3.443 \(\int \frac{(a+a \sin (e+f x))^2}{(c+d \sin (e+f x))^5} \, dx\)
Optimal. Leaf size=286 \[ \frac{a^2 \left (12 c^2-16 c d+7 d^2\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{4 f (c-d)^2 (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^2 \left (16 c^2 d+2 c^3-59 c d^2+32 d^3\right ) \cos (e+f x)}{24 d f (c-d)^2 (c+d)^4 (c+d \sin (e+f x))}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 d f (c-d) (c+d)^3 (c+d \sin (e+f x))^2}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{a^2 (c-d) \cos (e+f x)}{4 d f (c+d) (c+d \sin (e+f x))^4} \]
[Out]
(a^2*(12*c^2 - 16*c*d + 7*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(4*(c - d)^2*(c + d)^4*Sqrt[c
^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(4*d*(c + d)*f*(c + d*Sin[e + f*x])^4) - (a^2*(c + 8*d)*Cos[e + f*x]
)/(12*d*(c + d)^2*f*(c + d*Sin[e + f*x])^3) - (a^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x])/(24*(c - d)*d*(c +
d)^3*f*(c + d*Sin[e + f*x])^2) - (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*Cos[e + f*x])/(24*(c - d)^2*d*(c
+ d)^4*f*(c + d*Sin[e + f*x]))
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Rubi [A] time = 0.506972, antiderivative size = 286, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used =
{2762, 2754, 12, 2660, 618, 204} \[ \frac{a^2 \left (12 c^2-16 c d+7 d^2\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{4 f (c-d)^2 (c+d)^4 \sqrt{c^2-d^2}}-\frac{a^2 \left (16 c^2 d+2 c^3-59 c d^2+32 d^3\right ) \cos (e+f x)}{24 d f (c-d)^2 (c+d)^4 (c+d \sin (e+f x))}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 d f (c-d) (c+d)^3 (c+d \sin (e+f x))^2}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d f (c+d)^2 (c+d \sin (e+f x))^3}+\frac{a^2 (c-d) \cos (e+f x)}{4 d f (c+d) (c+d \sin (e+f x))^4} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^5,x]
[Out]
(a^2*(12*c^2 - 16*c*d + 7*d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(4*(c - d)^2*(c + d)^4*Sqrt[c
^2 - d^2]*f) + (a^2*(c - d)*Cos[e + f*x])/(4*d*(c + d)*f*(c + d*Sin[e + f*x])^4) - (a^2*(c + 8*d)*Cos[e + f*x]
)/(12*d*(c + d)^2*f*(c + d*Sin[e + f*x])^3) - (a^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x])/(24*(c - d)*d*(c +
d)^3*f*(c + d*Sin[e + f*x])^2) - (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*Cos[e + f*x])/(24*(c - d)^2*d*(c
+ d)^4*f*(c + d*Sin[e + f*x]))
Rule 2762
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && 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+a \sin (e+f x))^2}{(c+d \sin (e+f x))^5} \, dx &=\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a \int \frac{-8 a d-a (c+7 d) \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx}{4 d (c+d)}\\ &=\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d (c+d)^2 f (c+d \sin (e+f x))^3}+\frac{a \int \frac{21 a (c-d) d+2 a (c-d) (c+8 d) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{12 (c-d) d (c+d)^2}\\ &=\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d (c+d)^2 f (c+d \sin (e+f x))^3}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sin (e+f x))^2}-\frac{a \int \frac{-2 a (19 c-16 d) (c-d) d+a (c-d) \left (21 d^2-2 c (c+8 d)\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{24 (c-d)^2 d (c+d)^3}\\ &=\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d (c+d)^2 f (c+d \sin (e+f x))^3}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sin (e+f x))^2}-\frac{a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \cos (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sin (e+f x))}+\frac{a \int \frac{3 a (c-d) d \left (12 c^2-16 c d+7 d^2\right )}{c+d \sin (e+f x)} \, dx}{24 (c-d)^3 d (c+d)^4}\\ &=\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d (c+d)^2 f (c+d \sin (e+f x))^3}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sin (e+f x))^2}-\frac{a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \cos (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sin (e+f x))}+\frac{\left (a^2 \left (12 c^2-16 c d+7 d^2\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{8 (c-d)^2 (c+d)^4}\\ &=\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d (c+d)^2 f (c+d \sin (e+f x))^3}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sin (e+f x))^2}-\frac{a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \cos (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sin (e+f x))}+\frac{\left (a^2 \left (12 c^2-16 c d+7 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 )}{4 (c-d)^2 (c+d)^4 f}\\ &=\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d (c+d)^2 f (c+d \sin (e+f x))^3}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sin (e+f x))^2}-\frac{a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \cos (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sin (e+f x))}-\frac{\left (a^2 \left (12 c^2-16 c d+7 d^2\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 )}{2 (c-d)^2 (c+d)^4 f}\\ &=\frac{a^2 \left (12 c^2-16 c d+7 d^2\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{4 (c-d)^2 (c+d)^4 \sqrt{c^2-d^2} f}+\frac{a^2 (c-d) \cos (e+f x)}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac{a^2 (c+8 d) \cos (e+f x)}{12 d (c+d)^2 f (c+d \sin (e+f x))^3}-\frac{a^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x)}{24 (c-d) d (c+d)^3 f (c+d \sin (e+f x))^2}-\frac{a^2 \left (2 c^3+16 c^2 d-59 c d^2+32 d^3\right ) \cos (e+f x)}{24 (c-d)^2 d (c+d)^4 f (c+d \sin (e+f x))}\\ \end{align*}
Mathematica [A] time = 3.90137, size = 269, normalized size = 0.94 \[ \frac{a^2 \cos (e+f x) \left (\frac{\left (12 c^2-16 c d+7 d^2\right ) (c+d \sin (e+f x))^2 \left (\sqrt{-c-d} \sqrt{d-c} \sqrt{\cos ^2(e+f x)} ((c+4 d) \sin (e+f x)+4 c+d)-6 (c+d \sin (e+f x))^2 \tan ^{-1}\left (\frac{\sqrt{d-c} \sqrt{1-\sin (e+f x)}}{\sqrt{-c-d} \sqrt{\sin (e+f x)+1}}\right )\right )}{(-c-d)^{7/2} (d-c)^{3/2} \sqrt{\cos ^2(e+f x)}}-\frac{2 d (5 c-2 d) (\sin (e+f x)+1)^2 (c+d \sin (e+f x))}{(c-d) (c+d)}-6 d (\sin (e+f x)+1)^2\right )}{24 f (d-c) (c+d) (c+d \sin (e+f x))^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^5,x]
[Out]
(a^2*Cos[e + f*x]*(-6*d*(1 + Sin[e + f*x])^2 - (2*(5*c - 2*d)*d*(1 + Sin[e + f*x])^2*(c + d*Sin[e + f*x]))/((c
- d)*(c + d)) + ((12*c^2 - 16*c*d + 7*d^2)*(c + d*Sin[e + f*x])^2*(-6*ArcTan[(Sqrt[-c + d]*Sqrt[1 - Sin[e + f
*x]])/(Sqrt[-c - d]*Sqrt[1 + Sin[e + f*x]])]*(c + d*Sin[e + f*x])^2 + Sqrt[-c - d]*Sqrt[-c + d]*Sqrt[Cos[e + f
*x]^2]*(4*c + d + (c + 4*d)*Sin[e + f*x])))/((-c - d)^(7/2)*(-c + d)^(3/2)*Sqrt[Cos[e + f*x]^2])))/(24*(-c + d
)*(c + d)*f*(c + d*Sin[e + f*x])^4)
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Maple [B] time = 0.192, size = 6466, normalized size = 22.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^5,x)
[Out]
result too large to display
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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+a*sin(f*x+e))^2/(c+d*sin(f*x+e))^5,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.12127, size = 4629, normalized size = 16.19 \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/(c+d*sin(f*x+e))^5,x, algorithm="fricas")
[Out]
[1/48*(2*(8*a^2*c^6*d + 64*a^2*c^5*d^2 - 208*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 221*a^2*c^2*d^5 - 80*a^2*c*d^6 - 2
1*a^2*d^7)*cos(f*x + e)^3 - 3*(12*a^2*c^6 - 16*a^2*c^5*d + 79*a^2*c^4*d^2 - 96*a^2*c^3*d^3 + 54*a^2*c^2*d^4 -
16*a^2*c*d^5 + 7*a^2*d^6 + (12*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^4 - 2*(36*a^2*c^4*d^2 - 48
*a^2*c^3*d^3 + 33*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^2 + 4*(12*a^2*c^5*d - 16*a^2*c^4*d^2 +
19*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^2*c*d^5 - (12*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^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*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)) - 6*(16*a^2*c^7 - 20*a^2*c^6*d - 45*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 74*a^2*c^2*d^5 - 32*a^2*c*d^6 - 9*a^2*d
^7)*cos(f*x + e) + 2*((2*a^2*c^5*d^2 + 16*a^2*c^4*d^3 - 61*a^2*c^3*d^4 + 16*a^2*c^2*d^5 + 59*a^2*c*d^6 - 32*a^
2*d^7)*cos(f*x + e)^3 - 3*(4*a^2*c^7 + 32*a^2*c^6*d - 79*a^2*c^5*d^2 - 16*a^2*c^4*d^3 + 70*a^2*c^3*d^4 + 5*a^2
*c*d^6 - 16*a^2*d^7)*cos(f*x + e))*sin(f*x + e))/((c^8*d^4 + 2*c^7*d^5 - 2*c^6*d^6 - 6*c^5*d^7 + 6*c^3*d^9 + 2
*c^2*d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^4 - 2*(3*c^10*d^2 + 6*c^9*d^3 - 5*c^8*d^4 - 16*c^7*d^5 - 2*c^6*d^6
+ 12*c^5*d^7 + 6*c^4*d^8 - c^2*d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^2 + (c^12 + 2*c^11*d + 4*c^10*d^2 + 6*c
^9*d^3 - 11*c^8*d^4 - 28*c^7*d^5 + 28*c^5*d^7 + 11*c^4*d^8 - 6*c^3*d^9 - 4*c^2*d^10 - 2*c*d^11 - d^12)*f - 4*(
(c^9*d^3 + 2*c^8*d^4 - 2*c^7*d^5 - 6*c^6*d^6 + 6*c^4*d^8 + 2*c^3*d^9 - 2*c^2*d^10 - c*d^11)*f*cos(f*x + e)^2 -
(c^11*d + 2*c^10*d^2 - c^9*d^3 - 4*c^8*d^4 - 2*c^7*d^5 + 2*c^5*d^7 + 4*c^4*d^8 + c^3*d^9 - 2*c^2*d^10 - c*d^1
1)*f)*sin(f*x + e)), 1/24*((8*a^2*c^6*d + 64*a^2*c^5*d^2 - 208*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 221*a^2*c^2*d^5
- 80*a^2*c*d^6 - 21*a^2*d^7)*cos(f*x + e)^3 - 3*(12*a^2*c^6 - 16*a^2*c^5*d + 79*a^2*c^4*d^2 - 96*a^2*c^3*d^3 +
54*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6 + (12*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^4 - 2*(3
6*a^2*c^4*d^2 - 48*a^2*c^3*d^3 + 33*a^2*c^2*d^4 - 16*a^2*c*d^5 + 7*a^2*d^6)*cos(f*x + e)^2 + 4*(12*a^2*c^5*d -
16*a^2*c^4*d^2 + 19*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^2*c*d^5 - (12*a^2*c^3*d^3 - 16*a^2*c^2*d^4 + 7*a^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)))
- 3*(16*a^2*c^7 - 20*a^2*c^6*d - 45*a^2*c^4*d^3 + 16*a^2*c^3*d^4 + 74*a^2*c^2*d^5 - 32*a^2*c*d^6 - 9*a^2*d^7)
*cos(f*x + e) + ((2*a^2*c^5*d^2 + 16*a^2*c^4*d^3 - 61*a^2*c^3*d^4 + 16*a^2*c^2*d^5 + 59*a^2*c*d^6 - 32*a^2*d^7
)*cos(f*x + e)^3 - 3*(4*a^2*c^7 + 32*a^2*c^6*d - 79*a^2*c^5*d^2 - 16*a^2*c^4*d^3 + 70*a^2*c^3*d^4 + 5*a^2*c*d^
6 - 16*a^2*d^7)*cos(f*x + e))*sin(f*x + e))/((c^8*d^4 + 2*c^7*d^5 - 2*c^6*d^6 - 6*c^5*d^7 + 6*c^3*d^9 + 2*c^2*
d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^4 - 2*(3*c^10*d^2 + 6*c^9*d^3 - 5*c^8*d^4 - 16*c^7*d^5 - 2*c^6*d^6 + 12
*c^5*d^7 + 6*c^4*d^8 - c^2*d^10 - 2*c*d^11 - d^12)*f*cos(f*x + e)^2 + (c^12 + 2*c^11*d + 4*c^10*d^2 + 6*c^9*d^
3 - 11*c^8*d^4 - 28*c^7*d^5 + 28*c^5*d^7 + 11*c^4*d^8 - 6*c^3*d^9 - 4*c^2*d^10 - 2*c*d^11 - d^12)*f - 4*((c^9*
d^3 + 2*c^8*d^4 - 2*c^7*d^5 - 6*c^6*d^6 + 6*c^4*d^8 + 2*c^3*d^9 - 2*c^2*d^10 - c*d^11)*f*cos(f*x + e)^2 - (c^1
1*d + 2*c^10*d^2 - c^9*d^3 - 4*c^8*d^4 - 2*c^7*d^5 + 2*c^5*d^7 + 4*c^4*d^8 + c^3*d^9 - 2*c^2*d^10 - c*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+a*sin(f*x+e))**2/(c+d*sin(f*x+e))**5,x)
[Out]
Timed out
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Giac [B] time = 1.6169, size = 2099, normalized size = 7.34 \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/(c+d*sin(f*x+e))^5,x, algorithm="giac")
[Out]
1/12*(3*(12*a^2*c^2 - 16*a^2*c*d + 7*a^2*d^2)*(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 + 2*c^5*d - c^4*d^2 - 4*c^3*d^3 - c^2*d^4 + 2*c*d^5 + d^6)*sqrt(c^2 - d
^2)) + (12*a^2*c^9*tan(1/2*f*x + 1/2*e)^7 - 96*a^2*c^8*d*tan(1/2*f*x + 1/2*e)^7 + 45*a^2*c^7*d^2*tan(1/2*f*x +
1/2*e)^7 + 96*a^2*c^6*d^3*tan(1/2*f*x + 1/2*e)^7 + 24*a^2*c^5*d^4*tan(1/2*f*x + 1/2*e)^7 - 48*a^2*c^4*d^5*tan
(1/2*f*x + 1/2*e)^7 - 24*a^2*c^3*d^6*tan(1/2*f*x + 1/2*e)^7 - 48*a^2*c^9*tan(1/2*f*x + 1/2*e)^6 + 84*a^2*c^8*d
*tan(1/2*f*x + 1/2*e)^6 - 432*a^2*c^7*d^2*tan(1/2*f*x + 1/2*e)^6 + 411*a^2*c^6*d^3*tan(1/2*f*x + 1/2*e)^6 + 33
6*a^2*c^5*d^4*tan(1/2*f*x + 1/2*e)^6 - 24*a^2*c^4*d^5*tan(1/2*f*x + 1/2*e)^6 - 192*a^2*c^3*d^6*tan(1/2*f*x + 1
/2*e)^6 - 72*a^2*c^2*d^7*tan(1/2*f*x + 1/2*e)^6 + 12*a^2*c^9*tan(1/2*f*x + 1/2*e)^5 - 480*a^2*c^8*d*tan(1/2*f*
x + 1/2*e)^5 + 597*a^2*c^7*d^2*tan(1/2*f*x + 1/2*e)^5 - 480*a^2*c^6*d^3*tan(1/2*f*x + 1/2*e)^5 + 836*a^2*c^5*d
^4*tan(1/2*f*x + 1/2*e)^5 + 208*a^2*c^4*d^5*tan(1/2*f*x + 1/2*e)^5 - 152*a^2*c^3*d^6*tan(1/2*f*x + 1/2*e)^5 -
256*a^2*c^2*d^7*tan(1/2*f*x + 1/2*e)^5 - 96*a^2*c*d^8*tan(1/2*f*x + 1/2*e)^5 - 144*a^2*c^9*tan(1/2*f*x + 1/2*e
)^4 + 204*a^2*c^8*d*tan(1/2*f*x + 1/2*e)^4 - 1104*a^2*c^7*d^2*tan(1/2*f*x + 1/2*e)^4 + 1617*a^2*c^6*d^3*tan(1/
2*f*x + 1/2*e)^4 - 48*a^2*c^5*d^4*tan(1/2*f*x + 1/2*e)^4 + 406*a^2*c^4*d^5*tan(1/2*f*x + 1/2*e)^4 - 256*a^2*c^
3*d^6*tan(1/2*f*x + 1/2*e)^4 - 184*a^2*c^2*d^7*tan(1/2*f*x + 1/2*e)^4 - 128*a^2*c*d^8*tan(1/2*f*x + 1/2*e)^4 -
48*a^2*d^9*tan(1/2*f*x + 1/2*e)^4 - 12*a^2*c^9*tan(1/2*f*x + 1/2*e)^3 - 672*a^2*c^8*d*tan(1/2*f*x + 1/2*e)^3
+ 1035*a^2*c^7*d^2*tan(1/2*f*x + 1/2*e)^3 - 672*a^2*c^6*d^3*tan(1/2*f*x + 1/2*e)^3 + 1220*a^2*c^5*d^4*tan(1/2*
f*x + 1/2*e)^3 - 80*a^2*c^4*d^5*tan(1/2*f*x + 1/2*e)^3 - 152*a^2*c^3*d^6*tan(1/2*f*x + 1/2*e)^3 - 256*a^2*c^2*
d^7*tan(1/2*f*x + 1/2*e)^3 - 96*a^2*c*d^8*tan(1/2*f*x + 1/2*e)^3 - 144*a^2*c^9*tan(1/2*f*x + 1/2*e)^2 + 188*a^
2*c^8*d*tan(1/2*f*x + 1/2*e)^2 - 656*a^2*c^7*d^2*tan(1/2*f*x + 1/2*e)^2 + 1201*a^2*c^6*d^3*tan(1/2*f*x + 1/2*e
)^2 - 16*a^2*c^5*d^4*tan(1/2*f*x + 1/2*e)^2 - 120*a^2*c^4*d^5*tan(1/2*f*x + 1/2*e)^2 - 192*a^2*c^3*d^6*tan(1/2
*f*x + 1/2*e)^2 - 72*a^2*c^2*d^7*tan(1/2*f*x + 1/2*e)^2 - 12*a^2*c^9*tan(1/2*f*x + 1/2*e) - 288*a^2*c^8*d*tan(
1/2*f*x + 1/2*e) + 499*a^2*c^7*d^2*tan(1/2*f*x + 1/2*e) + 32*a^2*c^6*d^3*tan(1/2*f*x + 1/2*e) - 64*a^2*c^5*d^4
*tan(1/2*f*x + 1/2*e) - 80*a^2*c^4*d^5*tan(1/2*f*x + 1/2*e) - 24*a^2*c^3*d^6*tan(1/2*f*x + 1/2*e) - 48*a^2*c^9
+ 68*a^2*c^8*d + 16*a^2*c^7*d^2 - 5*a^2*c^6*d^3 - 16*a^2*c^5*d^4 - 6*a^2*c^4*d^5)/((c^10 + 2*c^9*d - c^8*d^2
- 4*c^7*d^3 - c^6*d^4 + 2*c^5*d^5 + c^4*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^4))/f