3.97 \(\int \frac{(a+a \sin (e+f x))^2 (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx\)
Optimal. Leaf size=222 \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac{a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 (3 A-13 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{256 \sqrt{2} c^{9/2} f}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}} \]
[Out]
(a^2*(3*A - 13*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(256*Sqrt[2]*c^(9/2)*f)
+ (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(8*f*(c - c*Sin[e + f*x])^(13/2)) + (a^2*(3*A - 13*B)*Cos[e + f*x]^3)/(48*f
*(c - c*Sin[e + f*x])^(9/2)) - (a^2*(3*A - 13*B)*Cos[e + f*x])/(64*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*(3
*A - 13*B)*Cos[e + f*x])/(256*c^3*f*(c - c*Sin[e + f*x])^(3/2))
________________________________________________________________________________________
Rubi [A] time = 0.51109, antiderivative size = 222, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used =
{2967, 2859, 2680, 2650, 2649, 206} \[ \frac{a^2 c^2 (A+B) \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac{a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 (3 A-13 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{256 \sqrt{2} c^{9/2} f}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}} \]
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])^(9/2),x]
[Out]
(a^2*(3*A - 13*B)*ArcTanh[(Sqrt[c]*Cos[e + f*x])/(Sqrt[2]*Sqrt[c - c*Sin[e + f*x]])])/(256*Sqrt[2]*c^(9/2)*f)
+ (a^2*(A + B)*c^2*Cos[e + f*x]^5)/(8*f*(c - c*Sin[e + f*x])^(13/2)) + (a^2*(3*A - 13*B)*Cos[e + f*x]^3)/(48*f
*(c - c*Sin[e + f*x])^(9/2)) - (a^2*(3*A - 13*B)*Cos[e + f*x])/(64*c^2*f*(c - c*Sin[e + f*x])^(5/2)) + (a^2*(3
*A - 13*B)*Cos[e + f*x])/(256*c^3*f*(c - c*Sin[e + f*x])^(3/2))
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 2680
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] && !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]
Rule 2650
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Rule 2649
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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))^{9/2}} \, 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))^{13/2}} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{1}{16} \left (a^2 (3 A-13 B) c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{11/2}} \, dx\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac{\left (a^2 (3 A-13 B)\right ) \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx}{32 c}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{\left (a^2 (3 A-13 B)\right ) \int \frac{1}{(c-c \sin (e+f x))^{3/2}} \, dx}{128 c^3}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{\left (a^2 (3 A-13 B)\right ) \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx}{512 c^4}\\ &=\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac{\left (a^2 (3 A-13 B)\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,-\frac{c \cos (e+f x)}{\sqrt{c-c \sin (e+f x)}}\right )}{256 c^4 f}\\ &=\frac{a^2 (3 A-13 B) \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{256 \sqrt{2} c^{9/2} f}+\frac{a^2 (A+B) c^2 \cos ^5(e+f x)}{8 f (c-c \sin (e+f x))^{13/2}}+\frac{a^2 (3 A-13 B) \cos ^3(e+f x)}{48 f (c-c \sin (e+f x))^{9/2}}-\frac{a^2 (3 A-13 B) \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac{a^2 (3 A-13 B) \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 2.67365, size = 357, normalized size = 1.61 \[ \frac{a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left ((-24-24 i) \sqrt [4]{-1} (3 A-13 B) \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac{1}{4} (e+f x)\right )+1\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8+2013 A \sin \left (\frac{1}{2} (e+f x)\right )+999 A \sin \left (\frac{3}{2} (e+f x)\right )-69 A \sin \left (\frac{5}{2} (e+f x)\right )+9 A \sin \left (\frac{7}{2} (e+f x)\right )+2013 A \cos \left (\frac{1}{2} (e+f x)\right )-999 A \cos \left (\frac{3}{2} (e+f x)\right )-69 A \cos \left (\frac{5}{2} (e+f x)\right )-9 A \cos \left (\frac{7}{2} (e+f x)\right )+1517 B \sin \left (\frac{1}{2} (e+f x)\right )+791 B \sin \left (\frac{3}{2} (e+f x)\right )-725 B \sin \left (\frac{5}{2} (e+f x)\right )-39 B \sin \left (\frac{7}{2} (e+f x)\right )+1517 B \cos \left (\frac{1}{2} (e+f x)\right )-791 B \cos \left (\frac{3}{2} (e+f x)\right )-725 B \cos \left (\frac{5}{2} (e+f x)\right )+39 B \cos \left (\frac{7}{2} (e+f x)\right )\right )}{6144 f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]
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])^(9/2),x]
[Out]
(a^2*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^2*(2013*A*Cos[(e + f*x)/2] + 1517*B*Cos[(e + f*x
)/2] - 999*A*Cos[(3*(e + f*x))/2] - 791*B*Cos[(3*(e + f*x))/2] - 69*A*Cos[(5*(e + f*x))/2] - 725*B*Cos[(5*(e +
f*x))/2] - 9*A*Cos[(7*(e + f*x))/2] + 39*B*Cos[(7*(e + f*x))/2] - (24 + 24*I)*(-1)^(1/4)*(3*A - 13*B)*ArcTan[
(1/2 + I/2)*(-1)^(1/4)*(1 + Tan[(e + f*x)/4])]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8 + 2013*A*Sin[(e + f*x)/
2] + 1517*B*Sin[(e + f*x)/2] + 999*A*Sin[(3*(e + f*x))/2] + 791*B*Sin[(3*(e + f*x))/2] - 69*A*Sin[(5*(e + f*x)
)/2] - 725*B*Sin[(5*(e + f*x))/2] + 9*A*Sin[(7*(e + f*x))/2] - 39*B*Sin[(7*(e + f*x))/2]))/(6144*f*(Cos[(e + f
*x)/2] + Sin[(e + f*x)/2])^4*(c - c*Sin[e + f*x])^(9/2))
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Maple [B] time = 1.628, size = 440, normalized size = 2. \begin{align*}{\frac{{a}^{2}}{1536\, \left ( -1+\sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) f} \left ( -12\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{5} \left ( 3\,A-13\,B \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+24\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{5} \left ( 3\,A-13\,B \right ) \sin \left ( fx+e \right ) -3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{5} \left ( 3\,A-13\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+24\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{5} \left ( 3\,A-13\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+144\,A\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{9/2}-264\,A \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}{c}^{7/2}-132\,A \left ( c+c\sin \left ( fx+e \right ) \right ) ^{5/2}{c}^{5/2}+18\,A \left ( c+c\sin \left ( fx+e \right ) \right ) ^{7/2}{c}^{3/2}-624\,B\sqrt{c+c\sin \left ( fx+e \right ) }{c}^{9/2}+1144\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{3/2}{c}^{7/2}-452\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{5/2}{c}^{5/2}-78\,B \left ( c+c\sin \left ( fx+e \right ) \right ) ^{7/2}{c}^{3/2}-72\,A\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{5}+312\,B\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c+c\sin \left ( fx+e \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{5} \right ) \sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{-{\frac{19}{2}}}{\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))^(9/2),x)
[Out]
1/1536/c^(19/2)*a^2*(-12*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5*(3*A-13*B)*sin(f*x+e)
*cos(f*x+e)^2+24*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5*(3*A-13*B)*sin(f*x+e)-3*2^(1/
2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5*(3*A-13*B)*cos(f*x+e)^4+24*2^(1/2)*arctanh(1/2*(c+c
*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5*(3*A-13*B)*cos(f*x+e)^2+144*A*(c+c*sin(f*x+e))^(1/2)*c^(9/2)-264*A*(c+
c*sin(f*x+e))^(3/2)*c^(7/2)-132*A*(c+c*sin(f*x+e))^(5/2)*c^(5/2)+18*A*(c+c*sin(f*x+e))^(7/2)*c^(3/2)-624*B*(c+
c*sin(f*x+e))^(1/2)*c^(9/2)+1144*B*(c+c*sin(f*x+e))^(3/2)*c^(7/2)-452*B*(c+c*sin(f*x+e))^(5/2)*c^(5/2)-78*B*(c
+c*sin(f*x+e))^(7/2)*c^(3/2)-72*A*2^(1/2)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5+312*B*2^(1/2
)*arctanh(1/2*(c+c*sin(f*x+e))^(1/2)*2^(1/2)/c^(1/2))*c^5)*(c*(1+sin(f*x+e)))^(1/2)/(-1+sin(f*x+e))^3/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 \frac{{\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{9}{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))^(9/2),x, algorithm="maxima")
[Out]
integrate((B*sin(f*x + e) + A)*(a*sin(f*x + e) + a)^2/(-c*sin(f*x + e) + c)^(9/2), x)
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Fricas [B] time = 1.63159, size = 1670, normalized size = 7.52 \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))^(9/2),x, algorithm="fricas")
[Out]
-1/3072*(3*sqrt(2)*((3*A - 13*B)*a^2*cos(f*x + e)^5 + 5*(3*A - 13*B)*a^2*cos(f*x + e)^4 - 8*(3*A - 13*B)*a^2*c
os(f*x + e)^3 - 20*(3*A - 13*B)*a^2*cos(f*x + e)^2 + 8*(3*A - 13*B)*a^2*cos(f*x + e) + 16*(3*A - 13*B)*a^2 - (
(3*A - 13*B)*a^2*cos(f*x + e)^4 - 4*(3*A - 13*B)*a^2*cos(f*x + e)^3 - 12*(3*A - 13*B)*a^2*cos(f*x + e)^2 + 8*(
3*A - 13*B)*a^2*cos(f*x + e) + 16*(3*A - 13*B)*a^2)*sin(f*x + e))*sqrt(c)*log(-(c*cos(f*x + e)^2 - 2*sqrt(2)*s
qrt(-c*sin(f*x + e) + c)*sqrt(c)*(cos(f*x + e) + sin(f*x + e) + 1) + 3*c*cos(f*x + e) + (c*cos(f*x + e) - 2*c)
*sin(f*x + e) + 2*c)/(cos(f*x + e)^2 + (cos(f*x + e) + 2)*sin(f*x + e) - cos(f*x + e) - 2)) + 4*(3*(3*A - 13*B
)*a^2*cos(f*x + e)^4 + (39*A + 343*B)*a^2*cos(f*x + e)^3 + 2*(129*A + 209*B)*a^2*cos(f*x + e)^2 - 12*(13*A + 2
9*B)*a^2*cos(f*x + e) - 384*(A + B)*a^2 - (3*(3*A - 13*B)*a^2*cos(f*x + e)^3 - 2*(15*A + 191*B)*a^2*cos(f*x +
e)^2 + 12*(19*A + 3*B)*a^2*cos(f*x + e) + 384*(A + B)*a^2)*sin(f*x + e))*sqrt(-c*sin(f*x + e) + c))/(c^5*f*cos
(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e)
+ 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e)
+ 16*c^5*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))**(9/2),x)
[Out]
Timed out
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Giac [B] time = 6.21462, size = 2141, normalized size = 9.64 \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))^(9/2),x, algorithm="giac")
[Out]
1/768*(3*sqrt(2)*(3*A*a^2 - 13*B*a^2)*arctan(-1/2*sqrt(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))/sqrt(-c))/(sqrt(-c)*c^4*sgn(tan(1/2*f*x + 1/2*e) - 1)) + 2*(1527*(sqrt(c)*tan(1/2*f*
x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^15*A*a^2 + 39*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*
f*x + 1/2*e)^2 + c))^15*B*a^2 - 4473*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^14*A*
a^2*sqrt(c) + 2487*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^14*B*a^2*sqrt(c) + 2223
3*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^13*A*a^2*c + 7593*(sqrt(c)*tan(1/2*f*x +
1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^13*B*a^2*c - 23811*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/
2*f*x + 1/2*e)^2 + c))^12*A*a^2*c^(3/2) + 1293*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 +
c))^12*B*a^2*c^(3/2) - 2133*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^11*A*a^2*c^2
+ 1563*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^11*B*a^2*c^2 + 68019*(sqrt(c)*tan(1
/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^10*A*a^2*c^(5/2) - 10589*(sqrt(c)*tan(1/2*f*x + 1/2*e) -
sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^10*B*a^2*c^(5/2) - 25371*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f
*x + 1/2*e)^2 + c))^9*A*a^2*c^3 - 9355*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^9*B
*a^2*c^3 - 71487*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^8*A*a^2*c^(7/2) - 3055*(s
qrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^8*B*a^2*c^(7/2) + 25173*(sqrt(c)*tan(1/2*f*x
+ 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^7*A*a^2*c^4 - 7195*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(
1/2*f*x + 1/2*e)^2 + c))^7*B*a^2*c^4 + 56469*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c
))^6*A*a^2*c^(9/2) + 15909*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^6*B*a^2*c^(9/2)
+ 10971*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^5*A*a^2*c^5 + 2123*(sqrt(c)*tan(1
/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^5*B*a^2*c^5 - 31881*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt
(c*tan(1/2*f*x + 1/2*e)^2 + c))^4*A*a^2*c^(11/2) - 3673*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1
/2*e)^2 + c))^4*B*a^2*c^(11/2) - 17079*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^3*A
*a^2*c^6 + 5913*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^3*B*a^2*c^6 - 7695*(sqrt(c
)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^2*A*a^2*c^(13/2) - 3007*(sqrt(c)*tan(1/2*f*x + 1/
2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))^2*B*a^2*c^(13/2) - 345*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1
/2*f*x + 1/2*e)^2 + c))*A*a^2*c^7 - 41*(sqrt(c)*tan(1/2*f*x + 1/2*e) - sqrt(c*tan(1/2*f*x + 1/2*e)^2 + c))*B*a
^2*c^7 - 117*A*a^2*c^(15/2) - 5*B*a^2*c^(15/2))/(((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)^8*c^4*sgn(tan(1
/2*f*x + 1/2*e) - 1)))/f