\(\int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx\) [38]

Optimal result

Integrand size = 38, antiderivative size = 239 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2 c f (c-c \sin (e+f x))^{5/2}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {24 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {12 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {3 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c^3 f \sqrt {c-c \sin (e+f x)}} \]

[Out]

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)-3*a^2*cos(f*x+e)*(a+a*sin(f*x+e))^(3/2)/c^3/f/(c-c*sin(f*x+e))^(1/2)-24*a^4*cos(f*x+e)*l
n(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)

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2920, 2818, 2819, 2816, 2746, 31} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {24 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c^3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {12 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {3 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c^3 f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2 c f (c-c \sin (e+f x))^{5/2}} \]

[In]

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

[Out]

(Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(2*c*f*(c - c*Sin[e + f*x])^(5/2)) - (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)) - (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]]) - (12*a^3*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(c^3*f*Sqrt[c - c
*Sin[e + f*x]]) - (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]])

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 2819

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] + Dist[a*((2*m - 1)/(
m + n)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[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])

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))^{5/2}} \, dx}{a c} \\ & = \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 \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx}{c^2} \\ & = \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 {(6 a) \int \frac {(a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^3} \\ & = \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 {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)}}+\frac {\left (12 a^2\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^3} \\ & = \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 {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)}}+\frac {\left (24 a^3\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c^3} \\ & = \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 {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)}}+\frac {\left (24 a^4 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{c^2 \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}}{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 {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)}}-\frac {\left (24 a^4 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c^3 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}}{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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.63 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.93 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {a (1+\sin (e+f x))} \left (273+\cos (4 (e+f x))+\cos (2 (e+f x)) \left (106-384 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+1152 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-320 \sin (e+f x)-1536 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-24 \sin (3 (e+f x))\right )}{16 c^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^3 \sqrt {c-c \sin (e+f x)}} \]

[In]

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

[Out]

(a^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*Sqrt[a*(1 + Sin[e + f*x])]*(273 + 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]] - 32
0*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]])

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {\left (-\left (\cos ^{4}\left (f x +e \right )\right )+12 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+96 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-48 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )-96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )-73 \left (\cos ^{2}\left (f x +e \right )\right )-58 \sin \left (f x +e \right )-192 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )+96 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )+74\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \sec \left (f x +e \right )}{2 f \left (\sin \left (f x +e \right )-1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{3}}\)	\(229\)

[In]

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

[Out]

-1/2/f*(-cos(f*x+e)^4+12*cos(f*x+e)^2*sin(f*x+e)+96*cos(f*x+e)^2*ln(csc(f*x+e)-cot(f*x+e)-1)-48*cos(f*x+e)^2*l
n(2/(1+cos(f*x+e)))+192*ln(csc(f*x+e)-cot(f*x+e)-1)*sin(f*x+e)-96*ln(2/(1+cos(f*x+e)))*sin(f*x+e)-73*cos(f*x+e
)^2-58*sin(f*x+e)-192*ln(csc(f*x+e)-cot(f*x+e)-1)+96*ln(2/(1+cos(f*x+e)))+74)*(a*(1+sin(f*x+e)))^(1/2)*a^3/(si
n(f*x+e)-1)/(-c*(sin(f*x+e)-1))^(1/2)/c^3*sec(f*x+e)

Fricas [F]

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

[In]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1396 vs. \(2 (217) = 434\).

Time = 0.36 (sec) , antiderivative size = 1396, normalized size of antiderivative = 5.84 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 46*c^(7/2)*sin(f*x + e)^6/(co
s(f*x + e) + 1)^6 - 32*c^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 17*c^(7/2)*sin(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 - 10
698*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)*s
in(f*x + e)^7/(cos(f*x + e) + 1)^7 + 2985*a^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 555*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/(c
os(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)*sin(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)*sin(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) + 10*(75*a^(7/2)*sin(f*x + e)/
(cos(f*x + e) + 1) - 375*a^(7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 854*a^(7/2)*sin(f*x + e)^3/(cos(f*x + e
) + 1)^3 - 1257*a^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1534*a^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5
- 1257*a^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 854*a^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 375*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)*sin(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)*sin(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))/f

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {2 \, a^{\frac {7}{2}} \sqrt {c} {\left (\frac {12 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {8 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} c^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {c^{4} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6 \, c^{4} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c^{8}}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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