3.469 \(\int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=411 \[ \frac {3 a \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{11/2}}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^2}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^3}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

[Out]

3/8*a*(2*a^2+b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^(11/2)/d-1/7*cos(d*x+c)^3/b/d/(a+
b*sin(d*x+c))^7-1/140*(a^2-3*b^2)*cos(d*x+c)/b^3/(a^2-b^2)/d/(a+b*sin(d*x+c))^5-1/280*a*(2*a^2-11*b^2)*cos(d*x
+c)/b^3/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^4-1/280*(2*a^4-15*a^2*b^2-8*b^4)*cos(d*x+c)/b^3/(a^2-b^2)^3/d/(a+b*sin(
d*x+c))^3-1/560*a*(4*a^4-36*a^2*b^2-73*b^4)*cos(d*x+c)/b^3/(a^2-b^2)^4/d/(a+b*sin(d*x+c))^2-1/560*(4*a^6-40*a^
4*b^2-247*a^2*b^4-32*b^6)*cos(d*x+c)/b^3/(a^2-b^2)^5/d/(a+b*sin(d*x+c))+1/28*cos(d*x+c)*(a+3*b*sin(d*x+c))/b^3
/d/(a+b*sin(d*x+c))^6

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Rubi [A]  time = 0.79, antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2693, 2863, 2754, 12, 2660, 618, 204} \[ \frac {3 a \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{11/2}}-\frac {\left (-40 a^4 b^2-247 a^2 b^4+4 a^6-32 b^6\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))}-\frac {a \left (-36 a^2 b^2+4 a^4-73 b^4\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^2}-\frac {\left (-15 a^2 b^2+2 a^4-8 b^4\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^3}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]

[Out]

(3*a*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(11/2)*d) - Cos[c + d*x]^3
/(7*b*d*(a + b*Sin[c + d*x])^7) - ((a^2 - 3*b^2)*Cos[c + d*x])/(140*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5)
- (a*(2*a^2 - 11*b^2)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4) - ((2*a^4 - 15*a^2*b^2 -
8*b^4)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^3) - (a*(4*a^4 - 36*a^2*b^2 - 73*b^4)*Cos[c
+ d*x])/(560*b^3*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^2) - ((4*a^6 - 40*a^4*b^2 - 247*a^2*b^4 - 32*b^6)*Cos[c
+ d*x])/(560*b^3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*(a + 3*b*Sin[c + d*x]))/(28*b^3*d*(a +
b*Sin[c + d*x])^6)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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])

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 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 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

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 2863

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) -
a*d*p + b*d*(m + 1)*Sin[e + f*x]))/(b^2*f*(m + 1)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + 1)*(m + p +
1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Si
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && N
eQ[m + p + 1, 0] && IntegerQ[2*m]

Rubi steps

\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {3 \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^7} \, dx}{7 b}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac {\int \frac {-6 b-2 a \sin (c+d x)}{(a+b \sin (c+d x))^6} \, dx}{56 b^3}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac {\int \frac {20 a b+8 \left (a^2-3 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5} \, dx}{280 b^3 \left (a^2-b^2\right )}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac {\int \frac {-48 b \left (a^2+2 b^2\right )-12 a \left (2 a^2-11 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4} \, dx}{1120 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac {\int \frac {36 a b \left (2 a^2+19 b^2\right )+24 \left (a^2-8 b^2\right ) \left (2 a^2+b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{3360 b^3 \left (a^2-b^2\right )^3}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac {\int \frac {-24 b \left (2 a^4+87 a^2 b^2+16 b^4\right )-12 a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{6720 b^3 \left (a^2-b^2\right )^4}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac {\int \frac {1260 a b^3 \left (2 a^2+b^2\right )}{a+b \sin (c+d x)} \, dx}{6720 b^3 \left (a^2-b^2\right )^5}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac {\left (3 a \left (2 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^5}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac {\left (3 a \left (2 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{8 \left (a^2-b^2\right )^5 d}\\ &=-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac {\left (3 a \left (2 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{4 \left (a^2-b^2\right )^5 d}\\ &=\frac {3 a \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{11/2} d}-\frac {\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac {a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac {\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac {a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac {\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac {\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}\\ \end {align*}

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Mathematica [B]  time = 6.08, size = 1167, normalized size = 2.84 \[ \frac {\cos ^5(c+d x)}{5 (a-b) d (a+b \sin (c+d x))^7}+\frac {a \left (-\frac {b (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{7 (b-a) (a+b) (a+b \sin (c+d x))^7}-\frac {-\frac {(a b+(7 a-b) b) (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{6 (b-a) (a+b) (a+b \sin (c+d x))^6}-\frac {7 \left (6 a^2-2 b a+b^2\right ) \left (-\frac {(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{7/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac {3 \left (-\frac {\sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (b-a) (a+b \sin (c+d x))^4}-\frac {\frac {5 \left (\frac {3 \left (\frac {\sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \sqrt {1-\sin (c+d x)}}{\sqrt {-a-b} \sqrt {\sin (c+d x)+1}}\right )}{(-a-b)^{3/2} \sqrt {a-b}}\right )}{2 (a+b)}-\frac {\sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right )}{3 (a+b)}-\frac {\sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}}{4 (b-a)}\right )}{5 (b-a)}\right )}{6 (b-a) (a+b)}}{7 (b-a) (a+b)}\right ) \cos (c+d x)}{(a-b) d \sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}+\frac {2 b \left (\frac {\cos ^7(c+d x)}{7 (a-b) d (a+b \sin (c+d x))^7}+\frac {a \left (-\frac {(1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{9/2}}{7 (b-a) (a+b \sin (c+d x))^7}-\frac {5 \left (-\frac {(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{9/2}}{6 (b-a) (a+b \sin (c+d x))^6}-\frac {-\frac {\sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)^{9/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac {\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {\sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a-b} \sqrt {1-\sin (c+d x)}}{\sqrt {-a-b} \sqrt {\sin (c+d x)+1}}\right )}{(-a-b)^{3/2} \sqrt {a-b}}\right )}{2 (a+b)}-\frac {\sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right )}{3 (a+b)}-\frac {\sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}\right )}{4 (a+b)}-\frac {\sqrt {1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (a+b) (a+b \sin (c+d x))^4}}{5 (b-a)}}{2 (b-a)}\right )}{7 (b-a)}\right ) \cos (c+d x)}{(a-b) d \sqrt {1-\sin (c+d x)} \sqrt {\sin (c+d x)+1}}\right )}{5 (a-b)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]

[Out]

Cos[c + d*x]^5/(5*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7*(b*(1 - Sin[c + d*x])^(5/2)*(1 + S
in[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-1/6*((a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^
(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (7*(6*a^2 - 2*a*b + b^2)*(-1/5*((1
- Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (3*(-1/4*(Sqrt[1 - Sin[c
+ d*x]]*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^4) - (-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c
+ d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3) + (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/
((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 +
Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a
+ b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b)))/(4*(-a + b))))/(5*(-a + b))))/(6*(-a + b)*(a + b)))/(7*(-a + b
)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]) + (2*b*(Cos[c + d*x]^7/(7*(a - b)*d*(a
+ b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7*((1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a
+ b*Sin[c + d*x])^7) - (5*(-1/6*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d
*x])^6) - (-1/5*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (-1/4*(S
qrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/((a + b)*(a + b*Sin[c + d*x])^4) + (7*(-1/3*(Sqrt[1 - Sin[c +
d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3) + (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c
+ d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[
-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]
])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b))))/(4*(a + b)))/(5*(-a + b)))/(2*(-a + b))))/(7*
(-a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])))/(5*(a - b))

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fricas [B]  time = 1.38, size = 2657, normalized size = 6.46 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

[-1/1120*(2*(4*a^8*b^3 - 44*a^6*b^5 - 207*a^4*b^7 + 215*a^2*b^9 + 32*b^11)*cos(d*x + c)^7 - 28*(6*a^10*b - 65*
a^8*b^3 - 224*a^6*b^5 + 222*a^4*b^7 + 53*a^2*b^9 + 8*b^11)*cos(d*x + c)^5 - 70*(14*a^10*b + 173*a^8*b^3 - 3*a^
6*b^5 - 137*a^4*b^7 - 47*a^2*b^9)*cos(d*x + c)^3 + 105*(2*a^10 + 43*a^8*b^2 + 91*a^6*b^4 + 49*a^4*b^6 + 7*a^2*
b^8 - 7*(2*a^4*b^6 + a^2*b^8)*cos(d*x + c)^6 + 7*(10*a^6*b^4 + 11*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^4 - 7*(6*a
^8*b^2 + 23*a^6*b^4 + 16*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^2 + (14*a^9*b + 77*a^7*b^3 + 77*a^5*b^5 + 23*a^3*b^
7 + a*b^9 - (2*a^3*b^7 + a*b^9)*cos(d*x + c)^6 + 3*(14*a^5*b^5 + 9*a^3*b^7 + a*b^9)*cos(d*x + c)^4 - (70*a^7*b
^3 + 119*a^5*b^5 + 48*a^3*b^7 + 3*a*b^9)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*co
s(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b
^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 420*(6*a^10*b + 17*a^8*b^3 - 7*a^6*b^5 - 13*a^4*
b^7 - 3*a^2*b^9)*cos(d*x + c) - 14*((4*a^9*b^2 - 44*a^7*b^4 - 177*a^5*b^6 + 200*a^3*b^8 + 17*a*b^10)*cos(d*x +
c)^5 - 10*(2*a^11 - 21*a^9*b^2 - 61*a^7*b^4 + 37*a^5*b^6 + 39*a^3*b^8 + 4*a*b^10)*cos(d*x + c)^3 - 15*(2*a^11
+ 29*a^9*b^2 + 14*a^7*b^4 - 28*a^5*b^6 - 16*a^3*b^8 - a*b^10)*cos(d*x + c))*sin(d*x + c))/(7*(a^13*b^6 - 6*a^
11*b^8 + 15*a^9*b^10 - 20*a^7*b^12 + 15*a^5*b^14 - 6*a^3*b^16 + a*b^18)*d*cos(d*x + c)^6 - 7*(5*a^15*b^4 - 27*
a^13*b^6 + 57*a^11*b^8 - 55*a^9*b^10 + 15*a^7*b^12 + 15*a^5*b^14 - 13*a^3*b^16 + 3*a*b^18)*d*cos(d*x + c)^4 +
7*(3*a^17*b^2 - 8*a^15*b^4 - 12*a^13*b^6 + 72*a^11*b^8 - 110*a^9*b^10 + 72*a^7*b^12 - 12*a^5*b^14 - 8*a^3*b^16
+ 3*a*b^18)*d*cos(d*x + c)^2 - (a^19 + 15*a^17*b^2 - 76*a^15*b^4 + 92*a^13*b^6 + 78*a^11*b^8 - 286*a^9*b^10 +
260*a^7*b^12 - 84*a^5*b^14 - 7*a^3*b^16 + 7*a*b^18)*d + ((a^12*b^7 - 6*a^10*b^9 + 15*a^8*b^11 - 20*a^6*b^13 +
15*a^4*b^15 - 6*a^2*b^17 + b^19)*d*cos(d*x + c)^6 - 3*(7*a^14*b^5 - 41*a^12*b^7 + 99*a^10*b^9 - 125*a^8*b^11
+ 85*a^6*b^13 - 27*a^4*b^15 + a^2*b^17 + b^19)*d*cos(d*x + c)^4 + (35*a^16*b^3 - 168*a^14*b^5 + 276*a^12*b^7 -
88*a^10*b^9 - 270*a^8*b^11 + 360*a^6*b^13 - 172*a^4*b^15 + 24*a^2*b^17 + 3*b^19)*d*cos(d*x + c)^2 - (7*a^18*b
- 7*a^16*b^3 - 84*a^14*b^5 + 260*a^12*b^7 - 286*a^10*b^9 + 78*a^8*b^11 + 92*a^6*b^13 - 76*a^4*b^15 + 15*a^2*b
^17 + b^19)*d)*sin(d*x + c)), -1/560*((4*a^8*b^3 - 44*a^6*b^5 - 207*a^4*b^7 + 215*a^2*b^9 + 32*b^11)*cos(d*x +
c)^7 - 14*(6*a^10*b - 65*a^8*b^3 - 224*a^6*b^5 + 222*a^4*b^7 + 53*a^2*b^9 + 8*b^11)*cos(d*x + c)^5 - 35*(14*a
^10*b + 173*a^8*b^3 - 3*a^6*b^5 - 137*a^4*b^7 - 47*a^2*b^9)*cos(d*x + c)^3 - 105*(2*a^10 + 43*a^8*b^2 + 91*a^6
*b^4 + 49*a^4*b^6 + 7*a^2*b^8 - 7*(2*a^4*b^6 + a^2*b^8)*cos(d*x + c)^6 + 7*(10*a^6*b^4 + 11*a^4*b^6 + 3*a^2*b^
8)*cos(d*x + c)^4 - 7*(6*a^8*b^2 + 23*a^6*b^4 + 16*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^2 + (14*a^9*b + 77*a^7*b^
3 + 77*a^5*b^5 + 23*a^3*b^7 + a*b^9 - (2*a^3*b^7 + a*b^9)*cos(d*x + c)^6 + 3*(14*a^5*b^5 + 9*a^3*b^7 + a*b^9)*
cos(d*x + c)^4 - (70*a^7*b^3 + 119*a^5*b^5 + 48*a^3*b^7 + 3*a*b^9)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^
2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 210*(6*a^10*b + 17*a^8*b^3 - 7*a^6*b^5 - 13*
a^4*b^7 - 3*a^2*b^9)*cos(d*x + c) - 7*((4*a^9*b^2 - 44*a^7*b^4 - 177*a^5*b^6 + 200*a^3*b^8 + 17*a*b^10)*cos(d*
x + c)^5 - 10*(2*a^11 - 21*a^9*b^2 - 61*a^7*b^4 + 37*a^5*b^6 + 39*a^3*b^8 + 4*a*b^10)*cos(d*x + c)^3 - 15*(2*a
^11 + 29*a^9*b^2 + 14*a^7*b^4 - 28*a^5*b^6 - 16*a^3*b^8 - a*b^10)*cos(d*x + c))*sin(d*x + c))/(7*(a^13*b^6 - 6
*a^11*b^8 + 15*a^9*b^10 - 20*a^7*b^12 + 15*a^5*b^14 - 6*a^3*b^16 + a*b^18)*d*cos(d*x + c)^6 - 7*(5*a^15*b^4 -
27*a^13*b^6 + 57*a^11*b^8 - 55*a^9*b^10 + 15*a^7*b^12 + 15*a^5*b^14 - 13*a^3*b^16 + 3*a*b^18)*d*cos(d*x + c)^4
+ 7*(3*a^17*b^2 - 8*a^15*b^4 - 12*a^13*b^6 + 72*a^11*b^8 - 110*a^9*b^10 + 72*a^7*b^12 - 12*a^5*b^14 - 8*a^3*b
^16 + 3*a*b^18)*d*cos(d*x + c)^2 - (a^19 + 15*a^17*b^2 - 76*a^15*b^4 + 92*a^13*b^6 + 78*a^11*b^8 - 286*a^9*b^1
0 + 260*a^7*b^12 - 84*a^5*b^14 - 7*a^3*b^16 + 7*a*b^18)*d + ((a^12*b^7 - 6*a^10*b^9 + 15*a^8*b^11 - 20*a^6*b^1
3 + 15*a^4*b^15 - 6*a^2*b^17 + b^19)*d*cos(d*x + c)^6 - 3*(7*a^14*b^5 - 41*a^12*b^7 + 99*a^10*b^9 - 125*a^8*b^
11 + 85*a^6*b^13 - 27*a^4*b^15 + a^2*b^17 + b^19)*d*cos(d*x + c)^4 + (35*a^16*b^3 - 168*a^14*b^5 + 276*a^12*b^
7 - 88*a^10*b^9 - 270*a^8*b^11 + 360*a^6*b^13 - 172*a^4*b^15 + 24*a^2*b^17 + 3*b^19)*d*cos(d*x + c)^2 - (7*a^1
8*b - 7*a^16*b^3 - 84*a^14*b^5 + 260*a^12*b^7 - 286*a^10*b^9 + 78*a^8*b^11 + 92*a^6*b^13 - 76*a^4*b^15 + 15*a^
2*b^17 + b^19)*d)*sin(d*x + c))]

________________________________________________________________________________________

giac [B]  time = 5.09, size = 1932, normalized size = 4.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/280*(105*(2*a^3 + a*b^2)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt
(a^2 - b^2)))/((a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*sqrt(a^2 - b^2)) - (350*a^16*ta
n(1/2*d*x + 1/2*c)^13 - 2905*a^14*b^2*tan(1/2*d*x + 1/2*c)^13 + 5600*a^12*b^4*tan(1/2*d*x + 1/2*c)^13 - 5600*a
^10*b^6*tan(1/2*d*x + 1/2*c)^13 + 2800*a^8*b^8*tan(1/2*d*x + 1/2*c)^13 - 560*a^6*b^10*tan(1/2*d*x + 1/2*c)^13
+ 630*a^15*b*tan(1/2*d*x + 1/2*c)^12 - 18165*a^13*b^3*tan(1/2*d*x + 1/2*c)^12 + 33600*a^11*b^5*tan(1/2*d*x + 1
/2*c)^12 - 33600*a^9*b^7*tan(1/2*d*x + 1/2*c)^12 + 16800*a^7*b^9*tan(1/2*d*x + 1/2*c)^12 - 3360*a^5*b^11*tan(1
/2*d*x + 1/2*c)^12 + 840*a^16*tan(1/2*d*x + 1/2*c)^11 - 15680*a^14*b^2*tan(1/2*d*x + 1/2*c)^11 - 41090*a^12*b^
4*tan(1/2*d*x + 1/2*c)^11 + 89600*a^10*b^6*tan(1/2*d*x + 1/2*c)^11 - 100800*a^8*b^8*tan(1/2*d*x + 1/2*c)^11 +
53760*a^6*b^10*tan(1/2*d*x + 1/2*c)^11 - 11200*a^4*b^12*tan(1/2*d*x + 1/2*c)^11 - 840*a^15*b*tan(1/2*d*x + 1/2
*c)^10 - 102760*a^13*b^3*tan(1/2*d*x + 1/2*c)^10 + 11270*a^11*b^5*tan(1/2*d*x + 1/2*c)^10 + 78400*a^9*b^7*tan(
1/2*d*x + 1/2*c)^10 - 151200*a^7*b^9*tan(1/2*d*x + 1/2*c)^10 + 97440*a^5*b^11*tan(1/2*d*x + 1/2*c)^10 - 22400*
a^3*b^13*tan(1/2*d*x + 1/2*c)^10 + 630*a^16*tan(1/2*d*x + 1/2*c)^9 - 51905*a^14*b^2*tan(1/2*d*x + 1/2*c)^9 - 2
49410*a^12*b^4*tan(1/2*d*x + 1/2*c)^9 + 202244*a^10*b^6*tan(1/2*d*x + 1/2*c)^9 - 129360*a^8*b^8*tan(1/2*d*x +
1/2*c)^9 - 62832*a^6*b^10*tan(1/2*d*x + 1/2*c)^9 + 92288*a^4*b^12*tan(1/2*d*x + 1/2*c)^9 - 26880*a^2*b^14*tan(
1/2*d*x + 1/2*c)^9 - 8330*a^15*b*tan(1/2*d*x + 1/2*c)^8 - 248745*a^13*b^3*tan(1/2*d*x + 1/2*c)^8 - 190610*a^11
*b^5*tan(1/2*d*x + 1/2*c)^8 + 253736*a^9*b^7*tan(1/2*d*x + 1/2*c)^8 - 338240*a^7*b^9*tan(1/2*d*x + 1/2*c)^8 +
120512*a^5*b^11*tan(1/2*d*x + 1/2*c)^8 + 24192*a^3*b^13*tan(1/2*d*x + 1/2*c)^8 - 17920*a*b^15*tan(1/2*d*x + 1/
2*c)^8 - 96040*a^14*b^2*tan(1/2*d*x + 1/2*c)^7 - 452340*a^12*b^4*tan(1/2*d*x + 1/2*c)^7 + 164528*a^10*b^6*tan(
1/2*d*x + 1/2*c)^7 - 99344*a^8*b^8*tan(1/2*d*x + 1/2*c)^7 - 177664*a^6*b^10*tan(1/2*d*x + 1/2*c)^7 + 153088*a^
4*b^12*tan(1/2*d*x + 1/2*c)^7 - 27648*a^2*b^14*tan(1/2*d*x + 1/2*c)^7 - 5120*b^16*tan(1/2*d*x + 1/2*c)^7 - 156
80*a^15*b*tan(1/2*d*x + 1/2*c)^6 - 296520*a^13*b^3*tan(1/2*d*x + 1/2*c)^6 - 247940*a^11*b^5*tan(1/2*d*x + 1/2*
c)^6 + 232736*a^9*b^7*tan(1/2*d*x + 1/2*c)^6 - 339920*a^7*b^9*tan(1/2*d*x + 1/2*c)^6 + 120512*a^5*b^11*tan(1/2
*d*x + 1/2*c)^6 + 24192*a^3*b^13*tan(1/2*d*x + 1/2*c)^6 - 17920*a*b^15*tan(1/2*d*x + 1/2*c)^6 - 630*a^16*tan(1
/2*d*x + 1/2*c)^5 - 92155*a^14*b^2*tan(1/2*d*x + 1/2*c)^5 - 333060*a^12*b^4*tan(1/2*d*x + 1/2*c)^5 + 151144*a^
10*b^6*tan(1/2*d*x + 1/2*c)^5 - 133280*a^8*b^8*tan(1/2*d*x + 1/2*c)^5 - 62832*a^6*b^10*tan(1/2*d*x + 1/2*c)^5
+ 92288*a^4*b^12*tan(1/2*d*x + 1/2*c)^5 - 26880*a^2*b^14*tan(1/2*d*x + 1/2*c)^5 - 13566*a^15*b*tan(1/2*d*x + 1
/2*c)^4 - 166775*a^13*b^3*tan(1/2*d*x + 1/2*c)^4 - 41412*a^11*b^5*tan(1/2*d*x + 1/2*c)^4 + 72128*a^9*b^7*tan(1
/2*d*x + 1/2*c)^4 - 150640*a^7*b^9*tan(1/2*d*x + 1/2*c)^4 + 97440*a^5*b^11*tan(1/2*d*x + 1/2*c)^4 - 22400*a^3*
b^13*tan(1/2*d*x + 1/2*c)^4 - 840*a^16*tan(1/2*d*x + 1/2*c)^3 - 41944*a^14*b^2*tan(1/2*d*x + 1/2*c)^3 - 76650*
a^12*b^4*tan(1/2*d*x + 1/2*c)^3 + 87472*a^10*b^6*tan(1/2*d*x + 1/2*c)^3 - 100688*a^8*b^8*tan(1/2*d*x + 1/2*c)^
3 + 53760*a^6*b^10*tan(1/2*d*x + 1/2*c)^3 - 11200*a^4*b^12*tan(1/2*d*x + 1/2*c)^3 - 5432*a^15*b*tan(1/2*d*x +
1/2*c)^2 - 33264*a^13*b^3*tan(1/2*d*x + 1/2*c)^2 + 34846*a^11*b^5*tan(1/2*d*x + 1/2*c)^2 - 34272*a^9*b^7*tan(1
/2*d*x + 1/2*c)^2 + 16912*a^7*b^9*tan(1/2*d*x + 1/2*c)^2 - 3360*a^5*b^11*tan(1/2*d*x + 1/2*c)^2 - 350*a^16*tan
(1/2*d*x + 1/2*c) - 6699*a^14*b^2*tan(1/2*d*x + 1/2*c) + 6790*a^12*b^4*tan(1/2*d*x + 1/2*c) - 6188*a^10*b^6*ta
n(1/2*d*x + 1/2*c) + 2912*a^8*b^8*tan(1/2*d*x + 1/2*c) - 560*a^6*b^10*tan(1/2*d*x + 1/2*c) - 686*a^15*b + 885*
a^13*b^3 - 842*a^11*b^5 + 408*a^9*b^7 - 80*a^7*b^9)/((a^17 - 5*a^15*b^2 + 10*a^13*b^4 - 10*a^11*b^6 + 5*a^9*b^
8 - a^7*b^10)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^7))/d

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maple [B]  time = 0.39, size = 9171, normalized size = 22.31 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4/(a+b*sin(d*x+c))^8,x)

[Out]

result too large to display

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 10.85, size = 2184, normalized size = 5.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4/(a + b*sin(c + d*x))^8,x)

[Out]

((686*a^8*b + 80*b^9 - 408*a^2*b^7 + 842*a^4*b^5 - 885*a^6*b^3)/(280*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 1
0*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)*(50*a^10 + 80*b^10 - 416*a^2*b^8 + 884*a^4*b^6 - 970*a^6*b^4 + 9
57*a^8*b^2))/(40*a*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^9*(3
840*b^14 - 90*a^14 - 13184*a^2*b^12 + 8976*a^4*b^10 + 18480*a^6*b^8 - 28892*a^8*b^6 + 35630*a^10*b^4 + 7415*a^
12*b^2))/(40*a^5*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^5*(90*
a^14 + 3840*b^14 - 13184*a^2*b^12 + 8976*a^4*b^10 + 19040*a^6*b^8 - 21592*a^8*b^6 + 47580*a^10*b^4 + 13165*a^1
2*b^2))/(40*a^5*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^11*(160
*b^12 - 12*a^12 - 768*a^2*b^10 + 1440*a^4*b^8 - 1280*a^6*b^6 + 587*a^8*b^4 + 224*a^10*b^2))/(4*a^3*(a^10 - b^1
0 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^3*(60*a^12 + 800*b^12 - 3840*a^2*b
^10 + 7192*a^4*b^8 - 6248*a^6*b^6 + 5475*a^8*b^4 + 2996*a^10*b^2))/(20*a^3*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b
^6 + 10*a^6*b^4 - 5*a^8*b^2)) - (tan(c/2 + (d*x)/2)^13*(10*a^10 - 16*b^10 + 80*a^2*b^8 - 160*a^4*b^6 + 160*a^6
*b^4 - 83*a^8*b^2))/(8*a*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2
)^6*(560*a^14*b + 640*b^15 - 864*a^2*b^13 - 4304*a^4*b^11 + 12140*a^6*b^9 - 8312*a^8*b^7 + 8855*a^10*b^5 + 105
90*a^12*b^3))/(10*a^6*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^8
*(1190*a^14*b + 2560*b^15 - 3456*a^2*b^13 - 17216*a^4*b^11 + 48320*a^6*b^9 - 36248*a^8*b^7 + 27230*a^10*b^5 +
35535*a^12*b^3))/(40*a^6*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2
)^10*(12*a^12*b + 320*b^13 - 1392*a^2*b^11 + 2160*a^4*b^9 - 1120*a^6*b^7 - 161*a^8*b^5 + 1468*a^10*b^3))/(4*a^
4*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (tan(c/2 + (d*x)/2)^4*(1938*a^12*b + 3200
*b^13 - 13920*a^2*b^11 + 21520*a^4*b^9 - 10304*a^6*b^7 + 5916*a^8*b^5 + 23825*a^10*b^3))/(40*a^4*(a^10 - b^10
+ 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) - (3*tan(c/2 + (d*x)/2)^12*(6*a^10*b - 32*b^11 + 160*a^2*b
^9 - 320*a^4*b^7 + 320*a^6*b^5 - 173*a^8*b^3))/(8*a^2*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a
^8*b^2)) + (tan(c/2 + (d*x)/2)^2*(388*a^10*b + 240*b^11 - 1208*a^2*b^9 + 2448*a^4*b^7 - 2489*a^6*b^5 + 2376*a^
8*b^3))/(20*a^2*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)) + (b*tan(c/2 + (d*x)/2)^7*(35
*a^6 + 16*b^6 + 168*a^2*b^4 + 210*a^4*b^2)*(686*a^8*b + 80*b^9 - 408*a^2*b^7 + 842*a^4*b^5 - 885*a^6*b^3))/(70
*a^7*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)))/(d*(tan(c/2 + (d*x)/2)^5*(210*a^6*b + 6
72*a^2*b^5 + 1120*a^4*b^3) + tan(c/2 + (d*x)/2)^9*(210*a^6*b + 672*a^2*b^5 + 1120*a^4*b^3) + a^7*tan(c/2 + (d*
x)/2)^14 + tan(c/2 + (d*x)/2)^3*(84*a^6*b + 280*a^4*b^3) + tan(c/2 + (d*x)/2)^11*(84*a^6*b + 280*a^4*b^3) + ta
n(c/2 + (d*x)/2)^6*(448*a*b^6 + 35*a^7 + 1680*a^3*b^4 + 840*a^5*b^2) + tan(c/2 + (d*x)/2)^8*(448*a*b^6 + 35*a^
7 + 1680*a^3*b^4 + 840*a^5*b^2) + tan(c/2 + (d*x)/2)^7*(280*a^6*b + 128*b^7 + 1344*a^2*b^5 + 1680*a^4*b^3) + a
^7 + tan(c/2 + (d*x)/2)^4*(21*a^7 + 560*a^3*b^4 + 420*a^5*b^2) + tan(c/2 + (d*x)/2)^10*(21*a^7 + 560*a^3*b^4 +
420*a^5*b^2) + tan(c/2 + (d*x)/2)^2*(7*a^7 + 84*a^5*b^2) + tan(c/2 + (d*x)/2)^12*(7*a^7 + 84*a^5*b^2) + 14*a^
6*b*tan(c/2 + (d*x)/2) + 14*a^6*b*tan(c/2 + (d*x)/2)^13)) + (3*a*atan((8*((3*a^2*tan(c/2 + (d*x)/2)*(2*a^2 + b
^2))/(8*(a + b)^(11/2)*(a - b)^(11/2)) + (3*a*(2*a^2 + b^2)*(16*a^10*b - 16*b^11 + 80*a^2*b^9 - 160*a^4*b^7 +
160*a^6*b^5 - 80*a^8*b^3))/(128*(a + b)^(11/2)*(a - b)^(11/2)*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b
^4 - 5*a^8*b^2)))*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2))/(3*a*b^2 + 6*a^3))*(2*a^2 +
b^2))/(8*d*(a + b)^(11/2)*(a - b)^(11/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4/(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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