3.433 \(\int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx\)
Optimal. Leaf size=192 \[ \frac{a \left (2 c^2-2 c d+d^2\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f (c+d) \left (c^2-d^2\right )^{5/2}}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sin (e+f x))}-\frac{a (2 c-3 d) \cos (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sin (e+f x))^2}-\frac{a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^3} \]
[Out]
(a*(2*c^2 - 2*c*d + d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(5/2)*f) - (a*
Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^3) - (a*(2*c - 3*d)*Cos[e + f*x])/(6*(c - d)*(c + d)^2*f*(c +
d*Sin[e + f*x])^2) - (a*(c - 4*d)*(2*c - d)*Cos[e + f*x])/(6*(c - d)^2*(c + d)^3*f*(c + d*Sin[e + f*x]))
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Rubi [A] time = 0.334907, antiderivative size = 192, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{2754, 12, 2660, 618, 204} \[ \frac{a \left (2 c^2-2 c d+d^2\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f (c+d) \left (c^2-d^2\right )^{5/2}}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sin (e+f x))}-\frac{a (2 c-3 d) \cos (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sin (e+f x))^2}-\frac{a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^3} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^4,x]
[Out]
(a*(2*c^2 - 2*c*d + d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(5/2)*f) - (a*
Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^3) - (a*(2*c - 3*d)*Cos[e + f*x])/(6*(c - d)*(c + d)^2*f*(c +
d*Sin[e + f*x])^2) - (a*(c - 4*d)*(2*c - d)*Cos[e + f*x])/(6*(c - d)^2*(c + d)^3*f*(c + d*Sin[e + f*x]))
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)}{(c+d \sin (e+f x))^4} \, dx &=-\frac{a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac{\int \frac{-3 a (c-d)-2 a (c-d) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 \left (c^2-d^2\right )}\\ &=-\frac{a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac{a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}+\frac{\int \frac{2 a (3 c-2 d) (c-d)+a (2 c-3 d) (c-d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 \left (c^2-d^2\right )^2}\\ &=-\frac{a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac{a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}-\frac{\int -\frac{3 a (c-d) \left (2 c^2-2 c d+d^2\right )}{c+d \sin (e+f x)} \, dx}{6 \left (c^2-d^2\right )^3}\\ &=-\frac{a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac{a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}+\frac{\left (a \left (2 c^2-2 c d+d^2\right )\right ) \int \frac{1}{c+d \sin (e+f x)} \, dx}{2 (c-d)^2 (c+d)^3}\\ &=-\frac{a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac{a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}+\frac{\left (a \left (2 c^2-2 c d+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 )}{(c-d)^2 (c+d)^3 f}\\ &=-\frac{a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac{a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}-\frac{\left (2 a \left (2 c^2-2 c d+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 )}{(c-d)^2 (c+d)^3 f}\\ &=\frac{a \left (2 c^2-2 c d+d^2\right ) \tan ^{-1}\left (\frac{d+c \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c^2-d^2}}\right )}{(c-d)^2 (c+d)^3 \sqrt{c^2-d^2} f}-\frac{a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac{a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac{a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}\\ \end{align*}
Mathematica [C] time = 2.69252, size = 428, normalized size = 2.23 \[ \frac{a (\sin (e+f x)+1) \left (\frac{2 c \left (14 c^2 d^2-18 c^3 d+4 c^4-27 c d^3+12 d^4\right ) \cot (e)-d \csc (e) \left (-30 c^2 d^2 \sin (2 e+f x)+2 c^2 d^2 \sin (2 e+3 f x)+3 d \left (-16 c^2 d+4 c^3+6 c d^2+d^3\right ) \cos (e+2 f x)-3 d^2 \left (2 c^2-2 c d+d^2\right ) \cos (3 e+2 f x)-24 c^2 d^2 \sin (f x)+30 c^3 d \sin (2 e+f x)+78 c^3 d \sin (f x)-24 c^4 \sin (f x)+15 c d^3 \sin (2 e+f x)-9 c d^3 \sin (2 e+3 f x)+12 c d^3 \sin (f x)+4 d^4 \sin (2 e+3 f x)-12 d^4 \sin (f x)\right )}{d (c+d \sin (e+f x))^3}+\frac{24 \left (2 c^2-2 c d+d^2\right ) (\cos (e)-i \sin (e)) \tan ^{-1}\left (\frac{(\cos (e)-i \sin (e)) \sec \left (\frac{f x}{2}\right ) \left (c \sin \left (\frac{f x}{2}\right )+d \cos \left (e+\frac{f x}{2}\right )\right )}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right )}{\sqrt{c^2-d^2} \sqrt{(\cos (e)-i \sin (e))^2}}\right )}{24 f (c-d)^2 (c+d)^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^4,x]
[Out]
(a*(1 + Sin[e + f*x])*((24*(2*c^2 - 2*c*d + d^2)*ArcTan[(Sec[(f*x)/2]*(Cos[e] - I*Sin[e])*(d*Cos[e + (f*x)/2]
+ c*Sin[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(
Cos[e] - I*Sin[e])^2]) + (2*c*(4*c^4 - 18*c^3*d + 14*c^2*d^2 - 27*c*d^3 + 12*d^4)*Cot[e] - d*Csc[e]*(3*d*(4*c^
3 - 16*c^2*d + 6*c*d^2 + d^3)*Cos[e + 2*f*x] - 3*d^2*(2*c^2 - 2*c*d + d^2)*Cos[3*e + 2*f*x] - 24*c^4*Sin[f*x]
+ 78*c^3*d*Sin[f*x] - 24*c^2*d^2*Sin[f*x] + 12*c*d^3*Sin[f*x] - 12*d^4*Sin[f*x] + 30*c^3*d*Sin[2*e + f*x] - 30
*c^2*d^2*Sin[2*e + f*x] + 15*c*d^3*Sin[2*e + f*x] + 2*c^2*d^2*Sin[2*e + 3*f*x] - 9*c*d^3*Sin[2*e + 3*f*x] + 4*
d^4*Sin[2*e + 3*f*x]))/(d*(c + d*Sin[e + f*x])^3)))/(24*(c - d)^2*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x
)/2])^2)
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Maple [B] time = 0.145, size = 3104, normalized size = 16.2 \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))/(c+d*sin(f*x+e))^4,x)
[Out]
-2/3/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*d^4+10/
f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*d^4/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*
f*x+1/2*e)^3-4/f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*d^4/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^
4+d^5)*tan(1/2*f*x+1/2*e)-2/f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*d^4/(c^5+c^4*d-2*c^3*d^2-2
*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^5-1/f*a/(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^4*d-
2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c*d^3+2/f*a/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/(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))*c^2-4/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/c^2*tan(1/2*f*x+1/2*e)^4*d^6-12/f*a/(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/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^3+24/f*a/(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^2/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^3-4
/f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*c*d^3/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1
/2*f*x+1/2*e)^3-4/3/f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3/c*d^5/(c^5+c^4*d-2*c^3*d^2-2*c^2*d
^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^3-2/f*a/(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^4*d-2*c^3*
d^2-2*c^2*d^3+c*d^4+d^5)*c^4+1/f*a/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*t
an(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))*d^2-2/f*a/(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^4
*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c^4*tan(1/2*f*x+1/2*e)^4-4/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c^4*tan(1/2*f*x+1/2*e)^2+4/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c^3*d+2/3/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c^2*d^2+6/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^4*d^4-4/f*a/(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^6/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^3-8/3/f*
a/(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^7/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/
2*f*x+1/2*e)^3+8/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+
d^5)*c^3*tan(1/2*f*x+1/2*e)^2*d-12/f*a/(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^4*d-2*c^3*d^
2-2*c^2*d^3+c*d^4+d^5)*c^2*tan(1/2*f*x+1/2*e)^2*d^2+28/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c*tan(1/2*f*x+1/2*e)^2*d^3-6/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/c*tan(1/2*f*x+1/2*e)^2*d^5-4/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/c^2*tan(1/2*f*x+1/2*e)^2*d^6-8/f*
a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*d*c^3/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*
f*x+1/2*e)+19/f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*d^2*c^2/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c
*d^4+d^5)*tan(1/2*f*x+1/2*e)-2/f*a/(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/(c^5+c^4*d-2*c^3*
d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)-4/f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e)*d+c)^3*d*c^3/(c
^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^5+5/f*a/(c*tan(1/2*f*x+1/2*e)^2+2*tan(1/2*f*x+1/2*e
)*d+c)^3*d^2*c^2/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^5+4/f*a/(c*tan(1/2*f*x+1/2*e)^2+
2*tan(1/2*f*x+1/2*e)*d+c)^3*d^3*c/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^5-2/f*a/(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/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e
)^5+4/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c^3*ta
n(1/2*f*x+1/2*e)^4*d-10/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3
+c*d^4+d^5)*c^2*tan(1/2*f*x+1/2*e)^4*d^2+17/f*a/(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^4*d
-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*c*tan(1/2*f*x+1/2*e)^4*d^3-2/f*a/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/(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))*c*d-6/f*a/(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^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/c*tan(1/2*f*x+1/2*e)^4*d^5
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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))/(c+d*sin(f*x+e))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.28681, size = 2877, normalized size = 14.98 \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))/(c+d*sin(f*x+e))^4,x, algorithm="fricas")
[Out]
[-1/12*(2*(2*a*c^4*d^2 - 9*a*c^3*d^3 + 2*a*c^2*d^4 + 9*a*c*d^5 - 4*a*d^6)*cos(f*x + e)^3 - 6*(2*a*c^5*d - 7*a*
c^4*d^2 + 8*a*c^2*d^4 - 2*a*c*d^5 - a*d^6)*cos(f*x + e)*sin(f*x + e) - 3*(2*a*c^5 - 2*a*c^4*d + 7*a*c^3*d^2 -
6*a*c^2*d^3 + 3*a*c*d^4 - 3*(2*a*c^3*d^2 - 2*a*c^2*d^3 + a*c*d^4)*cos(f*x + e)^2 + (6*a*c^4*d - 6*a*c^3*d^2 +
5*a*c^2*d^3 - 2*a*c*d^4 + a*d^5 - (2*a*c^2*d^3 - 2*a*c*d^4 + a*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*c
os(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*(a*c^6 - 2*a*c^5*d
- a*c^4*d^2 + a*c^3*d^3 + a*c^2*d^4 + a*c*d^5 - a*d^6)*cos(f*x + e))/(3*(c^8*d^2 + c^7*d^3 - 3*c^6*d^4 - 3*c^5
*d^5 + 3*c^4*d^6 + 3*c^3*d^7 - c^2*d^8 - c*d^9)*f*cos(f*x + e)^2 - (c^10 + c^9*d - 6*c^6*d^4 - 6*c^5*d^5 + 8*c
^4*d^6 + 8*c^3*d^7 - 3*c^2*d^8 - 3*c*d^9)*f + ((c^7*d^3 + c^6*d^4 - 3*c^5*d^5 - 3*c^4*d^6 + 3*c^3*d^7 + 3*c^2*
d^8 - c*d^9 - d^10)*f*cos(f*x + e)^2 - (3*c^9*d + 3*c^8*d^2 - 8*c^7*d^3 - 8*c^6*d^4 + 6*c^5*d^5 + 6*c^4*d^6 -
c*d^9 - d^10)*f)*sin(f*x + e)), -1/6*((2*a*c^4*d^2 - 9*a*c^3*d^3 + 2*a*c^2*d^4 + 9*a*c*d^5 - 4*a*d^6)*cos(f*x
+ e)^3 - 3*(2*a*c^5*d - 7*a*c^4*d^2 + 8*a*c^2*d^4 - 2*a*c*d^5 - a*d^6)*cos(f*x + e)*sin(f*x + e) - 3*(2*a*c^5
- 2*a*c^4*d + 7*a*c^3*d^2 - 6*a*c^2*d^3 + 3*a*c*d^4 - 3*(2*a*c^3*d^2 - 2*a*c^2*d^3 + a*c*d^4)*cos(f*x + e)^2 +
(6*a*c^4*d - 6*a*c^3*d^2 + 5*a*c^2*d^3 - 2*a*c*d^4 + a*d^5 - (2*a*c^2*d^3 - 2*a*c*d^4 + a*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*(a*c^6 - 2*a*
c^5*d - a*c^4*d^2 + a*c^3*d^3 + a*c^2*d^4 + a*c*d^5 - a*d^6)*cos(f*x + e))/(3*(c^8*d^2 + c^7*d^3 - 3*c^6*d^4 -
3*c^5*d^5 + 3*c^4*d^6 + 3*c^3*d^7 - c^2*d^8 - c*d^9)*f*cos(f*x + e)^2 - (c^10 + c^9*d - 6*c^6*d^4 - 6*c^5*d^5
+ 8*c^4*d^6 + 8*c^3*d^7 - 3*c^2*d^8 - 3*c*d^9)*f + ((c^7*d^3 + c^6*d^4 - 3*c^5*d^5 - 3*c^4*d^6 + 3*c^3*d^7 +
3*c^2*d^8 - c*d^9 - d^10)*f*cos(f*x + e)^2 - (3*c^9*d + 3*c^8*d^2 - 8*c^7*d^3 - 8*c^6*d^4 + 6*c^5*d^5 + 6*c^4*
d^6 - c*d^9 - d^10)*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))/(c+d*sin(f*x+e))**4,x)
[Out]
Timed out
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Giac [B] time = 1.43983, size = 1091, normalized size = 5.68 \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))/(c+d*sin(f*x+e))^4,x, algorithm="giac")
[Out]
1/3*(3*(2*a*c^2 - 2*a*c*d + a*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^5 + c^4*d - 2*c^3*d^2 - 2*c^2*d^3 + c*d^4 + d^5)*sqrt(c^2 - d^2)) - (12*a*c^6*d*tan(
1/2*f*x + 1/2*e)^5 - 15*a*c^5*d^2*tan(1/2*f*x + 1/2*e)^5 - 12*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^5 + 6*a*c^3*d^4*t
an(1/2*f*x + 1/2*e)^5 + 6*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^5 + 6*a*c^7*tan(1/2*f*x + 1/2*e)^4 - 12*a*c^6*d*tan(1
/2*f*x + 1/2*e)^4 + 30*a*c^5*d^2*tan(1/2*f*x + 1/2*e)^4 - 51*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^4 - 18*a*c^3*d^4*t
an(1/2*f*x + 1/2*e)^4 + 18*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^4 + 12*a*c*d^6*tan(1/2*f*x + 1/2*e)^4 + 36*a*c^6*d*t
an(1/2*f*x + 1/2*e)^3 - 72*a*c^5*d^2*tan(1/2*f*x + 1/2*e)^3 + 12*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^3 - 30*a*c^3*d
^4*tan(1/2*f*x + 1/2*e)^3 + 4*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^3 + 12*a*c*d^6*tan(1/2*f*x + 1/2*e)^3 + 8*a*d^7*t
an(1/2*f*x + 1/2*e)^3 + 12*a*c^7*tan(1/2*f*x + 1/2*e)^2 - 24*a*c^6*d*tan(1/2*f*x + 1/2*e)^2 + 36*a*c^5*d^2*tan
(1/2*f*x + 1/2*e)^2 - 84*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 + 18*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 + 12*a*c*d^6*t
an(1/2*f*x + 1/2*e)^2 + 24*a*c^6*d*tan(1/2*f*x + 1/2*e) - 57*a*c^5*d^2*tan(1/2*f*x + 1/2*e) + 12*a*c^3*d^4*tan
(1/2*f*x + 1/2*e) + 6*a*c^2*d^5*tan(1/2*f*x + 1/2*e) + 6*a*c^7 - 12*a*c^6*d - 2*a*c^5*d^2 + 3*a*c^4*d^3 + 2*a*
c^3*d^4)/((c^8 + c^7*d - 2*c^6*d^2 - 2*c^5*d^3 + c^4*d^4 + c^3*d^5)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*
x + 1/2*e) + c)^3))/f