3.98 \(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx\)

Optimal. Leaf size=210 \[ \frac{2 a^3 c^4 (15 A-B) \cos ^7(e+f x)}{195 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a^3 c^5 (15 A-B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^3 c^6 (15 A-B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac{256 a^3 c^7 (15 A-B) \cos ^7(e+f x)}{45045 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f} \]

[Out]

(256*a^3*(15*A - B)*c^7*Cos[e + f*x]^7)/(45045*f*(c - c*Sin[e + f*x])^(7/2)) + (64*a^3*(15*A - B)*c^6*Cos[e +
f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^(5/2)) + (8*a^3*(15*A - B)*c^5*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x]
)^(3/2)) + (2*a^3*(15*A - B)*c^4*Cos[e + f*x]^7)/(195*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*B*c^3*Cos[e + f*x]^
7*Sqrt[c - c*Sin[e + f*x]])/(15*f)

________________________________________________________________________________________

Rubi [A]  time = 0.534393, antiderivative size = 210, 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, 2856, 2674, 2673} \[ \frac{2 a^3 c^4 (15 A-B) \cos ^7(e+f x)}{195 f \sqrt{c-c \sin (e+f x)}}+\frac{8 a^3 c^5 (15 A-B) \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac{64 a^3 c^6 (15 A-B) \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac{256 a^3 c^7 (15 A-B) \cos ^7(e+f x)}{45045 f (c-c \sin (e+f x))^{7/2}}-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

(256*a^3*(15*A - B)*c^7*Cos[e + f*x]^7)/(45045*f*(c - c*Sin[e + f*x])^(7/2)) + (64*a^3*(15*A - B)*c^6*Cos[e +
f*x]^7)/(6435*f*(c - c*Sin[e + f*x])^(5/2)) + (8*a^3*(15*A - B)*c^5*Cos[e + f*x]^7)/(715*f*(c - c*Sin[e + f*x]
)^(3/2)) + (2*a^3*(15*A - B)*c^4*Cos[e + f*x]^7)/(195*f*Sqrt[c - c*Sin[e + f*x]]) - (2*a^3*B*c^3*Cos[e + f*x]^
7*Sqrt[c - c*Sin[e + f*x]])/(15*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 2856

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[(d*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(f*g*(m + p + 1)), x]
+ Dist[(a*d*m + b*c*(m + p + 1))/(b*(m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^m, x], x] /; Fre
eQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[Simplify[(2*m + p + 1)/2], 0] && NeQ[m + p + 1
, 0]

Rule 2674

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[Simplify[(2*m + p - 1)/2], 0] && NeQ[m + p, 0]

Rule 2673

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx\\ &=-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{1}{15} \left (a^3 (15 A-B) c^3\right ) \int \cos ^6(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{1}{65} \left (4 a^3 (15 A-B) c^4\right ) \int \frac{\cos ^6(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{8 a^3 (15 A-B) c^5 \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{1}{715} \left (32 a^3 (15 A-B) c^5\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac{64 a^3 (15 A-B) c^6 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac{8 a^3 (15 A-B) c^5 \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f}+\frac{\left (128 a^3 (15 A-B) c^6\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx}{6435}\\ &=\frac{256 a^3 (15 A-B) c^7 \cos ^7(e+f x)}{45045 f (c-c \sin (e+f x))^{7/2}}+\frac{64 a^3 (15 A-B) c^6 \cos ^7(e+f x)}{6435 f (c-c \sin (e+f x))^{5/2}}+\frac{8 a^3 (15 A-B) c^5 \cos ^7(e+f x)}{715 f (c-c \sin (e+f x))^{3/2}}+\frac{2 a^3 (15 A-B) c^4 \cos ^7(e+f x)}{195 f \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 B c^3 \cos ^7(e+f x) \sqrt{c-c \sin (e+f x)}}{15 f}\\ \end{align*}

Mathematica [B]  time = 6.88615, size = 1569, normalized size = 7.47 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^(7/2),x]

[Out]

(5*(8*A - B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(64*f*(Cos[(e + f*x)/2] - Sin
[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (5*(6*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f
*x])^3*(c - c*Sin[e + f*x])^(7/2))/(192*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])^6) + (3*(10*A - 3*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(320*f*
(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - (3*(4*A + 3*B)*Cos[(7*(e +
f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(448*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((12*A - 5*B)*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e
+ f*x])^(7/2))/(576*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) - ((2*
A + 5*B)*Cos[(11*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(704*f*(Cos[(e + f*x)/2] - S
in[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A - B)*Cos[(13*(e + f*x))/2]*(a + a*Sin[e +
f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(832*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^6) - (B*Cos[(15*(e + f*x))/2]*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(960*f*(Cos[(e + f
*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (5*(8*A - B)*Sin[(e + f*x)/2]*(a + a*S
in[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2))/(64*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + S
in[(e + f*x)/2])^6) + (5*(6*A + B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(3*(e + f*x))/2])/(19
2*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (3*(10*A - 3*B)*(a + a*
Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(320*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^
7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (3*(4*A + 3*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*
Sin[(7*(e + f*x))/2])/(448*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)
+ ((12*A - 5*B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(9*(e + f*x))/2])/(576*f*(Cos[(e + f*x)/
2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + ((2*A + 5*B)*(a + a*Sin[e + f*x])^3*(c - c
*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(704*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2])^6) + ((2*A - B)*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(13*(e + f*x))/2])/(8
32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6) + (B*(a + a*Sin[e + f*x]
)^3*(c - c*Sin[e + f*x])^(7/2)*Sin[(15*(e + f*x))/2])/(960*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e +
f*x)/2] + Sin[(e + f*x)/2])^6)

________________________________________________________________________________________

Maple [A]  time = 1.05, size = 121, normalized size = 0.6 \begin{align*}{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ){c}^{4} \left ( 1+\sin \left ( fx+e \right ) \right ) ^{4}{a}^{3} \left ( \left ( -3465\,A+12243\,B \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+ \left ( 24780\,A-25676\,B \right ) \sin \left ( fx+e \right ) +3003\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( 14175\,A-24969\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}-26700\,A+25804\,B \right ) }{45045\,f\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+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x)

[Out]

2/45045*(-1+sin(f*x+e))*c^4*(1+sin(f*x+e))^4*a^3*((-3465*A+12243*B)*sin(f*x+e)*cos(f*x+e)^2+(24780*A-25676*B)*
sin(f*x+e)+3003*B*cos(f*x+e)^4+(14175*A-24969*B)*cos(f*x+e)^2-26700*A+25804*B)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/
2)/f

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(-c*sin(f*x + e) + c)^(7/2), x)

________________________________________________________________________________________

Fricas [B]  time = 1.52881, size = 972, normalized size = 4.63 \begin{align*} -\frac{2 \,{\left (3003 \, B a^{3} c^{3} \cos \left (f x + e\right )^{8} - 231 \,{\left (15 \, A - 14 \, B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{7} + 21 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{6} - 28 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{5} + 40 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{4} - 64 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{3} + 128 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{2} - 512 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right ) - 1024 \,{\left (15 \, A - B\right )} a^{3} c^{3} -{\left (3003 \, B a^{3} c^{3} \cos \left (f x + e\right )^{7} + 231 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{6} + 252 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{5} + 280 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{4} + 320 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{3} + 384 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right )^{2} + 512 \,{\left (15 \, A - B\right )} a^{3} c^{3} \cos \left (f x + e\right ) + 1024 \,{\left (15 \, A - B\right )} a^{3} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{45045 \,{\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

-2/45045*(3003*B*a^3*c^3*cos(f*x + e)^8 - 231*(15*A - 14*B)*a^3*c^3*cos(f*x + e)^7 + 21*(15*A - B)*a^3*c^3*cos
(f*x + e)^6 - 28*(15*A - B)*a^3*c^3*cos(f*x + e)^5 + 40*(15*A - B)*a^3*c^3*cos(f*x + e)^4 - 64*(15*A - B)*a^3*
c^3*cos(f*x + e)^3 + 128*(15*A - B)*a^3*c^3*cos(f*x + e)^2 - 512*(15*A - B)*a^3*c^3*cos(f*x + e) - 1024*(15*A
- B)*a^3*c^3 - (3003*B*a^3*c^3*cos(f*x + e)^7 + 231*(15*A - B)*a^3*c^3*cos(f*x + e)^6 + 252*(15*A - B)*a^3*c^3
*cos(f*x + e)^5 + 280*(15*A - B)*a^3*c^3*cos(f*x + e)^4 + 320*(15*A - B)*a^3*c^3*cos(f*x + e)^3 + 384*(15*A -
B)*a^3*c^3*cos(f*x + e)^2 + 512*(15*A - B)*a^3*c^3*cos(f*x + e) + 1024*(15*A - B)*a^3*c^3)*sin(f*x + e))*sqrt(
-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{3}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^3*(-c*sin(f*x + e) + c)^(7/2), x)