3.173 \(\int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx\)

Optimal. Leaf size=246 \[ \frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}} \]

[Out]

1/16*(A+B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/f/(c-c*sin(f*x+e))^(17/2)+1/56*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e)
)^(7/2)/c/f/(c-c*sin(f*x+e))^(15/2)+1/224*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^2/f/(c-c*sin(f*x+e))^(13
/2)+1/1120*(A-3*B)*cos(f*x+e)*(a+a*sin(f*x+e))^(7/2)/c^3/f/(c-c*sin(f*x+e))^(11/2)+1/8960*(A-3*B)*cos(f*x+e)*(
a+a*sin(f*x+e))^(7/2)/c^4/f/(c-c*sin(f*x+e))^(9/2)

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Rubi [A]  time = 0.59, antiderivative size = 246, normalized size of antiderivative = 1.00, 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 = {2972, 2743, 2742} \[ \frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}} \]

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])^(17/2),x]

[Out]

((A + B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(16*f*(c - c*Sin[e + f*x])^(17/2)) + ((A - 3*B)*Cos[e + f*x]
*(a + a*Sin[e + f*x])^(7/2))/(56*c*f*(c - c*Sin[e + f*x])^(15/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x
])^(7/2))/(224*c^2*f*(c - c*Sin[e + f*x])^(13/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(1120*
c^3*f*(c - c*Sin[e + f*x])^(11/2)) + ((A - 3*B)*Cos[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(8960*c^4*f*(c - c*Si
n[e + f*x])^(9/2))

Rule 2742

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(a*f*(2*m + 1)), x] /; FreeQ[{a, b, c, d, e, f
, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && NeQ[m, -2^(-1)]

Rule 2743

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((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)/(a*f*(2*m + 1)), x] + Dist[(m + n + 1)/(a*(2*m
+ 1)), 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, m, n}, x]
&& EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m
, 1] ||  !SumSimplerQ[n, 1])

Rule 2972

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] + Dist[(a*B*(m - n) + A*b*(m + n + 1))/(a*b*(2*m + 1)), 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, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2
- b^2, 0] && (LtQ[m, -2^(-1)] || (ILtQ[m + n, 0] &&  !SumSimplerQ[n, 1])) && NeQ[2*m + 1, 0]

Rubi steps

\begin {align*} \int \frac {(a+a \sin (e+f x))^{7/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{17/2}} \, dx &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac {(A-3 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx}{4 c}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {(3 (A-3 B)) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{56 c^2}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{112 c^3}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{1120 c^4}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{16 f (c-c \sin (e+f x))^{17/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{15/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{224 c^2 f (c-c \sin (e+f x))^{13/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{1120 c^3 f (c-c \sin (e+f x))^{11/2}}+\frac {(A-3 B) \cos (e+f x) (a+a \sin (e+f x))^{7/2}}{8960 c^4 f (c-c \sin (e+f x))^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 7.14, size = 436, normalized size = 1.77 \[ \frac {(-A-7 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}{5 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {(A+3 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {4 (3 A+5 B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{7 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {(A+B) (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {B (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}{4 f (c-c \sin (e+f x))^{17/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7} \]

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])^(17/2),x]

[Out]

((A + B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f
*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) - (4*(3*A + 5*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e
+ f*x]))^(7/2))/(7*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + ((A + 3*B)*(Cos[(e
+ f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c -
c*Sin[e + f*x])^(17/2)) + ((-A - 7*B)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(
5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)) + (B*(Cos[(e + f*x)/2] - Sin[(e + f*x
)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2))/(4*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(17/2)
)

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fricas [A]  time = 0.54, size = 243, normalized size = 0.99 \[ \frac {{\left (35 \, B a^{3} \cos \left (f x + e\right )^{4} - 56 \, {\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (17 \, A + 19 \, B\right )} a^{3} - 4 \, {\left (7 \, {\left (A + 2 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 2 \, {\left (9 \, A + 8 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{140 \, {\left (c^{9} f \cos \left (f x + e\right )^{9} - 32 \, c^{9} f \cos \left (f x + e\right )^{7} + 160 \, c^{9} f \cos \left (f x + e\right )^{5} - 256 \, c^{9} f \cos \left (f x + e\right )^{3} + 128 \, c^{9} f \cos \left (f x + e\right ) + 8 \, {\left (c^{9} f \cos \left (f x + e\right )^{7} - 10 \, c^{9} f \cos \left (f x + e\right )^{5} + 24 \, c^{9} f \cos \left (f x + e\right )^{3} - 16 \, c^{9} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]

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))^(17/2),x, algorithm="fricas")

[Out]

1/140*(35*B*a^3*cos(f*x + e)^4 - 56*(A + 2*B)*a^3*cos(f*x + e)^2 + 4*(17*A + 19*B)*a^3 - 4*(7*(A + 2*B)*a^3*co
s(f*x + e)^2 - 2*(9*A + 8*B)*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^9*f*cos(
f*x + e)^9 - 32*c^9*f*cos(f*x + e)^7 + 160*c^9*f*cos(f*x + e)^5 - 256*c^9*f*cos(f*x + e)^3 + 128*c^9*f*cos(f*x
+ e) + 8*(c^9*f*cos(f*x + e)^7 - 10*c^9*f*cos(f*x + e)^5 + 24*c^9*f*cos(f*x + e)^3 - 16*c^9*f*cos(f*x + e))*s
in(f*x + e))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))^(17/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.81, size = 560, normalized size = 2.28 \[ -\frac {\sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} \left (-3076 A \sin \left (f x +e \right )+1468 A \sin \left (f x +e \right ) \cos \left (f x +e \right )-300 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3076 A -268 B +40 B \left (\cos ^{6}\left (f x +e \right )\right )-204 B \sin \left (f x +e \right ) \cos \left (f x +e \right )-1332 A \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+372 A \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-9 B \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+504 B \left (\cos ^{2}\left (f x +e \right )\right )-B \left (\cos ^{8}\left (f x +e \right )\right )+164 B \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+96 A \left (\cos ^{7}\left (f x +e \right )\right )+111 B \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+B \left (\cos ^{7}\left (f x +e \right )\right ) \sin \left (f x +e \right )-12 A \left (\cos ^{7}\left (f x +e \right )\right ) \sin \left (f x +e \right )-1548 A \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+268 B \sin \left (f x +e \right )-8 B \left (\cos ^{7}\left (f x +e \right )\right )+3880 A \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-31 B \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )-1608 A \cos \left (f x +e \right )-275 B \left (\cos ^{4}\left (f x +e \right )\right )+64 B \cos \left (f x +e \right )+108 A \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+12 A \left (\cos ^{8}\left (f x +e \right )\right )+2880 A \left (\cos ^{4}\left (f x +e \right )\right )-960 A \left (\cos ^{5}\left (f x +e \right )\right )+80 B \left (\cos ^{5}\left (f x +e \right )\right )+2332 A \left (\cos ^{3}\left (f x +e \right )\right )-136 B \left (\cos ^{3}\left (f x +e \right )\right )-5348 A \left (\cos ^{2}\left (f x +e \right )\right )-480 A \left (\cos ^{6}\left (f x +e \right )\right )\right )}{140 f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {17}{2}} \left (\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+\cos ^{4}\left (f x +e \right )-4 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \left (\cos ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )-4 \cos \left (f x +e \right )+8\right )} \]

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))^(17/2),x)

[Out]

-1/140/f*sin(f*x+e)*(a*(1+sin(f*x+e)))^(7/2)*(164*B*cos(f*x+e)^3*sin(f*x+e)-3076*A*sin(f*x+e)+1468*A*sin(f*x+e
)*cos(f*x+e)+108*A*cos(f*x+e)^6*sin(f*x+e)-1548*A*cos(f*x+e)^3*sin(f*x+e)-1332*A*cos(f*x+e)^4*sin(f*x+e)+111*B
*sin(f*x+e)*cos(f*x+e)^4+3880*A*cos(f*x+e)^2*sin(f*x+e)-300*B*cos(f*x+e)^2*sin(f*x+e)-275*B*cos(f*x+e)^4+3076*
A-268*B+2332*A*cos(f*x+e)^3-204*B*sin(f*x+e)*cos(f*x+e)-9*B*cos(f*x+e)^6*sin(f*x+e)-960*A*cos(f*x+e)^5+80*B*co
s(f*x+e)^5+40*B*cos(f*x+e)^6-31*B*cos(f*x+e)^5*sin(f*x+e)+372*A*cos(f*x+e)^5*sin(f*x+e)-B*cos(f*x+e)^8+504*B*c
os(f*x+e)^2+2880*A*cos(f*x+e)^4-136*B*cos(f*x+e)^3+268*B*sin(f*x+e)-1608*A*cos(f*x+e)-12*A*cos(f*x+e)^7*sin(f*
x+e)+B*cos(f*x+e)^7*sin(f*x+e)+64*B*cos(f*x+e)-480*A*cos(f*x+e)^6-5348*A*cos(f*x+e)^2+96*A*cos(f*x+e)^7-8*B*co
s(f*x+e)^7+12*A*cos(f*x+e)^8)/(-c*(sin(f*x+e)-1))^(17/2)/(cos(f*x+e)^4+sin(f*x+e)*cos(f*x+e)^3+3*cos(f*x+e)^3-
4*cos(f*x+e)^2*sin(f*x+e)-8*cos(f*x+e)^2-4*sin(f*x+e)*cos(f*x+e)-4*cos(f*x+e)+8*sin(f*x+e)+8)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))^(17/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 28.34, size = 841, normalized size = 3.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(17/2),x)

[Out]

((c - c*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*((8*B*a^3*exp(e*5i + f*x*5i)*(a + a*(
(exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(c^9*f) + (8*B*a^3*exp(e*13i + f*x*13i)*(a +
a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2))/(c^9*f) - (64*a^3*exp(e*6i + f*x*6i)*(a +
a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*1i + B*2i))/(5*c^9*f) - (32*a^3*exp(e*7i
+ f*x*7i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(8*A + 11*B))/(5*c^9*f) + (
64*a^3*exp(e*12i + f*x*12i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*1i + B*
2i))/(5*c^9*f) - (32*a^3*exp(e*11i + f*x*11i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2)
)^(1/2)*(8*A + 11*B))/(5*c^9*f) + (64*a^3*exp(e*8i + f*x*8i)*(a + a*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i +
f*x*1i)*1i)/2))^(1/2)*(A*13i + B*10i))/(7*c^9*f) - (64*a^3*exp(e*10i + f*x*10i)*(a + a*((exp(- e*1i - f*x*1i)
*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(A*13i + B*10i))/(7*c^9*f) + (16*a^3*exp(e*9i + f*x*9i)*(a + a*((ex
p(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i)*1i)/2))^(1/2)*(64*A + 53*B))/(7*c^9*f)))/(exp(e*1i + f*x*1i)*16
i - 119*exp(e*2i + f*x*2i) - exp(e*3i + f*x*3i)*544i + 1700*exp(e*4i + f*x*4i) + exp(e*5i + f*x*5i)*3808i - 61
88*exp(e*6i + f*x*6i) - exp(e*7i + f*x*7i)*7072i + 4862*exp(e*8i + f*x*8i) + 4862*exp(e*10i + f*x*10i) + exp(e
*11i + f*x*11i)*7072i - 6188*exp(e*12i + f*x*12i) - exp(e*13i + f*x*13i)*3808i + 1700*exp(e*14i + f*x*14i) + e
xp(e*15i + f*x*15i)*544i - 119*exp(e*16i + f*x*16i) - exp(e*17i + f*x*17i)*16i + exp(e*18i + f*x*18i) + 1)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**(17/2),x)

[Out]

Timed out

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