3.685 \(\int \frac{(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx\)
Optimal. Leaf size=305 \[ -\frac{\left (a^2 \left (-\left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{7/2}}+\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )+b^2 \left (-\left (-10 c^2 d^2+c^4-6 d^4\right )\right )\right ) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac{\left (5 a c d+b \left (c^2-6 d^2\right )\right ) (b c-a d) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2} \]
[Out]
-(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[
c^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3)
- ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*Cos[e + f*x])/(6*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^2) + ((a^2*
d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(6*d*(c^2 - d^
2)^3*f*(c + d*Sin[e + f*x]))
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Rubi [A] time = 0.563126, antiderivative size = 305, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used =
{2790, 2754, 12, 2660, 618, 204} \[ -\frac{\left (a^2 \left (-\left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{7/2}}+\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )+b^2 \left (-\left (-10 c^2 d^2+c^4-6 d^4\right )\right )\right ) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))}+\frac{(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac{\left (5 a c d+b \left (c^2-6 d^2\right )\right ) (b c-a d) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]
[Out]
-(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[
c^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3)
- ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*Cos[e + f*x])/(6*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^2) + ((a^2*
d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(6*d*(c^2 - d^
2)^3*f*(c + d*Sin[e + f*x]))
Rule 2790
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> -Simp[
((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 - b^2)), x] - Di
st[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*(2*b*c*d - a*(c^2 + d^2)) + (a^2
*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1) + c^2*(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]
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))^2}{(c+d \sin (e+f x))^4} \, dx &=\frac{(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac{\int \frac{3 d \left (\left (a^2+b^2\right ) c-2 a b d\right )+\left (4 a b c d-2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 d \left (c^2-d^2\right )}\\ &=\frac{(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac{(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac{\int \frac{2 d \left (10 a b c d-a^2 \left (3 c^2+2 d^2\right )-b^2 \left (2 c^2+3 d^2\right )\right )-(b c-a d) \left (b c^2+5 a c d-6 b d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 d \left (c^2-d^2\right )^2}\\ &=\frac{(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac{(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}+\frac{\int -\frac{3 d \left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right )}{c+d \sin (e+f x)} \, dx}{6 d \left (c^2-d^2\right )^3}\\ &=\frac{(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac{(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac{\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{2 \left (c^2-d^2\right )^3}\\ &=\frac{(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac{(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac{\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\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 )}{\left (c^2-d^2\right )^3 f}\\ &=\frac{(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac{(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}+\frac{\left (2 \left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\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 )}{\left (c^2-d^2\right )^3 f}\\ &=-\frac{\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac{(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac{(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 1.38977, size = 346, normalized size = 1.13 \[ \frac{\frac{12 \left (a^2 \left (2 c^3+3 c d^2\right )-2 a b d \left (4 c^2+d^2\right )+b^2 c \left (c^2+4 d^2\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+\frac{\cos (e+f x) \left (-6 \left (-a^2 c d^2 \left (9 c^2+d^2\right )-2 a b d \left (-9 c^2 d^2-2 c^4+d^4\right )+b^2 \left (-9 c^3 d^2+c^5-2 c d^4\right )\right ) \sin (e+f x)+d \left (-a^2 d^2 \left (11 c^2+4 d^2\right )+a b \left (4 c^3 d+26 c d^3\right )+b^2 \left (-10 c^2 d^2+c^4-6 d^4\right )\right ) \cos (2 (e+f x))+a^2 c^2 d^3+36 a^2 c^4 d+8 a^2 d^5-44 a b c^3 d^2-24 a b c^5-22 a b c d^4+14 b^2 c^2 d^3+25 b^2 c^4 d+6 b^2 d^5\right )}{\left (c^2-d^2\right )^3 (c+d \sin (e+f x))^3}}{12 f} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]
[Out]
((12*(-2*a*b*d*(4*c^2 + d^2) + b^2*c*(c^2 + 4*d^2) + a^2*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sq
rt[c^2 - d^2]])/(c^2 - d^2)^(7/2) + (Cos[e + f*x]*(-24*a*b*c^5 + 36*a^2*c^4*d + 25*b^2*c^4*d - 44*a*b*c^3*d^2
+ a^2*c^2*d^3 + 14*b^2*c^2*d^3 - 22*a*b*c*d^4 + 8*a^2*d^5 + 6*b^2*d^5 + d*(-(a^2*d^2*(11*c^2 + 4*d^2)) + a*b*(
4*c^3*d + 26*c*d^3) + b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[2*(e + f*x)] - 6*(-(a^2*c*d^2*(9*c^2 + d^2)) - 2*a*b
*d*(-2*c^4 - 9*c^2*d^2 + d^4) + b^2*(c^5 - 9*c^3*d^2 - 2*c*d^4))*Sin[e + f*x]))/((c^2 - d^2)^3*(c + d*Sin[e +
f*x])^3))/(12*f)
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Maple [B] time = 0.105, size = 4818, normalized size = 15.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x)
[Out]
8/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*b^2
*d^5+1/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^5/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2
*e)^5*b^2-1/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^5/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*
x+1/2*e)*b^2+6/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*a^2*c^4*d-5
/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*a^2*c^2*d^3-4/f/(c*tan(
1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*a*b*c^5+13/3/f/(c*tan(1/2*f*x+1/2*e
)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*b^2*c^4*d+2/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/
2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*a^2*d^5-56/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+
c)^3*c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*a*b*d^4+4/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/
2*e)*d+c)^3/c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*a*b*d^6-16/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/
2*f*x+1/2*e)*d+c)^3*c^4/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*a*b*d-38/f/(c*tan(1/2*f*x+1/2*e)^2+2*
tan(1/2*f*x+1/2*e)*d+c)^3*c^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*a*b*d^3-8/f/(c^6-3*c^4*d^2+3*c^
2*d^4-d^6)/(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*b*c^2*d-8/f/(c*tan(1/2*f
*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^4/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5*a*b*d-2/f/(c*ta
n(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5*a*b*d^3-
28/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*c^3*tan(1/2*f*x+1/2*e)^
4*a*b*d^2-22/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*c*tan(1/2*f*x
+1/2*e)^4*a*b*d^4+4/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/c*tan(
1/2*f*x+1/2*e)^4*a*b*d^6-24/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^4*d/(c^6-3*c^4*d^2+3*c^2*d
^4-d^6)*tan(1/2*f*x+1/2*e)^3*a*b-56/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^2*d^3/(c^6-3*c^4*d
^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3*a*b+8/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c^2*d^7/(
c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3*a*b-40/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3
*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*a*b*d^2+36/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2
*e)*d+c)^3*c^3*d^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3*a^2+14/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1
/2*f*x+1/2*e)*d+c)^3*c*d^4/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3*a^2-8/3/f/(c*tan(1/2*f*x+1/2*e)^
2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c*d^6/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3*a^2+8/3/f/(c*tan(1/2*f*
x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c^3*d^8/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3*a^2+26/f/(c*
tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^3*d^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3*b^
2+64/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c*d^4/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1
/2*e)^3*b^2+2/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*a*b*c*d^4+
2/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5*a
^2*d^6+4/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1
/2*e)^5*b^2*d^2+6/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*c^4*tan(
1/2*f*x+1/2*e)^4*a^2*d+27/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*
c^2*tan(1/2*f*x+1/2*e)^4*a^2*d^3+4/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*
d^4-d^6)/c^2*tan(1/2*f*x+1/2*e)^4*a^2*d^7-4/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d
^2+3*c^2*d^4-d^6)*c^5*tan(1/2*f*x+1/2*e)^4*a*b+2/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-
3*c^4*d^2+3*c^2*d^4-d^6)*b^2*c^2*d^3-12/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3
*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^4*a^2*d^5-6/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*
d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*a^2*d^5+2/f/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/(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^2*c^3+1/f/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/(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^2*c^3+4/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c
)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*a*b*d^5+8/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*
d+c)^3*c^4/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*b^2*d+34/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x
+1/2*e)*d+c)^3*c^2/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*b^2*d^3+27/f/(c*tan(1/2*f*x+1/2*e)^2+2*t
an(1/2*f*x+1/2*e)*d+c)^3*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*a^2*d^2-4/f/(c*tan(1/2*f*x+1/2*e
)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*a^2*d^4+2/f/(c*tan(1/2*f*x+
1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*a^2*d^6+22/f/(c*tan(1/
2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*b^2*d^2+4/f/(c
*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)*b^2*d^4-2
0/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*a*b*c^3*d^2+9/f/(c*tan
(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5*a^2*d^2-6
/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5*a^
2*d^4-68/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*d^5/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x
+1/2*e)^3*a*b+8/3/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c*d^6/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*ta
n(1/2*f*x+1/2*e)^3*b^2+12/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^4/(c^6-3*c^4*d^2+3*c^2*d^4-d
^6)*tan(1/2*f*x+1/2*e)^2*a^2*d+40/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^2/(c^6-3*c^4*d^2+3*c
^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*a^2*d^3+4/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c^2/(c^6-3*c^
4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*a^2*d^7-8/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c^5/
(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2*a*b+3/f/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/(c^2-d^2)^(1/2)*arcta
n(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*a^2*c*d^2-2/f/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/(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*b*d^3+4/f/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/(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^2*c*d^2+5/f/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/
2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*c^4*tan(1/2*f*x+1/2*e)^4*b^2*d+20/f/(c*tan(1/2*f*x+1/2*e)^2+
2*tan(1/2*f*x+1/2*e)*d+c)^3/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*c^2*tan(1/2*f*x+1/2*e)^4*b^2*d^3
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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))^2/(c+d*sin(f*x+e))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.5338, size = 3687, normalized size = 12.09 \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))^2/(c+d*sin(f*x+e))^4,x, algorithm="fricas")
[Out]
[-1/12*(2*(b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3*d^4 - 26*a*b*c*d^6 - 11*(a^2 + b^2)*c^4*d^3 + (7*a^2 + 4*b^2
)*c^2*d^5 + 2*(2*a^2 + 3*b^2)*d^7)*cos(f*x + e)^3 - 6*(b^2*c^7 + 4*a*b*c^6*d + 14*a*b*c^4*d^3 - 20*a*b*c^2*d^5
+ 2*a*b*d^7 - (9*a^2 + 10*b^2)*c^5*d^2 + (8*a^2 + 7*b^2)*c^3*d^4 + (a^2 + 2*b^2)*c*d^6)*cos(f*x + e)*sin(f*x
+ e) + 3*(8*a*b*c^5*d + 26*a*b*c^3*d^3 + 6*a*b*c*d^5 - (2*a^2 + b^2)*c^6 - (9*a^2 + 7*b^2)*c^4*d^2 - 3*(3*a^2
+ 4*b^2)*c^2*d^4 - 3*(8*a*b*c^3*d^3 + 2*a*b*c*d^5 - (2*a^2 + b^2)*c^4*d^2 - (3*a^2 + 4*b^2)*c^2*d^4)*cos(f*x +
e)^2 + (24*a*b*c^4*d^2 + 14*a*b*c^2*d^4 + 2*a*b*d^6 - 3*(2*a^2 + b^2)*c^5*d - (11*a^2 + 13*b^2)*c^3*d^3 - (3*
a^2 + 4*b^2)*c*d^5 - (8*a*b*c^2*d^4 + 2*a*b*d^6 - (2*a^2 + b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5)*cos(f*x + e)^
2)*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*co
s(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)) - 12*(2*a*b*c^7 + 2*a*b*c^5*d^2 + 2*a^2*c^4*d^3 + b^2*c^2*d^5 - 4*a*b*c*d^6 - (3*a^2 + 2*b^2)*c^6*d + (a^
2 + b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11
- c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9
+ d^11)*f*cos(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e))
, -1/6*((b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3*d^4 - 26*a*b*c*d^6 - 11*(a^2 + b^2)*c^4*d^3 + (7*a^2 + 4*b^2)*
c^2*d^5 + 2*(2*a^2 + 3*b^2)*d^7)*cos(f*x + e)^3 - 3*(b^2*c^7 + 4*a*b*c^6*d + 14*a*b*c^4*d^3 - 20*a*b*c^2*d^5 +
2*a*b*d^7 - (9*a^2 + 10*b^2)*c^5*d^2 + (8*a^2 + 7*b^2)*c^3*d^4 + (a^2 + 2*b^2)*c*d^6)*cos(f*x + e)*sin(f*x +
e) + 3*(8*a*b*c^5*d + 26*a*b*c^3*d^3 + 6*a*b*c*d^5 - (2*a^2 + b^2)*c^6 - (9*a^2 + 7*b^2)*c^4*d^2 - 3*(3*a^2 +
4*b^2)*c^2*d^4 - 3*(8*a*b*c^3*d^3 + 2*a*b*c*d^5 - (2*a^2 + b^2)*c^4*d^2 - (3*a^2 + 4*b^2)*c^2*d^4)*cos(f*x + e
)^2 + (24*a*b*c^4*d^2 + 14*a*b*c^2*d^4 + 2*a*b*d^6 - 3*(2*a^2 + b^2)*c^5*d - (11*a^2 + 13*b^2)*c^3*d^3 - (3*a^
2 + 4*b^2)*c*d^5 - (8*a*b*c^2*d^4 + 2*a*b*d^6 - (2*a^2 + b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5)*cos(f*x + e)^2)
*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))) - 6*(2*a*b*c^7 + 2
*a*b*c^5*d^2 + 2*a^2*c^4*d^3 + b^2*c^2*d^5 - 4*a*b*c*d^6 - (3*a^2 + 2*b^2)*c^6*d + (a^2 + b^2)*d^7)*cos(f*x +
e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 +
14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos(f*x + e)^
2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*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))**2/(c+d*sin(f*x+e))**4,x)
[Out]
Timed out
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Giac [B] time = 1.49687, size = 1777, normalized size = 5.83 \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))^2/(c+d*sin(f*x+e))^4,x, algorithm="giac")
[Out]
1/3*(3*(2*a^2*c^3 + b^2*c^3 - 8*a*b*c^2*d + 3*a^2*c*d^2 + 4*b^2*c*d^2 - 2*a*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)))/((c^6 - 3*c^4*d^2 + 3*c^2*d^4 - d^6)*sqr
t(c^2 - d^2)) + (3*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 24*a*b*c^7*d*tan(1/2*f*x + 1/2*e)^5 + 27*a^2*c^6*d^2*tan(1
/2*f*x + 1/2*e)^5 + 12*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 - 6*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 18*a^2*c^4*
d^4*tan(1/2*f*x + 1/2*e)^5 + 6*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 12*a*b*c^8*tan(1/2*f*x + 1/2*e)^4 + 18*a^2
*c^7*d*tan(1/2*f*x + 1/2*e)^4 + 15*b^2*c^7*d*tan(1/2*f*x + 1/2*e)^4 - 84*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^4 +
81*a^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 + 60*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 - 66*a*b*c^4*d^4*tan(1/2*f*x + 1
/2*e)^4 - 36*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^4 + 12*a*b*c^2*d^6*tan(1/2*f*x + 1/2*e)^4 + 12*a^2*c*d^7*tan(1/2
*f*x + 1/2*e)^4 - 72*a*b*c^7*d*tan(1/2*f*x + 1/2*e)^3 + 108*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 + 78*b^2*c^6*d^
2*tan(1/2*f*x + 1/2*e)^3 - 168*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 + 42*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 64
*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 68*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 8*a^2*c^2*d^6*tan(1/2*f*x + 1/2*
e)^3 + 8*b^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 8*a*b*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 8*a^2*d^8*tan(1/2*f*x + 1/2
*e)^3 - 24*a*b*c^8*tan(1/2*f*x + 1/2*e)^2 + 36*a^2*c^7*d*tan(1/2*f*x + 1/2*e)^2 + 24*b^2*c^7*d*tan(1/2*f*x + 1
/2*e)^2 - 120*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^2 + 120*a^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^2 + 102*b^2*c^5*d^3*ta
n(1/2*f*x + 1/2*e)^2 - 168*a*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 - 18*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 24*b^2
*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 12*a*b*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 + 12*a^2*c*d^7*tan(1/2*f*x + 1/2*e)^2
- 3*b^2*c^8*tan(1/2*f*x + 1/2*e) - 48*a*b*c^7*d*tan(1/2*f*x + 1/2*e) + 81*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e) + 6
6*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e) - 114*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e) - 12*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e
) + 12*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e) + 12*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e) + 6*a^2*c^2*d^6*tan(1/2*f*x + 1/
2*e) - 12*a*b*c^8 + 18*a^2*c^7*d + 13*b^2*c^7*d - 20*a*b*c^6*d^2 - 5*a^2*c^5*d^3 + 2*b^2*c^5*d^3 + 2*a*b*c^4*d
^4 + 2*a^2*c^3*d^5)/((c^9 - 3*c^7*d^2 + 3*c^5*d^4 - c^3*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2
*e) + c)^3))/f