3.271 \(\int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx\)
Optimal. Leaf size=283 \[ -\frac{\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (4 c^2 d+2 c^3+7 c d^2+2 d^3\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{a f (c-d) \left (c^2-d^2\right )^{5/2}}-\frac{d \left (2 A c^2+9 A c d+4 A d^2-5 B c^2-6 B c d-4 B d^2\right ) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac{d (2 A c+3 A d-3 B c-2 B d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2} \]
[Out]
-(((3*A*d*(2*c^2 + 2*c*d + d^2) - B*(2*c^3 + 4*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[
c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(5/2)*f)) - (d*(2*A*c - 3*B*c + 3*A*d - 2*B*d)*Cos[e + f*x])/(2*a*(c - d)^
2*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*
x])^2) - (d*(2*A*c^2 - 5*B*c^2 + 9*A*c*d - 6*B*c*d + 4*A*d^2 - 4*B*d^2)*Cos[e + f*x])/(2*a*(c - d)^3*(c + d)^2
*f*(c + d*Sin[e + f*x]))
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Rubi [A] time = 0.550317, antiderivative size = 283, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used =
{2978, 2754, 12, 2660, 618, 204} \[ -\frac{\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (4 c^2 d+2 c^3+7 c d^2+2 d^3\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{a f (c-d) \left (c^2-d^2\right )^{5/2}}-\frac{d \left (2 A c^2+9 A c d+4 A d^2-5 B c^2-6 B c d-4 B d^2\right ) \cos (e+f x)}{2 a f (c-d)^3 (c+d)^2 (c+d \sin (e+f x))}-\frac{d (2 A c+3 A d-3 B c-2 B d) \cos (e+f x)}{2 a f (c-d)^2 (c+d) (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{f (c-d) (a \sin (e+f x)+a) (c+d \sin (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3),x]
[Out]
-(((3*A*d*(2*c^2 + 2*c*d + d^2) - B*(2*c^3 + 4*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[
c^2 - d^2]])/(a*(c - d)*(c^2 - d^2)^(5/2)*f)) - (d*(2*A*c - 3*B*c + 3*A*d - 2*B*d)*Cos[e + f*x])/(2*a*(c - d)^
2*(c + d)*f*(c + d*Sin[e + f*x])^2) - ((A - B)*Cos[e + f*x])/((c - d)*f*(a + a*Sin[e + f*x])*(c + d*Sin[e + f*
x])^2) - (d*(2*A*c^2 - 5*B*c^2 + 9*A*c*d - 6*B*c*d + 4*A*d^2 - 4*B*d^2)*Cos[e + f*x])/(2*a*(c - d)^3*(c + d)^2
*f*(c + d*Sin[e + f*x]))
Rule 2978
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*
x])^(n + 1))/(a*f*(2*m + 1)*(b*c - a*d)), x] + Dist[1/(a*(2*m + 1)*(b*c - a*d)), Int[(a + b*Sin[e + f*x])^(m +
1)*(c + d*Sin[e + f*x])^n*Simp[B*(a*c*m + b*d*(n + 1)) + A*(b*c*(m + 1) - a*d*(2*m + n + 2)) + d*(A*b - a*B)*
(m + n + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2
- b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -2^(-1)] && !GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c,
0])
Rule 2754
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 2660
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx &=-\frac{(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac{\int \frac{a (3 A d-B (c+2 d))-2 a (A-B) d \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{a^2 (c-d)}\\ &=-\frac{d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac{\int \frac{-2 a \left (2 (A-B) d^2+c (3 A d-B (c+2 d))\right )+a d (2 A c-3 B c+3 A d-2 B d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 a^2 (c-d)^2 (c+d)}\\ &=-\frac{d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\int \frac{a \left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right )}{c+d \sin (e+f x)} \, dx}{2 a^2 (c-d)^3 (c+d)^2}\\ &=-\frac{d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{2 a (c-d)^3 (c+d)^2}\\ &=-\frac{d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}-\frac{\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac{1}{2} (e+f x)\right )\right )}{a (c-d)^3 (c+d)^2 f}\\ &=-\frac{d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}+\frac{\left (2 \left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac{1}{2} (e+f x)\right )\right )}{a (c-d)^3 (c+d)^2 f}\\ &=-\frac{\left (3 A d \left (2 c^2+2 c d+d^2\right )-B \left (2 c^3+4 c^2 d+7 c d^2+2 d^3\right )\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{a (c-d)^3 (c+d)^2 \sqrt{c^2-d^2} f}-\frac{d (2 A c-3 B c+3 A d-2 B d) \cos (e+f x)}{2 a (c-d)^2 (c+d) f (c+d \sin (e+f x))^2}-\frac{(A-B) \cos (e+f x)}{(c-d) f (a+a \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac{d \left (2 A c^2-5 B c^2+9 A c d-6 B c d+4 A d^2-4 B d^2\right ) \cos (e+f x)}{2 a (c-d)^3 (c+d)^2 f (c+d \sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 1.38614, size = 313, normalized size = 1.11 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (\frac{d \left (B \left (3 c^2+2 c d+2 d^2\right )-A d (5 c+2 d)\right ) \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}{(c+d)^2 (c+d \sin (e+f x))}+\frac{2 \left (B \left (4 c^2 d+2 c^3+7 c d^2+2 d^3\right )-3 A d \left (2 c^2+2 c d+d^2\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{(c+d)^2 \sqrt{c^2-d^2}}+\frac{d (c-d) (B c-A d) \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}{(c+d) (c+d \sin (e+f x))^2}+4 (A-B) \sin \left (\frac{1}{2} (e+f x)\right )\right )}{2 a f (c-d)^3 (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^3),x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(4*(A - B)*Sin[(e + f*x)/2] + (2*(-3*A*d*(2*c^2 + 2*c*d + d^2) + B*(2*c
^3 + 4*c^2*d + 7*c*d^2 + 2*d^3))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2]))/((c + d)^2*Sqrt[c^2 - d^2]) + ((c - d)*d*(B*c - A*d)*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)
/2]))/((c + d)*(c + d*Sin[e + f*x])^2) + (d*(-(A*d*(5*c + 2*d)) + B*(3*c^2 + 2*c*d + 2*d^2))*Cos[e + f*x]*(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2]))/((c + d)^2*(c + d*Sin[e + f*x]))))/(2*a*(c - d)^3*f*(1 + Sin[e + f*x]))
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Maple [B] time = 0.165, size = 2482, normalized size = 8.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^3,x)
[Out]
-2/a/f/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)*A-2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^3/
(c^2+2*c*d+d^2)*c*tan(1/2*f*x+1/2*e)^2*A-4/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^5
/(c^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)^2*A-6/a/f/(c-d)^3/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/
2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*A*c^2*d-7/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d
^3*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*A+2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*
d^5/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*A+5/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2
*d^2*c^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*B+2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c
)^2*d^3*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*B-6/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+
c)^2*d^2/(c^2+2*c*d+d^2)*c^2*tan(1/2*f*x+1/2*e)^2*A-11/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e
)*d+c)^2*d^4/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^2*A-3/a/f/(c-d)^3/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(
2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*A*d^3-6/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*
d+c)^2*d^2/(c^2+2*c*d+d^2)*A*c^2+1/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^3/(c^2+2*
c*d+d^2)*B*c+4/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^4/(c^2+2*c*d+d^2)*tan(1/2*f*x
+1/2*e)^2*B+2/a/f/(c-d)^3/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1
/2))*B*c^3+2/a/f/(c-d)^3/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/
2))*B*d^3+2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^2/(c^2+2*c*d+d^2)*B*c^2-6/a/f/(c
-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^4/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*A+4/a/f/(c-d)
^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^4/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*B-2/a/f/(c-d)^3/
(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^4/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3*A-2/a/f/(c-d)^3/(
c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^3/(c^2+2*c*d+d^2)*A*c+4/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)
^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d/(c^2+2*c*d+d^2)*B*c^3+1/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2
*e)*d+c)^2*d^4/(c^2+2*c*d+d^2)*A+4/a/f/(c-d)^3/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2
*e)+2*d)/(c^2-d^2)^(1/2))*B*c^2*d+7/a/f/(c-d)^3/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/
2*e)+2*d)/(c^2-d^2)^(1/2))*B*c*d^2-6/a/f/(c-d)^3/(c^2+2*c*d+d^2)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1
/2*e)+2*d)/(c^2-d^2)^(1/2))*A*c*d^2+4/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d/(c^2+2
*c*d+d^2)*c^3*tan(1/2*f*x+1/2*e)^2*B+2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^6/(c^
2+2*c*d+d^2)/c^2*tan(1/2*f*x+1/2*e)^2*A+2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d^2/
(c^2+2*c*d+d^2)*c^2*tan(1/2*f*x+1/2*e)^2*B+9/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*d
^3/(c^2+2*c*d+d^2)*c*tan(1/2*f*x+1/2*e)^2*B+2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2*
d^5/(c^2+2*c*d+d^2)/c*tan(1/2*f*x+1/2*e)^2*B-17/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^
2*d^3*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*A+2/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2
*d^5/c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*A+11/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^2
*d^2*c^2/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*B+6/a/f/(c-d)^3/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^
2*d^3*c/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)*B+2/a/f/(c-d)^3/(tan(1/2*f*x+1/2*e)+1)*B
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.19584, size = 7112, normalized size = 25.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="fricas")
[Out]
[1/4*(4*(A - B)*c^6 - 12*(A - B)*c^4*d^2 + 12*(A - B)*c^2*d^4 - 4*(A - B)*d^6 - 2*((2*A - 5*B)*c^4*d^2 + 3*(3*
A - 2*B)*c^3*d^3 + (2*A + B)*c^2*d^4 - 3*(3*A - 2*B)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^3 + 2*(4*(A - 2*B)*c^
5*d + 4*(3*A - 2*B)*c^4*d^2 - (2*A - 7*B)*c^3*d^3 - 5*(3*A - 2*B)*c^2*d^4 - (2*A - B)*c*d^5 + (3*A - 2*B)*d^6)
*cos(f*x + e)^2 - (2*B*c^5 - 2*(3*A - 4*B)*c^4*d - (18*A - 17*B)*c^3*d^2 - (21*A - 20*B)*c^2*d^3 - (12*A - 11*
B)*c*d^4 - (3*A - 2*B)*d^5 - (2*B*c^3*d^2 - 2*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f
*x + e)^3 - (4*B*c^4*d - 2*(6*A - 5*B)*c^3*d^2 - 18*(A - B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5)*c
os(f*x + e)^2 + (2*B*c^5 - 2*(3*A - 2*B)*c^4*d - 3*(2*A - 3*B)*c^3*d^2 - 3*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c
*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e) + (2*B*c^5 - 2*(3*A - 4*B)*c^4*d - (18*A - 17*B)*c^3*d^2 - (21*A - 20*B)*
c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5 - (2*B*c^3*d^2 - 2*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (
3*A - 2*B)*d^5)*cos(f*x + e)^2 + 2*(2*B*c^4*d - 2*(3*A - 2*B)*c^3*d^2 - (6*A - 7*B)*c^2*d^3 - (3*A - 2*B)*c*d^
4)*cos(f*x + e))*sin(f*x + e))*sqrt(-c^2 + d^2)*log(-((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2
- d^2 - 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x
+ e) - c^2 - d^2)) + 2*(2*(A - B)*c^6 + 4*(A - 2*B)*c^5*d + (8*A - 7*B)*c^4*d^2 + (7*A + B)*c^3*d^3 - (7*A -
5*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (3*A - 4*B)*d^6)*cos(f*x + e) - 2*(2*(A - B)*c^6 - 6*(A - B)*c^4*d^2 + 6*(
A - B)*c^2*d^4 - 2*(A - B)*d^6 - ((2*A - 5*B)*c^4*d^2 + 3*(3*A - 2*B)*c^3*d^3 + (2*A + B)*c^2*d^4 - 3*(3*A - 2
*B)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^2 - (4*(A - 2*B)*c^5*d + (14*A - 13*B)*c^4*d^2 + (7*A + B)*c^3*d^3 - (
13*A - 11*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (A - 2*B)*d^6)*cos(f*x + e))*sin(f*x + e))/((a*c^7*d^2 - a*c^6*d^3
- 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^3 + (2*a*c^8*d - a*
c^7*d^2 - 7*a*c^6*d^3 + 3*a*c^5*d^4 + 9*a*c^4*d^5 - 3*a*c^3*d^6 - 5*a*c^2*d^7 + a*c*d^8 + a*d^9)*f*cos(f*x + e
)^2 - (a*c^9 - a*c^8*d - 2*a*c^7*d^2 + 2*a*c^6*d^3 + 2*a*c^3*d^6 - 2*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x +
e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*
c*d^8 + a*d^9)*f + ((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 +
a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^8*d - a*c^7*d^2 - 3*a*c^6*d^3 + 3*a*c^5*d^4 + 3*a*c^4*d^5 - 3*a*c^3*d^6 - a*
c^2*d^7 + a*c*d^8)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 -
4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f)*sin(f*x + e)), 1/2*(2*(A - B)*c^6 - 6*(A - B)*c^4*d^2 + 6*(A
- B)*c^2*d^4 - 2*(A - B)*d^6 - ((2*A - 5*B)*c^4*d^2 + 3*(3*A - 2*B)*c^3*d^3 + (2*A + B)*c^2*d^4 - 3*(3*A - 2*B
)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^3 + (4*(A - 2*B)*c^5*d + 4*(3*A - 2*B)*c^4*d^2 - (2*A - 7*B)*c^3*d^3 - 5
*(3*A - 2*B)*c^2*d^4 - (2*A - B)*c*d^5 + (3*A - 2*B)*d^6)*cos(f*x + e)^2 + (2*B*c^5 - 2*(3*A - 4*B)*c^4*d - (1
8*A - 17*B)*c^3*d^2 - (21*A - 20*B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5 - (2*B*c^3*d^2 - 2*(3*A -
2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e)^3 - (4*B*c^4*d - 2*(6*A - 5*B)*c^3*d^2 - 18*(
A - B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e)^2 + (2*B*c^5 - 2*(3*A - 2*B)*c^4*d - 3*(2
*A - 3*B)*c^3*d^2 - 3*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e) + (2*B*c^5 - 2*(
3*A - 4*B)*c^4*d - (18*A - 17*B)*c^3*d^2 - (21*A - 20*B)*c^2*d^3 - (12*A - 11*B)*c*d^4 - (3*A - 2*B)*d^5 - (2*
B*c^3*d^2 - 2*(3*A - 2*B)*c^2*d^3 - (6*A - 7*B)*c*d^4 - (3*A - 2*B)*d^5)*cos(f*x + e)^2 + 2*(2*B*c^4*d - 2*(3*
A - 2*B)*c^3*d^2 - (6*A - 7*B)*c^2*d^3 - (3*A - 2*B)*c*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(c^2 - d^2)*arctan
(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) + (2*(A - B)*c^6 + 4*(A - 2*B)*c^5*d + (8*A - 7*B)*c^4*
d^2 + (7*A + B)*c^3*d^3 - (7*A - 5*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (3*A - 4*B)*d^6)*cos(f*x + e) - (2*(A - B
)*c^6 - 6*(A - B)*c^4*d^2 + 6*(A - B)*c^2*d^4 - 2*(A - B)*d^6 - ((2*A - 5*B)*c^4*d^2 + 3*(3*A - 2*B)*c^3*d^3 +
(2*A + B)*c^2*d^4 - 3*(3*A - 2*B)*c*d^5 - 4*(A - B)*d^6)*cos(f*x + e)^2 - (4*(A - 2*B)*c^5*d + (14*A - 13*B)*
c^4*d^2 + (7*A + B)*c^3*d^3 - (13*A - 11*B)*c^2*d^4 - (11*A - 7*B)*c*d^5 - (A - 2*B)*d^6)*cos(f*x + e))*sin(f*
x + e))/((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*c
os(f*x + e)^3 + (2*a*c^8*d - a*c^7*d^2 - 7*a*c^6*d^3 + 3*a*c^5*d^4 + 9*a*c^4*d^5 - 3*a*c^3*d^6 - 5*a*c^2*d^7 +
a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - (a*c^9 - a*c^8*d - 2*a*c^7*d^2 + 2*a*c^6*d^3 + 2*a*c^3*d^6 - 2*a*c^2*d^7
- a*c*d^8 + a*d^9)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 -
4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*f + ((a*c^7*d^2 - a*c^6*d^3 - 3*a*c^5*d^4 + 3*a*c^4*d^5 + 3*a*c^
3*d^6 - 3*a*c^2*d^7 - a*c*d^8 + a*d^9)*f*cos(f*x + e)^2 - 2*(a*c^8*d - a*c^7*d^2 - 3*a*c^6*d^3 + 3*a*c^5*d^4 +
3*a*c^4*d^5 - 3*a*c^3*d^6 - a*c^2*d^7 + a*c*d^8)*f*cos(f*x + e) - (a*c^9 + a*c^8*d - 4*a*c^7*d^2 - 4*a*c^6*d^
3 + 6*a*c^5*d^4 + 6*a*c^4*d^5 - 4*a*c^3*d^6 - 4*a*c^2*d^7 + a*c*d^8 + a*d^9)*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+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))**3,x)
[Out]
Timed out
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Giac [B] time = 1.33618, size = 1017, normalized size = 3.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c+d*sin(f*x+e))^3,x, algorithm="giac")
[Out]
((2*B*c^3 - 6*A*c^2*d + 4*B*c^2*d - 6*A*c*d^2 + 7*B*c*d^2 - 3*A*d^3 + 2*B*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/
2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((a*c^5 - a*c^4*d - 2*a*c^3*d^2 + 2*a*c^2*d^
3 + a*c*d^4 - a*d^5)*sqrt(c^2 - d^2)) - 2*(A - B)/((a*c^3 - 3*a*c^2*d + 3*a*c*d^2 - a*d^3)*(tan(1/2*f*x + 1/2*
e) + 1)) + (5*B*c^4*d^2*tan(1/2*f*x + 1/2*e)^3 - 7*A*c^3*d^3*tan(1/2*f*x + 1/2*e)^3 + 2*B*c^3*d^3*tan(1/2*f*x
+ 1/2*e)^3 - 2*A*c^2*d^4*tan(1/2*f*x + 1/2*e)^3 + 2*A*c*d^5*tan(1/2*f*x + 1/2*e)^3 + 4*B*c^5*d*tan(1/2*f*x + 1
/2*e)^2 - 6*A*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 + 2*B*c^4*d^2*tan(1/2*f*x + 1/2*e)^2 - 2*A*c^3*d^3*tan(1/2*f*x +
1/2*e)^2 + 9*B*c^3*d^3*tan(1/2*f*x + 1/2*e)^2 - 11*A*c^2*d^4*tan(1/2*f*x + 1/2*e)^2 + 4*B*c^2*d^4*tan(1/2*f*x
+ 1/2*e)^2 - 4*A*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 2*B*c*d^5*tan(1/2*f*x + 1/2*e)^2 + 2*A*d^6*tan(1/2*f*x + 1/2*e
)^2 + 11*B*c^4*d^2*tan(1/2*f*x + 1/2*e) - 17*A*c^3*d^3*tan(1/2*f*x + 1/2*e) + 6*B*c^3*d^3*tan(1/2*f*x + 1/2*e)
- 6*A*c^2*d^4*tan(1/2*f*x + 1/2*e) + 4*B*c^2*d^4*tan(1/2*f*x + 1/2*e) + 2*A*c*d^5*tan(1/2*f*x + 1/2*e) + 4*B*
c^5*d - 6*A*c^4*d^2 + 2*B*c^4*d^2 - 2*A*c^3*d^3 + B*c^3*d^3 + A*c^2*d^4)/((a*c^7 - a*c^6*d - 2*a*c^5*d^2 + 2*a
*c^4*d^3 + a*c^3*d^4 - a*c^2*d^5)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2))/f