^5/f/(a+a*sin(f*x+e))^(1/
2)/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2920, 2818, 2816, 2746, 31} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}} \]

[In]

Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(4*c*f*(c - c*Sin[e + f*x])^(9/2)) - (a*Cos[e + f*x]*(a + a*Sin[e +
f*x])^(5/2))/(3*c^2*f*(c - c*Sin[e + f*x])^(7/2)) + (a^2*Cos[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(2*c^3*f*(c
- c*Sin[e + f*x])^(5/2)) - (a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^4*f*(c - c*Sin[e + f*x])^(3/2)) - (a
^4*Cos[e + f*x]*Log[1 - Sin[e + f*x]])/(c^5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[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]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 2816

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[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]

Rule 2818

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] - Dist[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] && G
tQ[2*m + n + 1, 0])

Rule 2920

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+a \sin (e+f x))^{9/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{a c} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx}{c^2} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {a \int \frac {(a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{c^3} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^2 \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{c^4} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {a^3 \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^5} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (a^4 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^4 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (a^4 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f (c-c \sin (e+f x))^{9/2}}-\frac {a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c^2 f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^4 f (c-c \sin (e+f x))^{3/2}}-\frac {a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.65 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.80 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {4 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}-\frac {32 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}+\frac {12 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}-\frac {8 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}}-\frac {2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{11/2}} \]

[In]

Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

(4*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/
2])^7*(c - c*Sin[e + f*x])^(11/2)) - (32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))
/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) + (12*(Cos[(e + f*x)/2] - Sin[(e +
f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2
)) - (8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2)) - (2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(Cos[(e + f*x)/2] - Sin[
(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])
^(11/2))

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.51

method	result	size
default	\(-\frac {\left (6 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-3 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+24 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-12 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-8 \left (\cos ^{4}\left (f x +e \right )\right )-8 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-48 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+24 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-48 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )+28 \left (\cos ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )+48 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-24 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-20\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \sec \left (f x +e \right )}{3 f \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 \left (\cos ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )+4\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{5}}\)	\(368\)

[In]

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/f*(6*cos(f*x+e)^4*ln(csc(f*x+e)-cot(f*x+e)-1)-3*cos(f*x+e)^4*ln(2/(1+cos(f*x+e)))+24*cos(f*x+e)^2*sin(f*x
+e)*ln(csc(f*x+e)-cot(f*x+e)-1)-12*cos(f*x+e)^2*sin(f*x+e)*ln(2/(1+cos(f*x+e)))-8*cos(f*x+e)^4-8*cos(f*x+e)^2*
sin(f*x+e)-48*cos(f*x+e)^2*ln(csc(f*x+e)-cot(f*x+e)-1)+24*cos(f*x+e)^2*ln(2/(1+cos(f*x+e)))-48*ln(csc(f*x+e)-c
ot(f*x+e)-1)*sin(f*x+e)+24*ln(2/(1+cos(f*x+e)))*sin(f*x+e)+28*cos(f*x+e)^2+8*sin(f*x+e)+48*ln(csc(f*x+e)-cot(f
*x+e)-1)-24*ln(2/(1+cos(f*x+e)))-20)*(a*(1+sin(f*x+e)))^(1/2)*a^3/(cos(f*x+e)^2*sin(f*x+e)-3*cos(f*x+e)^2-4*si
n(f*x+e)+4)/(-c*(sin(f*x+e)-1))^(1/2)/c^5*sec(f*x+e)

Fricas [F]

\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

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^6*cos(f*x + e)^6 - 18*c^6*cos(f*x + e)^4 + 48*c^6*co
s(f*x + e)^2 - 32*c^6 + 2*(3*c^6*cos(f*x + e)^4 - 16*c^6*cos(f*x + e)^2 + 16*c^6)*sin(f*x + e)), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(11/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(7/2)*cos(f*x + e)^2/(-c*sin(f*x + e) + c)^(11/2), x)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {{\left (24 \, a^{3} \sqrt {c} \log \left ({\left | \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + \frac {48 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 36 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 16 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}}\right )} \sqrt {a}}{12 \, c^{6} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="giac")

[Out]

1/12*(24*a^3*sqrt(c)*log(abs(sin(-1/4*pi + 1/2*f*x + 1/2*e)))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + (48*a^3*sq
rt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^6 - 36*a^3*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)^4 + 16*a^3*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 - 3*a^3*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)
^8)*sqrt(a)/(c^6*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]

[In]

int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(11/2),x)

[Out]

int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(11/2), x)