3.160 \(\int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx\)

Optimal. Leaf size=250 \[ -\frac{a^2 (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac{a^3 (9 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac{a^4 (9 A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{a (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f} \]

[Out]

-(a^4*(9*A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(9*A - B)*Cos
[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(126*f) - (a^2*(9*A - B)*Cos[e + f*x]*(a + a*Si
n[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(84*f) - (a*(9*A - B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c
- c*Sin[e + f*x])^(9/2))/(72*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(9*f
)

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Rubi [A]  time = 0.566807, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2973, 2740, 2738} \[ -\frac{a^2 (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac{a^3 (9 A-B) \cos (e+f x) \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac{a^4 (9 A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 f \sqrt{a \sin (e+f x)+a}}-\frac{a (9 A-B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac{B \cos (e+f x) (a \sin (e+f x)+a)^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2),x]

[Out]

-(a^4*(9*A - B)*Cos[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/(315*f*Sqrt[a + a*Sin[e + f*x]]) - (a^3*(9*A - B)*Cos
[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2))/(126*f) - (a^2*(9*A - B)*Cos[e + f*x]*(a + a*Si
n[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(84*f) - (a*(9*A - B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2)*(c
- c*Sin[e + f*x])^(9/2))/(72*f) - (B*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(9*f
)

Rule 2973

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((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)/(f*(
m + n + 1)), x] - Dist[(B*c*(m - n) - A*d*(m + n + 1))/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^m*(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)] && NeQ[m + n + 1, 0]

Rule 2740

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Sim
p[(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] && E
qQ[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 2738

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[
(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx &=-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac{1}{9} (9 A-B) \int (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2} \, dx\\ &=-\frac{a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac{1}{12} (a (9 A-B)) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2} \, dx\\ &=-\frac{a^2 (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac{a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac{1}{21} \left (a^2 (9 A-B)\right ) \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2} \, dx\\ &=-\frac{a^3 (9 A-B) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac{a^2 (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac{a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}+\frac{1}{63} \left (a^3 (9 A-B)\right ) \int \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2} \, dx\\ &=-\frac{a^4 (9 A-B) \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{315 f \sqrt{a+a \sin (e+f x)}}-\frac{a^3 (9 A-B) \cos (e+f x) \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}{126 f}-\frac{a^2 (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{84 f}-\frac{a (9 A-B) \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{9/2}}{72 f}-\frac{B \cos (e+f x) (a+a \sin (e+f x))^{7/2} (c-c \sin (e+f x))^{9/2}}{9 f}\\ \end{align*}

Mathematica [B]  time = 7.13863, size = 870, normalized size = 3.48 \[ \frac{7 (10 A-B) \sin (e+f x) (a (\sin (e+f x)+1))^{7/2} (c-c \sin (e+f x))^{9/2}}{128 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{7 (A-B) \cos (2 (e+f x)) (a (\sin (e+f x)+1))^{7/2} (c-c \sin (e+f x))^{9/2}}{128 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{7 (A-B) \cos (4 (e+f x)) (a (\sin (e+f x)+1))^{7/2} (c-c \sin (e+f x))^{9/2}}{256 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{(A-B) \cos (6 (e+f x)) (a (\sin (e+f x)+1))^{7/2} (c-c \sin (e+f x))^{9/2}}{128 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{(A-B) \cos (8 (e+f x)) (a (\sin (e+f x)+1))^{7/2} (c-c \sin (e+f x))^{9/2}}{1024 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{7 A (a (\sin (e+f x)+1))^{7/2} \sin (3 (e+f x)) (c-c \sin (e+f x))^{9/2}}{64 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{(7 A+2 B) (a (\sin (e+f x)+1))^{7/2} \sin (5 (e+f x)) (c-c \sin (e+f x))^{9/2}}{320 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{(4 A+5 B) (a (\sin (e+f x)+1))^{7/2} \sin (7 (e+f x)) (c-c \sin (e+f x))^{9/2}}{1792 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{B (a (\sin (e+f x)+1))^{7/2} \sin (9 (e+f x)) (c-c \sin (e+f x))^{9/2}}{2304 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9 \left (\cos \left (\frac{1}{2} (e+f x)\right )+\sin \left (\frac{1}{2} (e+f x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^(7/2)*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(7*(A - B)*Cos[2*(e + f*x)]*(a*(1 + Sin[e + f*x]))^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(128*f*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7) + (7*(A - B)*Cos[4*(e + f*x)]*(a*(1 + Sin[e + f
*x]))^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(256*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin
[(e + f*x)/2])^7) + ((A - B)*Cos[6*(e + f*x)]*(a*(1 + Sin[e + f*x]))^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(128*f*
(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7) + ((A - B)*Cos[8*(e + f*x)]*(
a*(1 + Sin[e + f*x]))^(7/2)*(c - c*Sin[e + f*x])^(9/2))/(1024*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(
e + f*x)/2] + Sin[(e + f*x)/2])^7) + (7*(10*A - B)*Sin[e + f*x]*(a*(1 + Sin[e + f*x]))^(7/2)*(c - c*Sin[e + f*
x])^(9/2))/(128*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7) + (7*A*(a*(
1 + Sin[e + f*x]))^(7/2)*(c - c*Sin[e + f*x])^(9/2)*Sin[3*(e + f*x)])/(64*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/
2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7) + ((7*A + 2*B)*(a*(1 + Sin[e + f*x]))^(7/2)*(c - c*Sin[e + f*x]
)^(9/2)*Sin[5*(e + f*x)])/(320*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])
^7) + ((4*A + 5*B)*(a*(1 + Sin[e + f*x]))^(7/2)*(c - c*Sin[e + f*x])^(9/2)*Sin[7*(e + f*x)])/(1792*f*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7) + (B*(a*(1 + Sin[e + f*x]))^(7/2)*(c -
c*Sin[e + f*x])^(9/2)*Sin[9*(e + f*x)])/(2304*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(Cos[(e + f*x)/2] + S
in[(e + f*x)/2])^7)

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Maple [A]  time = 0.306, size = 259, normalized size = 1. \begin{align*}{\frac{ \left ( -280\,B \left ( \cos \left ( fx+e \right ) \right ) ^{8}+315\,A \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \right ) -315\,B \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \right ) -360\,A \left ( \cos \left ( fx+e \right ) \right ) ^{6}+40\,B \left ( \cos \left ( fx+e \right ) \right ) ^{6}+315\,A \left ( \cos \left ( fx+e \right ) \right ) ^{4}\sin \left ( fx+e \right ) -315\,B\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-432\,A \left ( \cos \left ( fx+e \right ) \right ) ^{4}+48\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+315\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -315\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -576\,A \left ( \cos \left ( fx+e \right ) \right ) ^{2}+64\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}+315\,A\sin \left ( fx+e \right ) -315\,B\sin \left ( fx+e \right ) -1152\,A+128\,B \right ) \sin \left ( fx+e \right ) }{2520\,f \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{7}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{9}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x)

[Out]

1/2520/f*(-280*B*cos(f*x+e)^8+315*A*cos(f*x+e)^6*sin(f*x+e)-315*B*cos(f*x+e)^6*sin(f*x+e)-360*A*cos(f*x+e)^6+4
0*B*cos(f*x+e)^6+315*A*cos(f*x+e)^4*sin(f*x+e)-315*B*sin(f*x+e)*cos(f*x+e)^4-432*A*cos(f*x+e)^4+48*B*cos(f*x+e
)^4+315*A*cos(f*x+e)^2*sin(f*x+e)-315*B*cos(f*x+e)^2*sin(f*x+e)-576*A*cos(f*x+e)^2+64*B*cos(f*x+e)^2+315*A*sin
(f*x+e)-315*B*sin(f*x+e)-1152*A+128*B)*(-c*(-1+sin(f*x+e)))^(9/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(7/2)/(-1+sin(
f*x+e))/cos(f*x+e)^7

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.89381, size = 427, normalized size = 1.71 \begin{align*} \frac{{\left (315 \,{\left (A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{8} - 315 \,{\left (A - B\right )} a^{3} c^{4} + 8 \,{\left (35 \, B a^{3} c^{4} \cos \left (f x + e\right )^{8} + 5 \,{\left (9 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{6} + 6 \,{\left (9 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{4} + 8 \,{\left (9 \, A - B\right )} a^{3} c^{4} \cos \left (f x + e\right )^{2} + 16 \,{\left (9 \, A - B\right )} a^{3} c^{4}\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}}{2520 \, f \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

1/2520*(315*(A - B)*a^3*c^4*cos(f*x + e)^8 - 315*(A - B)*a^3*c^4 + 8*(35*B*a^3*c^4*cos(f*x + e)^8 + 5*(9*A - B
)*a^3*c^4*cos(f*x + e)^6 + 6*(9*A - B)*a^3*c^4*cos(f*x + e)^4 + 8*(9*A - B)*a^3*c^4*cos(f*x + e)^2 + 16*(9*A -
B)*a^3*c^4)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))**(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^(7/2)*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

sage2