3.108 \(\int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx\)
Optimal. Leaf size=200 \[ -\frac{12 c^2 (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac{32 c^3 (7 A-9 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{35 a f}-\frac{128 c^4 (7 A-9 B) \cos (e+f x)}{35 a f \sqrt{c-c \sin (e+f x)}}-\frac{c (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f} \]
[Out]
(-128*(7*A - 9*B)*c^4*Cos[e + f*x])/(35*a*f*Sqrt[c - c*Sin[e + f*x]]) - (32*(7*A - 9*B)*c^3*Cos[e + f*x]*Sqrt[
c - c*Sin[e + f*x]])/(35*a*f) - (12*(7*A - 9*B)*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(35*a*f) - ((7*A
- 9*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(7*a*f) - ((A - B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/
(a*c*f)
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Rubi [A] time = 0.384575, antiderivative size = 200, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used =
{2967, 2855, 2647, 2646} \[ -\frac{12 c^2 (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac{32 c^3 (7 A-9 B) \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{35 a f}-\frac{128 c^4 (7 A-9 B) \cos (e+f x)}{35 a f \sqrt{c-c \sin (e+f x)}}-\frac{c (7 A-9 B) \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x]),x]
[Out]
(-128*(7*A - 9*B)*c^4*Cos[e + f*x])/(35*a*f*Sqrt[c - c*Sin[e + f*x]]) - (32*(7*A - 9*B)*c^3*Cos[e + f*x]*Sqrt[
c - c*Sin[e + f*x]])/(35*a*f) - (12*(7*A - 9*B)*c^2*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(35*a*f) - ((7*A
- 9*B)*c*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(7*a*f) - ((A - B)*Sec[e + f*x]*(c - c*Sin[e + f*x])^(9/2))/
(a*c*f)
Rule 2967
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))
Rule 2855
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((b*c + a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(p +
1)), x] + Dist[(b*(a*d*m + b*c*(m + p + 1)))/(a*g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x]
)^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]
Rule 2647
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]
Rule 2646
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rubi steps
\begin{align*} \int \frac{(A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2}}{a+a \sin (e+f x)} \, dx &=\frac{\int \sec ^2(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{9/2} \, dx}{a c}\\ &=-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac{(7 A-9 B) \int (c-c \sin (e+f x))^{7/2} \, dx}{2 a}\\ &=-\frac{(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac{(6 (7 A-9 B) c) \int (c-c \sin (e+f x))^{5/2} \, dx}{7 a}\\ &=-\frac{12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac{(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac{\left (48 (7 A-9 B) c^2\right ) \int (c-c \sin (e+f x))^{3/2} \, dx}{35 a}\\ &=-\frac{32 (7 A-9 B) c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{35 a f}-\frac{12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac{(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}-\frac{\left (64 (7 A-9 B) c^3\right ) \int \sqrt{c-c \sin (e+f x)} \, dx}{35 a}\\ &=-\frac{128 (7 A-9 B) c^4 \cos (e+f x)}{35 a f \sqrt{c-c \sin (e+f x)}}-\frac{32 (7 A-9 B) c^3 \cos (e+f x) \sqrt{c-c \sin (e+f x)}}{35 a f}-\frac{12 (7 A-9 B) c^2 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{35 a f}-\frac{(7 A-9 B) c \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{7 a f}-\frac{(A-B) \sec (e+f x) (c-c \sin (e+f x))^{9/2}}{a c f}\\ \end{align*}
Mathematica [A] time = 5.70282, size = 157, normalized size = 0.78 \[ -\frac{c^3 \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (196 (A-2 B) \cos (2 (e+f x))+2450 A \sin (e+f x)-14 A \sin (3 (e+f x))+4900 A-3430 B \sin (e+f x)+58 B \sin (3 (e+f x))+5 B \cos (4 (e+f x))-6125 B)}{140 a f (\sin (e+f x)+1) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x]),x]
[Out]
-(c^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[c - c*Sin[e + f*x]]*(4900*A - 6125*B + 196*(A - 2*B)*Cos[2*(e
+ f*x)] + 5*B*Cos[4*(e + f*x)] + 2450*A*Sin[e + f*x] - 3430*B*Sin[e + f*x] - 14*A*Sin[3*(e + f*x)] + 58*B*Sin
[3*(e + f*x)]))/(140*a*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x]))
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Maple [A] time = 0.926, size = 111, normalized size = 0.6 \begin{align*}{\frac{2\,{c}^{4} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( \left ( -7\,A+29\,B \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+ \left ( 308\,A-436\,B \right ) \sin \left ( fx+e \right ) +5\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( 49\,A-103\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+588\,A-716\,B \right ) }{35\,af\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x)
[Out]
2/35*c^4/a*(-1+sin(f*x+e))*((-7*A+29*B)*sin(f*x+e)*cos(f*x+e)^2+(308*A-436*B)*sin(f*x+e)+5*B*cos(f*x+e)^4+(49*
A-103*B)*cos(f*x+e)^2+588*A-716*B)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)/f
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Maxima [B] time = 1.55037, size = 645, normalized size = 3.22 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x, algorithm="maxima")
[Out]
2/35*(7*(91*c^(7/2) + 86*c^(7/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 336*c^(7/2)*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 + 266*c^(7/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 490*c^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 266
*c^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 336*c^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 86*c^(7/2)*si
n(f*x + e)^7/(cos(f*x + e) + 1)^7 + 91*c^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8)*A/((a + a*sin(f*x + e)/(co
s(f*x + e) + 1))*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(7/2)) - 2*(407*c^(7/2) + 407*c^(7/2)*sin(f*x + e)/
(cos(f*x + e) + 1) + 1442*c^(7/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1337*c^(7/2)*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3 + 2030*c^(7/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 1337*c^(7/2)*sin(f*x + e)^5/(cos(f*x + e) + 1)
^5 + 1442*c^(7/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 407*c^(7/2)*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 407*
c^(7/2)*sin(f*x + e)^8/(cos(f*x + e) + 1)^8)*B/((a + a*sin(f*x + e)/(cos(f*x + e) + 1))*(sin(f*x + e)^2/(cos(f
*x + e) + 1)^2 + 1)^(7/2)))/f
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Fricas [A] time = 1.59043, size = 282, normalized size = 1.41 \begin{align*} -\frac{2 \,{\left (5 \, B c^{3} \cos \left (f x + e\right )^{4} +{\left (49 \, A - 103 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} + 4 \,{\left (147 \, A - 179 \, B\right )} c^{3} -{\left ({\left (7 \, A - 29 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 4 \,{\left (77 \, A - 109 \, B\right )} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{35 \, a f \cos \left (f x + e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x, algorithm="fricas")
[Out]
-2/35*(5*B*c^3*cos(f*x + e)^4 + (49*A - 103*B)*c^3*cos(f*x + e)^2 + 4*(147*A - 179*B)*c^3 - ((7*A - 29*B)*c^3*
cos(f*x + e)^2 - 4*(77*A - 109*B)*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(a*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+B*sin(f*x+e))*(c-c*sin(f*x+e))**(7/2)/(a+a*sin(f*x+e)),x)
[Out]
Timed out
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Giac [B] time = 2.08861, size = 1127, normalized size = 5.64 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e)),x, algorithm="giac")
[Out]
-1/105*(4*(210*sqrt(2)*A*c^(25/2) - 210*sqrt(2)*B*c^(25/2) - 77*sqrt(2)*A*a^8*sqrt(c) + 109*sqrt(2)*B*a^8*sqrt
(c) + 154*A*a^8*sqrt(c) - 218*B*a^8*sqrt(c))*sgn(tan(1/2*f*x + 1/2*e) - 1)/(sqrt(2)*a*c^9 - a*c^9) - 3360*((sq
rt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))*A*c^4*sgn(tan(1/2*f*x + 1/2*e) - 1) - (sqrt(c
)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))*B*c^4*sgn(tan(1/2*f*x + 1/2*e) - 1) - A*c^(9/2)*s
gn(tan(1/2*f*x + 1/2*e) - 1) + B*c^(9/2)*sgn(tan(1/2*f*x + 1/2*e) - 1))/(((sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt
(c*tan(1/2*f*x + 1/2*e)^2 + c))^2 + 2*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))*sqrt
(c) - c)*a) - (((((((3*(119*A*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1) - 178*B*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) -
1))*tan(1/2*f*x + 1/2*e)/c^12 + 35*(7*A*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1) - 8*B*a^7*c^7*sgn(tan(1/2*f*x +
1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 7*(141*A*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1) - 212*B*a^7*c^7*sgn(
tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 35*(25*A*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1) - 34*B*
a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 35*(25*A*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e)
- 1) - 34*B*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 7*(141*A*a^7*c^7*sgn(tan(1/2*f
*x + 1/2*e) - 1) - 212*B*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 35*(7*A*a^7*c^7*s
gn(tan(1/2*f*x + 1/2*e) - 1) - 8*B*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)*tan(1/2*f*x + 1/2*e) + 3*(119*
A*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1) - 178*B*a^7*c^7*sgn(tan(1/2*f*x + 1/2*e) - 1))/c^12)/(c*tan(1/2*f*x +
1/2*e)^2 + c)^(7/2))/f