∫ (a+a sin(e+fx))2(A+B sin(e+fx))________________________

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

Optimal. Leaf size=197 \[ \frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{15015 c^2 f (c-c \sin (e+f x))^5}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{3003 c f (c-c \sin (e+f x))^6}+\frac{a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac{a^2 c (4 A-9 B) \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8} \]

[Out]

(a^2*(A + B)*c^2*Cos[e + f*x]^5)/(13*f*(c - c*Sin[e + f*x])^9) + (a^2*(4*A - 9*B)*c*Cos[e + f*x]^5)/(143*f*(c

- c*Sin[e + f*x])^8) + (a^2*(4*A - 9*B)*Cos[e + f*x]^5)/(429*f*(c - c*Sin[e + f*x])^7) + (2*a^2*(4*A - 9*B)*Co

s[e + f*x]^5)/(3003*c*f*(c - c*Sin[e + f*x])^6) + (2*a^2*(4*A - 9*B)*Cos[e + f*x]^5)/(15015*c^2*f*(c - c*Sin[e

+ f*x])^5)

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Rubi [A] time = 0.464564, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2967, 2859, 2672, 2671} \[ \frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{15015 c^2 f (c-c \sin (e+f x))^5}+\frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{3003 c f (c-c \sin (e+f x))^6}+\frac{a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac{a^2 c (4 A-9 B) \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8} \]

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,x]

[Out]

(a^2*(A + B)*c^2*Cos[e + f*x]^5)/(13*f*(c - c*Sin[e + f*x])^9) + (a^2*(4*A - 9*B)*c*Cos[e + f*x]^5)/(143*f*(c

- c*Sin[e + f*x])^8) + (a^2*(4*A - 9*B)*Cos[e + f*x]^5)/(429*f*(c - c*Sin[e + f*x])^7) + (2*a^2*(4*A - 9*B)*Co

s[e + f*x]^5)/(3003*c*f*(c - c*Sin[e + f*x])^6) + (2*a^2*(4*A - 9*B)*Cos[e + f*x]^5)/(15015*c^2*f*(c - c*Sin[e

+ f*x])^5)

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 2859

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*(2*m +

p + 1)), x] + Dist[(a*d*m + b*c*(m + p + 1))/(a*b*(2*m + p + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^

(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && (LtQ[m, -1] || ILtQ[Simplify[

m + p], 0]) && NeQ[2*m + p + 1, 0]

Rule 2672

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)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*

Simplify[2*m + p + 1]), 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] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]

Rule 2671

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)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^

2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^7} \, dx &=\left (a^2 c^2\right ) \int \frac{\cos ^4(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^9} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{1}{13} \left (a^2 (4 A-9 B) c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^8} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{a^2 (4 A-9 B) c \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8}+\frac{1}{143} \left (3 a^2 (4 A-9 B)\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{a^2 (4 A-9 B) c \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8}+\frac{a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac{\left (2 a^2 (4 A-9 B)\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx}{429 c}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{a^2 (4 A-9 B) c \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8}+\frac{a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{3003 c f (c-c \sin (e+f x))^6}+\frac{\left (2 a^2 (4 A-9 B)\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx}{3003 c^2}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{13 f (c-c \sin (e+f x))^9}+\frac{a^2 (4 A-9 B) c \cos ^5(e+f x)}{143 f (c-c \sin (e+f x))^8}+\frac{a^2 (4 A-9 B) \cos ^5(e+f x)}{429 f (c-c \sin (e+f x))^7}+\frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{3003 c f (c-c \sin (e+f x))^6}+\frac{2 a^2 (4 A-9 B) \cos ^5(e+f x)}{15015 c^2 f (c-c \sin (e+f x))^5}\\ \end{align*}

Mathematica [A] time = 3.59157, size = 313, normalized size = 1.59 \[ -\frac{a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (6006 (8 A+7 B) \cos \left (\frac{1}{2} (e+f x)\right )-1716 (11 A+19 B) \cos \left (\frac{3}{2} (e+f x)\right )+54912 A \sin \left (\frac{1}{2} (e+f x)\right )+24024 A \sin \left (\frac{3}{2} (e+f x)\right )-2860 A \sin \left (\frac{5}{2} (e+f x)\right )+312 A \sin \left (\frac{9}{2} (e+f x)\right )-4 A \sin \left (\frac{13}{2} (e+f x)\right )-1144 A \cos \left (\frac{7}{2} (e+f x)\right )+52 A \cos \left (\frac{11}{2} (e+f x)\right )+26598 B \sin \left (\frac{1}{2} (e+f x)\right )+21021 B \sin \left (\frac{3}{2} (e+f x)\right )-8580 B \sin \left (\frac{5}{2} (e+f x)\right )-702 B \sin \left (\frac{9}{2} (e+f x)\right )+9 B \sin \left (\frac{13}{2} (e+f x)\right )-15015 B \cos \left (\frac{5}{2} (e+f x)\right )+2574 B \cos \left (\frac{7}{2} (e+f x)\right )-117 B \cos \left (\frac{11}{2} (e+f x)\right )\right )}{240240 c^7 f (\sin (e+f x)-1)^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]

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,x]

[Out]

-(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(6006*(8*A + 7*B)*Cos[(e + f*x)/2] - 1716*(11

*A + 19*B)*Cos[(3*(e + f*x))/2] - 15015*B*Cos[(5*(e + f*x))/2] - 1144*A*Cos[(7*(e + f*x))/2] + 2574*B*Cos[(7*(

e + f*x))/2] + 52*A*Cos[(11*(e + f*x))/2] - 117*B*Cos[(11*(e + f*x))/2] + 54912*A*Sin[(e + f*x)/2] + 26598*B*S

in[(e + f*x)/2] + 24024*A*Sin[(3*(e + f*x))/2] + 21021*B*Sin[(3*(e + f*x))/2] - 2860*A*Sin[(5*(e + f*x))/2] -

8580*B*Sin[(5*(e + f*x))/2] + 312*A*Sin[(9*(e + f*x))/2] - 702*B*Sin[(9*(e + f*x))/2] - 4*A*Sin[(13*(e + f*x))

/2] + 9*B*Sin[(13*(e + f*x))/2]))/(240240*c^7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4*(-1 + Sin[e + f*x])^7)

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Maple [A] time = 0.185, size = 293, normalized size = 1.5 \begin{align*} 2\,{\frac{{a}^{2}}{f{c}^{7}} \left ( -1/10\,{\frac{8320\,A+7680\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{10}}}-1/3\,{\frac{120\,A+30\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{3}}}-1/13\,{\frac{256\,A+256\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{13}}}-1/7\,{\frac{7744\,A+5368\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{7}}}-1/8\,{\frac{10560\,A+8256\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{8}}}-1/11\,{\frac{4480\,A+4352\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{11}}}-1/2\,{\frac{16\,A+2\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{2}}}-1/6\,{\frac{4320\,A+2568\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{6}}}-{\frac{A}{\tan \left ( 1/2\,fx+e/2 \right ) -1}}-1/9\,{\frac{10896\,A+9360\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{9}}}-1/4\,{\frac{560\,A+208\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{4}}}-1/12\,{\frac{1536\,A+1536\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{12}}}-1/5\,{\frac{1816\,A+884\,B}{ \left ( \tan \left ( 1/2\,fx+e/2 \right ) -1 \right ) ^{5}}} \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,x)

[Out]

2/f*a^2/c^7*(-1/10*(8320*A+7680*B)/(tan(1/2*f*x+1/2*e)-1)^10-1/3*(120*A+30*B)/(tan(1/2*f*x+1/2*e)-1)^3-1/13*(2

56*A+256*B)/(tan(1/2*f*x+1/2*e)-1)^13-1/7*(7744*A+5368*B)/(tan(1/2*f*x+1/2*e)-1)^7-1/8*(10560*A+8256*B)/(tan(1

/2*f*x+1/2*e)-1)^8-1/11*(4480*A+4352*B)/(tan(1/2*f*x+1/2*e)-1)^11-1/2*(16*A+2*B)/(tan(1/2*f*x+1/2*e)-1)^2-1/6*

(4320*A+2568*B)/(tan(1/2*f*x+1/2*e)-1)^6-A/(tan(1/2*f*x+1/2*e)-1)-1/9*(10896*A+9360*B)/(tan(1/2*f*x+1/2*e)-1)^

9-1/4*(560*A+208*B)/(tan(1/2*f*x+1/2*e)-1)^4-1/12*(1536*A+1536*B)/(tan(1/2*f*x+1/2*e)-1)^12-1/5*(1816*A+884*B)

/(tan(1/2*f*x+1/2*e)-1)^5)

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Maxima [B] time = 1.52729, size = 4212, normalized size = 21.38 \begin{align*} \text{result too large to display} \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,x, algorithm="maxima")

[Out]

-2/45045*(2*A*a^2*(4771*sin(f*x + e)/(cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*si

n(f*x + e)^3/(cos(f*x + e) + 1)^3 - 187330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 265122*sin(f*x + e)^5/(cos(f*

x + e) + 1)^5 - 353496*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 2072

07*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 30030*sin(f*x + e)^10/(co

s(f*x + e) + 1)^10 - 367)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e)

+ 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*

sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(

cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)

^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin

(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) + 4*B*a^2*(4771*sin(f*x + e)/(

cos(f*x + e) + 1) - 28626*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 74932*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 18

7330*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 265122*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 353496*sin(f*x + e)^6/

(cos(f*x + e) + 1)^6 + 276276*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 207207*sin(f*x + e)^8/(cos(f*x + e) + 1)^8

+ 75075*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 30030*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 367)/(c^7 - 13*c^

7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f

*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 +

1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x

+ e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x

+ e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c

^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) + 15*A*a^2*(3796*sin(f*x + e)/(cos(f*x + e) + 1) - 22776*sin(f*x + e

)^2/(cos(f*x + e) + 1)^2 + 77506*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 193765*sin(f*x + e)^4/(cos(f*x + e) + 1

)^4 + 339768*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 453024*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 444444*sin(f*x

+ e)^7/(cos(f*x + e) + 1)^7 - 333333*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 180180*sin(f*x + e)^9/(cos(f*x + e

) + 1)^9 - 72072*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 18018*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - 3003*si

n(f*x + e)^12/(cos(f*x + e) + 1)^12 - 523)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)

^2/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) +

1)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7

*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(

cos(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1

)^11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) - 70*A*a^2*(6

11*sin(f*x + e)/(cos(f*x + e) + 1) - 2379*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 8723*sin(f*x + e)^3/(cos(f*x +

e) + 1)^3 - 18590*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33462*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 40326*sin

(f*x + e)^6/(cos(f*x + e) + 1)^6 + 40326*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 27027*sin(f*x + e)^8/(cos(f*x +

e) + 1)^8 + 15015*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 4719*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 1287*sin

(f*x + e)^11/(cos(f*x + e) + 1)^11 - 47)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2

/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1

)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*s

in(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(co

s(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^

11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) - 35*B*a^2*(611

*sin(f*x + e)/(cos(f*x + e) + 1) - 2379*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 8723*sin(f*x + e)^3/(cos(f*x + e

) + 1)^3 - 18590*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 33462*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 40326*sin(f

*x + e)^6/(cos(f*x + e) + 1)^6 + 40326*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 27027*sin(f*x + e)^8/(cos(f*x + e

) + 1)^8 + 15015*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 4719*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 1287*sin(f

*x + e)^11/(cos(f*x + e) + 1)^11 - 47)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2/(

cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1)^

4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*sin

(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(cos(

f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^11

+ 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13) - 462*B*a^2*(13*s

in(f*x + e)/(cos(f*x + e) + 1) - 78*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 286*sin(f*x + e)^3/(cos(f*x + e) + 1

)^3 - 520*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 936*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 858*sin(f*x + e)^6/(

cos(f*x + e) + 1)^6 + 858*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 351*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 195*

sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 1)/(c^7 - 13*c^7*sin(f*x + e)/(cos(f*x + e) + 1) + 78*c^7*sin(f*x + e)^2

/(cos(f*x + e) + 1)^2 - 286*c^7*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 715*c^7*sin(f*x + e)^4/(cos(f*x + e) + 1

)^4 - 1287*c^7*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 1716*c^7*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1716*c^7*s

in(f*x + e)^7/(cos(f*x + e) + 1)^7 + 1287*c^7*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 715*c^7*sin(f*x + e)^9/(co

s(f*x + e) + 1)^9 + 286*c^7*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 - 78*c^7*sin(f*x + e)^11/(cos(f*x + e) + 1)^

11 + 13*c^7*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - c^7*sin(f*x + e)^13/(cos(f*x + e) + 1)^13))/f

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Fricas [B] time = 1.42552, size = 1204, normalized size = 6.11 \begin{align*} \frac{2 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{7} - 12 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} - 49 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} + 70 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} + 105 \,{\left (7 \, A + 20 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 105 \,{\left (25 \, A + 51 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2310 \,{\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 4620 \,{\left (A + B\right )} a^{2} +{\left (2 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{6} + 14 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{5} - 35 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{4} - 105 \,{\left (4 \, A - 9 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} + 105 \,{\left (3 \, A + 29 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2310 \,{\left (A + B\right )} a^{2} \cos \left (f x + e\right ) - 4620 \,{\left (A + B\right )} a^{2}\right )} \sin \left (f x + e\right )}{15015 \,{\left (c^{7} f \cos \left (f x + e\right )^{7} + 7 \, c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{5} - 56 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{3} + 112 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f -{\left (c^{7} f \cos \left (f x + e\right )^{6} - 6 \, c^{7} f \cos \left (f x + e\right )^{5} - 24 \, c^{7} f \cos \left (f x + e\right )^{4} + 32 \, c^{7} f \cos \left (f x + e\right )^{3} + 80 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f \cos \left (f x + e\right ) - 64 \, c^{7} f\right )} \sin \left (f x + e\right )\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,x, algorithm="fricas")

[Out]

1/15015*(2*(4*A - 9*B)*a^2*cos(f*x + e)^7 - 12*(4*A - 9*B)*a^2*cos(f*x + e)^6 - 49*(4*A - 9*B)*a^2*cos(f*x + e

)^5 + 70*(4*A - 9*B)*a^2*cos(f*x + e)^4 + 105*(7*A + 20*B)*a^2*cos(f*x + e)^3 + 105*(25*A + 51*B)*a^2*cos(f*x

+ e)^2 - 2310*(A + B)*a^2*cos(f*x + e) - 4620*(A + B)*a^2 + (2*(4*A - 9*B)*a^2*cos(f*x + e)^6 + 14*(4*A - 9*B)

*a^2*cos(f*x + e)^5 - 35*(4*A - 9*B)*a^2*cos(f*x + e)^4 - 105*(4*A - 9*B)*a^2*cos(f*x + e)^3 + 105*(3*A + 29*B

)*a^2*cos(f*x + e)^2 - 2310*(A + B)*a^2*cos(f*x + e) - 4620*(A + B)*a^2)*sin(f*x + e))/(c^7*f*cos(f*x + e)^7 +

7*c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x + e)^5 - 56*c^7*f*cos(f*x + e)^4 + 48*c^7*f*cos(f*x + e)^3 + 112*c^

7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f - (c^7*f*cos(f*x + e)^6 - 6*c^7*f*cos(f*x + e)^5 - 24*c^

7*f*cos(f*x + e)^4 + 32*c^7*f*cos(f*x + e)^3 + 80*c^7*f*cos(f*x + e)^2 - 32*c^7*f*cos(f*x + e) - 64*c^7*f)*sin

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] time = 1.29137, size = 601, normalized size = 3.05 \begin{align*} -\frac{2 \,{\left (15015 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{12} - 60060 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} + 15015 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11} + 270270 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 15015 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 600600 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 105105 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9} + 1174173 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 93093 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 1465464 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 234234 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 1559844 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 131274 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 1094808 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 181038 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 659945 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 47190 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 233948 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 45903 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 77454 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1599 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 7904 \, A a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2769 \, B a^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1763 \, A a^{2} - 213 \, B a^{2}\right )}}{15015 \, c^{7} f{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1\right )}^{13}} \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,x, algorithm="giac")

[Out]

-2/15015*(15015*A*a^2*tan(1/2*f*x + 1/2*e)^12 - 60060*A*a^2*tan(1/2*f*x + 1/2*e)^11 + 15015*B*a^2*tan(1/2*f*x

+ 1/2*e)^11 + 270270*A*a^2*tan(1/2*f*x + 1/2*e)^10 - 15015*B*a^2*tan(1/2*f*x + 1/2*e)^10 - 600600*A*a^2*tan(1/

2*f*x + 1/2*e)^9 + 105105*B*a^2*tan(1/2*f*x + 1/2*e)^9 + 1174173*A*a^2*tan(1/2*f*x + 1/2*e)^8 - 93093*B*a^2*ta

n(1/2*f*x + 1/2*e)^8 - 1465464*A*a^2*tan(1/2*f*x + 1/2*e)^7 + 234234*B*a^2*tan(1/2*f*x + 1/2*e)^7 + 1559844*A*

a^2*tan(1/2*f*x + 1/2*e)^6 - 131274*B*a^2*tan(1/2*f*x + 1/2*e)^6 - 1094808*A*a^2*tan(1/2*f*x + 1/2*e)^5 + 1810

38*B*a^2*tan(1/2*f*x + 1/2*e)^5 + 659945*A*a^2*tan(1/2*f*x + 1/2*e)^4 - 47190*B*a^2*tan(1/2*f*x + 1/2*e)^4 - 2

33948*A*a^2*tan(1/2*f*x + 1/2*e)^3 + 45903*B*a^2*tan(1/2*f*x + 1/2*e)^3 + 77454*A*a^2*tan(1/2*f*x + 1/2*e)^2 -

1599*B*a^2*tan(1/2*f*x + 1/2*e)^2 - 7904*A*a^2*tan(1/2*f*x + 1/2*e) + 2769*B*a^2*tan(1/2*f*x + 1/2*e) + 1763*

A*a^2 - 213*B*a^2)/(c^7*f*(tan(1/2*f*x + 1/2*e) - 1)^13)

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