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

Optimal. Leaf size=210 \[ \frac{256 a^2 c^6 (13 A-3 B) \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 (13 A-3 B) \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 (13 A-3 B) \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 (13 A-3 B) \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \]

[Out]

(256*a^2*(13*A - 3*B)*c^6*Cos[e + f*x]^5)/(15015*f*(c - c*Sin[e + f*x])^(5/2)) + (64*a^2*(13*A - 3*B)*c^5*Cos[
e + f*x]^5)/(3003*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^2*(13*A - 3*B)*c^4*Cos[e + f*x]^5)/(429*f*Sqrt[c - c*Si
n[e + f*x]]) + (2*a^2*(13*A - 3*B)*c^3*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(143*f) - (2*a^2*B*c^2*Cos[e +
f*x]^5*(c - c*Sin[e + f*x])^(3/2))/(13*f)

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Rubi [A]  time = 0.554384, 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} \[ \frac{256 a^2 c^6 (13 A-3 B) \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 c^5 (13 A-3 B) \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 c^4 (13 A-3 B) \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 c^3 (13 A-3 B) \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(256*a^2*(13*A - 3*B)*c^6*Cos[e + f*x]^5)/(15015*f*(c - c*Sin[e + f*x])^(5/2)) + (64*a^2*(13*A - 3*B)*c^5*Cos[
e + f*x]^5)/(3003*f*(c - c*Sin[e + f*x])^(3/2)) + (8*a^2*(13*A - 3*B)*c^4*Cos[e + f*x]^5)/(429*f*Sqrt[c - c*Si
n[e + f*x]]) + (2*a^2*(13*A - 3*B)*c^3*Cos[e + f*x]^5*Sqrt[c - c*Sin[e + f*x]])/(143*f) - (2*a^2*B*c^2*Cos[e +
f*x]^5*(c - c*Sin[e + f*x])^(3/2))/(13*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))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{1}{13} \left (a^2 (13 A-3 B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{1}{143} \left (12 a^2 (13 A-3 B) c^3\right ) \int \cos ^4(e+f x) \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{1}{429} \left (32 a^2 (13 A-3 B) c^4\right ) \int \frac{\cos ^4(e+f x)}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}+\frac{\left (128 a^2 (13 A-3 B) c^5\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx}{3003}\\ &=\frac{256 a^2 (13 A-3 B) c^6 \cos ^5(e+f x)}{15015 f (c-c \sin (e+f x))^{5/2}}+\frac{64 a^2 (13 A-3 B) c^5 \cos ^5(e+f x)}{3003 f (c-c \sin (e+f x))^{3/2}}+\frac{8 a^2 (13 A-3 B) c^4 \cos ^5(e+f x)}{429 f \sqrt{c-c \sin (e+f x)}}+\frac{2 a^2 (13 A-3 B) c^3 \cos ^5(e+f x) \sqrt{c-c \sin (e+f x)}}{143 f}-\frac{2 a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{13 f}\\ \end{align*}

Mathematica [B]  time = 6.70347, size = 1355, normalized size = 6.45 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*A - 2*B)*Cos[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[
(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((4*A + B)*Cos[(3*(e + f*x))/2]*(a + a*Sin[e + f*x]
)^2*(c - c*Sin[e + f*x])^(7/2))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x
)/2])^4) + ((22*A - 7*B)*Cos[(5*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(160*f*(Cos[(
e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((A - 4*B)*Cos[(7*(e + f*x))/2]*(
a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x
)/2] + Sin[(e + f*x)/2])^4) + (A*Cos[(9*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(48*f
*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2*A - 3*B)*Cos[(11*(e +
f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + (B*Cos[(13*(e + f*x))/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(
7/2))/(416*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((7*A - 2*B)*S
in[(e + f*x)/2]*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2))/(8*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^
7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((4*A + B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[
(3*(e + f*x))/2])/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) + ((2
2*A - 7*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(5*(e + f*x))/2])/(160*f*(Cos[(e + f*x)/2] -
Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((A - 4*B)*(a + a*Sin[e + f*x])^2*(c - c*Sin[e
+ f*x])^(7/2)*Sin[(7*(e + f*x))/2])/(112*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^4) + (A*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(9*(e + f*x))/2])/(48*f*(Cos[(e + f*x
)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4) - ((2*A - 3*B)*(a + a*Sin[e + f*x])^2*(c -
c*Sin[e + f*x])^(7/2)*Sin[(11*(e + f*x))/2])/(352*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2]
+ Sin[(e + f*x)/2])^4) + (B*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^(7/2)*Sin[(13*(e + f*x))/2])/(416*f*(
Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

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Maple [A]  time = 1.12, 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 ) ^{3}{a}^{2} \left ( \left ( -1365\,A+4935\,B \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+ \left ( 11180\,A-11820\,B \right ) \sin \left ( fx+e \right ) +1155\,B \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( 5915\,A-10605\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}-12844\,A+12204\,B \right ) }{15015\,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))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x)

[Out]

2/15015*(-1+sin(f*x+e))*c^4*(1+sin(f*x+e))^3*a^2*((-1365*A+4935*B)*sin(f*x+e)*cos(f*x+e)^2+(11180*A-11820*B)*s
in(f*x+e)+1155*B*cos(f*x+e)^4+(5915*A-10605*B)*cos(f*x+e)^2-12844*A+12204*B)/cos(f*x+e)/(c-c*sin(f*x+e))^(1/2)
/f

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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 )}^{2}{\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))^2*(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)^2*(-c*sin(f*x + e) + c)^(7/2), x)

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Fricas [A]  time = 1.54687, size = 895, normalized size = 4.26 \begin{align*} \frac{2 \,{\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{7} + 105 \,{\left (13 \, A - 14 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{6} + 35 \,{\left (91 \, A - 87 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} - 20 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 32 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} - 64 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} +{\left (1155 \, B a^{2} c^{3} \cos \left (f x + e\right )^{6} - 105 \,{\left (13 \, A - 25 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{5} + 140 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{4} + 160 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{3} + 192 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right )^{2} + 256 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3} \cos \left (f x + e\right ) + 512 \,{\left (13 \, A - 3 \, B\right )} a^{2} c^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{15015 \,{\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))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

2/15015*(1155*B*a^2*c^3*cos(f*x + e)^7 + 105*(13*A - 14*B)*a^2*c^3*cos(f*x + e)^6 + 35*(91*A - 87*B)*a^2*c^3*c
os(f*x + e)^5 - 20*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^4 + 32*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^3 - 64*(13*A - 3
*B)*a^2*c^3*cos(f*x + e)^2 + 256*(13*A - 3*B)*a^2*c^3*cos(f*x + e) + 512*(13*A - 3*B)*a^2*c^3 + (1155*B*a^2*c^
3*cos(f*x + e)^6 - 105*(13*A - 25*B)*a^2*c^3*cos(f*x + e)^5 + 140*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^4 + 160*(1
3*A - 3*B)*a^2*c^3*cos(f*x + e)^3 + 192*(13*A - 3*B)*a^2*c^3*cos(f*x + e)^2 + 256*(13*A - 3*B)*a^2*c^3*cos(f*x
+ e) + 512*(13*A - 3*B)*a^2*c^3)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c)/(f*cos(f*x + e) - f*sin(f*x + e) + f
)

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

[Out]

Timed out

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

[Out]

Timed out