\(\int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx\) [172]

Optimal result

Integrand size = 40, antiderivative size = 202 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}} \]

[Out]

1/14*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(15/2)+1/168*(3*A-11*B)*cos(f*x+e)*(a+a*sin(f*
x+e))^(7/2)/c/f/(c-c*sin(f*x+e))^(13/2)+1/840*(3*A-11*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^2/f/(c-c*sin(f*x+
e))^(11/2)+1/6720*(3*A-11*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^3/f/(c-c*sin(f*x+e))^(9/2)

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {3051, 2822, 2821} \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac {(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {(3 A-11 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}} \]

[In]

Int[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(15/2),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(14*f*(c - c*Sin[e + f*x])^(15/2)) + ((3*A - 11*B)*Cos[e + f
*x]*(a + a*Sin[e + f*x])^(7/2))/(168*c*f*(c - c*Sin[e + f*x])^(13/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[
e + f*x])^(7/2))/(840*c^2*f*(c - c*Sin[e + f*x])^(11/2)) + ((3*A - 11*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/
2))/(6720*c^3*f*(c - c*Sin[e + f*x])^(9/2))

Rule 2821

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] && NeQ[m, -2^(-1)]

Rule 2822

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] + Dist[(m + n + 1)/(a*(2*m
+ 1)), 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, m, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m
, 1] ||  !SumSimplerQ[n, 1])

Rule 3051

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[(A*b - a*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] + Dist[(a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1)), 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, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2
- b^2, 0] && (LtQ[m, -2^(-1)] || (ILtQ[m + n, 0] &&  !SumSimplerQ[n, 1])) && NeQ[2*m + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {(3 A-11 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{14 c} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac {(3 A-11 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{84 c^2} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {(3 A-11 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{840 c^3} \\ & = \frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{168 c f (c-c \sin (e+f x))^{13/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{840 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {(3 A-11 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{6720 c^3 f (c-c \sin (e+f x))^{9/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(442\) vs. \(2(202)=404\).

Time = 17.20 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.19 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {8 (A+B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{7/2}}{7 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {2 (3 A+5 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {6 (A+3 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {(-A-7 B) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{4 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {B \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}} \]

[In]

Integrate[((a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^(15/2),x]

[Out]

(8*(A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) - (2*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Si
n[e + f*x]))^(7/2))/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + (6*(A + 3*B)*(
Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^7*(c - c*Sin[e + f*x])^(15/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^
(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) + (B*(Cos[(e + f*x)/2] - Sin[
(e + f*x)/2])^9*(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]
)^(15/2))

Maple [A] (verified)

Time = 4.97 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.40

method	result	size
default	\(\frac {a^{3} \tan \left (f x +e \right ) \left (39 A \left (\cos ^{6}\left (f x +e \right )\right )+3 B \left (\cos ^{4}\left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )\right )+273 A \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+21 B \left (\cos ^{2}\left (f x +e \right )\right ) \left (\sin ^{3}\left (f x +e \right )\right )-936 A \left (\cos ^{4}\left (f x +e \right )\right )-69 B \left (\sin ^{2}\left (f x +e \right )\right ) \left (\cos ^{2}\left (f x +e \right )\right )-1911 A \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-266 B \left (\sin ^{3}\left (f x +e \right )\right )+3225 A \left (\cos ^{2}\left (f x +e \right )\right )-4 B \left (\sin ^{2}\left (f x +e \right )\right )+2268 A \sin \left (f x +e \right )-210 B \sin \left (f x +e \right )-2748 A \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{420 c^{7} f \left (\cos ^{6}\left (f x +e \right )+6 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-18 \left (\cos ^{4}\left (f x +e \right )\right )-32 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \left (\cos ^{2}\left (f x +e \right )\right )+32 \sin \left (f x +e \right )-32\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\)	\(283\)
parts	\(\frac {A \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \left (13 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-91 \left (\cos ^{5}\left (f x +e \right )\right )-312 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+728 \left (\cos ^{3}\left (f x +e \right )\right )+1075 \cos \left (f x +e \right ) \sin \left (f x +e \right )-1393 \cos \left (f x +e \right )-916 \tan \left (f x +e \right )+756 \sec \left (f x +e \right )\right )}{140 f \left (\cos ^{6}\left (f x +e \right )+6 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-18 \left (\cos ^{4}\left (f x +e \right )\right )-32 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \left (\cos ^{2}\left (f x +e \right )\right )+32 \sin \left (f x +e \right )-32\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{7}}-\frac {B \sec \left (f x +e \right ) \left (\cos \left (f x +e \right )-1\right ) \left (1+\cos \left (f x +e \right )\right ) \left (3 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-21 \left (\cos ^{4}\left (f x +e \right )\right )-69 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+287 \left (\cos ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )-476\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3}}{420 f \left (\cos ^{6}\left (f x +e \right )+6 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-18 \left (\cos ^{4}\left (f x +e \right )\right )-32 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+48 \left (\cos ^{2}\left (f x +e \right )\right )+32 \sin \left (f x +e \right )-32\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{7}}\)	\(395\)

[In]

int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2),x,method=_RETURNVERBOSE)

[Out]

1/420*a^3/c^7/f*tan(f*x+e)*(39*A*cos(f*x+e)^6+3*B*cos(f*x+e)^4*sin(f*x+e)^2+273*A*cos(f*x+e)^4*sin(f*x+e)+21*B
*cos(f*x+e)^2*sin(f*x+e)^3-936*A*cos(f*x+e)^4-69*B*sin(f*x+e)^2*cos(f*x+e)^2-1911*A*sin(f*x+e)*cos(f*x+e)^2-26
6*B*sin(f*x+e)^3+3225*A*cos(f*x+e)^2-4*B*sin(f*x+e)^2+2268*A*sin(f*x+e)-210*B*sin(f*x+e)-2748*A)*(a*(1+sin(f*x
+e)))^(1/2)/(cos(f*x+e)^6+6*cos(f*x+e)^4*sin(f*x+e)-18*cos(f*x+e)^4-32*cos(f*x+e)^2*sin(f*x+e)+48*cos(f*x+e)^2
+32*sin(f*x+e)-32)/(-c*(sin(f*x+e)-1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.16 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=-\frac {{\left (140 \, B a^{3} \cos \left (f x + e\right )^{4} - 7 \, {\left (27 \, A + 61 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (57 \, A + 71 \, B\right )} a^{3} - 7 \, {\left (5 \, {\left (3 \, A + 5 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (9 \, A + 7 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{420 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{7} - 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right ) - {\left (c^{8} f \cos \left (f x + e\right )^{7} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2),x, algorithm="fricas")

[Out]

-1/420*(140*B*a^3*cos(f*x + e)^4 - 7*(27*A + 61*B)*a^3*cos(f*x + e)^2 + 4*(57*A + 71*B)*a^3 - 7*(5*(3*A + 5*B)
*a^3*cos(f*x + e)^2 - 4*(9*A + 7*B)*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(7*c
^8*f*cos(f*x + e)^7 - 56*c^8*f*cos(f*x + e)^5 + 112*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e) - (c^8*f*cos(
f*x + e)^7 - 24*c^8*f*cos(f*x + e)^5 + 80*c^8*f*cos(f*x + e)^3 - 64*c^8*f*cos(f*x + e))*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**(15/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\int { \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(15/2), x)

Giac [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.69 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {{\left (280 \, B a^{3} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 105 \, A a^{3} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 385 \, B a^{3} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 63 \, A a^{3} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 231 \, B a^{3} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 21 \, A a^{3} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 77 \, B a^{3} \sqrt {c} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 3 \, A a^{3} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 11 \, B a^{3} \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}}{6720 \, {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{7} c^{8} f \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^(15/2),x, algorithm="giac")

[Out]

1/6720*(280*B*a^3*sqrt(c)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^8*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) + 105*A*a^3*sqr
t(c)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 385*B*a^3*sqrt(c)*cos(-1/4*pi + 1/
2*f*x + 1/2*e)^6*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 63*A*a^3*sqrt(c)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^4*sgn(c
os(-1/4*pi + 1/2*f*x + 1/2*e)) + 231*B*a^3*sqrt(c)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^4*sgn(cos(-1/4*pi + 1/2*f*x
+ 1/2*e)) + 21*A*a^3*sqrt(c)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 77*B*a^3*s
qrt(c)*cos(-1/4*pi + 1/2*f*x + 1/2*e)^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)) - 3*A*a^3*sqrt(c)*sgn(cos(-1/4*pi
+ 1/2*f*x + 1/2*e)) + 11*B*a^3*sqrt(c)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e)))*sqrt(a)/((cos(-1/4*pi + 1/2*f*x +
1/2*e)^2 - 1)^7*c^8*f*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e)))

Mupad [B] (verification not implemented)

Time = 25.38 (sec) , antiderivative size = 827, normalized size of antiderivative = 4.09 \[ \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Too large to display} \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(15/2),x)

[Out]

-((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*((B*a^3*exp(e*4i + f*x*4i)*(a + a*((
exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*16i)/(3*c^8*f) + (B*a^3*exp(e*12i + f*x*12i)*(a
+ a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*16i)/(3*c^8*f) - (a^3*exp(e*5i + f*x*5i)
*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*3i + B*5i)*8i)/(3*c^8*f) + (a^3*ex
p(e*11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*3i + B*5i)*8i)/(
3*c^8*f) - (a^3*exp(e*6i + f*x*6i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(27
*A + 41*B)*16i)/(15*c^8*f) - (a^3*exp(e*10i + f*x*10i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1
i)*1i)/2))^(1/2)*(27*A + 41*B)*16i)/(15*c^8*f) + (a^3*exp(e*7i + f*x*7i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 -
(exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*43i + B*29i)*8i)/(5*c^8*f) - (a^3*exp(e*9i + f*x*9i)*(a + a*((exp(- e*1i
- f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*43i + B*29i)*8i)/(5*c^8*f) + (a^3*exp(e*8i + f*x*8i)*(
a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(89*A + 82*B)*32i)/(35*c^8*f)))/(exp(e*
1i + f*x*1i)*14i - 90*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)*350i + 910*exp(e*4i + f*x*4i) + exp(e*5i + f*x*5
i)*1638i - 2002*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7i)*1430i - exp(e*9i + f*x*9i)*1430i + 2002*exp(e*10i + f*
x*10i) + exp(e*11i + f*x*11i)*1638i - 910*exp(e*12i + f*x*12i) - exp(e*13i + f*x*13i)*350i + 90*exp(e*14i + f*
x*14i) + exp(e*15i + f*x*15i)*14i - exp(e*16i + f*x*16i) + 1)