3.1322 \(\int \frac{\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=228 \[ \frac{2 a \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac{\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (-14 a^2 b^2+8 a^4+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}-\frac{x \left (-40 a^4 b^2+30 a^2 b^4+16 a^6-5 b^6\right )}{16 b^7}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d} \]

[Out]

-((16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^7) + (2*a*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/
2])/Sqrt[a^2 - b^2]])/(b^7*d) - (Cos[c + d*x]^5*(6*a - 5*b*Sin[c + d*x]))/(30*b^2*d) + (Cos[c + d*x]^3*(8*a*(a
^2 - b^2) - b*(6*a^2 - 5*b^2)*Sin[c + d*x]))/(24*b^4*d) - (Cos[c + d*x]*(16*a*(a^2 - b^2)^2 - b*(8*a^4 - 14*a^
2*b^2 + 5*b^4)*Sin[c + d*x]))/(16*b^6*d)

________________________________________________________________________________________

Rubi [A]  time = 0.515406, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2865, 2735, 2660, 618, 204} \[ \frac{2 a \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac{\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (-14 a^2 b^2+8 a^4+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}-\frac{x \left (-40 a^4 b^2+30 a^2 b^4+16 a^6-5 b^6\right )}{16 b^7}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]

[Out]

-((16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*x)/(16*b^7) + (2*a*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/
2])/Sqrt[a^2 - b^2]])/(b^7*d) - (Cos[c + d*x]^5*(6*a - 5*b*Sin[c + d*x]))/(30*b^2*d) + (Cos[c + d*x]^3*(8*a*(a
^2 - b^2) - b*(6*a^2 - 5*b^2)*Sin[c + d*x]))/(24*b^4*d) - (Cos[c + d*x]*(16*a*(a^2 - b^2)^2 - b*(8*a^4 - 14*a^
2*b^2 + 5*b^4)*Sin[c + d*x]))/(16*b^6*d)

Rule 2865

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 + p)*Sin[e + f*x]))/(b^2*f*(m + p)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + p)*(m + p +
1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 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 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{\cos ^6(c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac{\int \frac{\cos ^4(c+d x) \left (-a b-\left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 b^2}\\ &=-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}+\frac{\int \frac{\cos ^2(c+d x) \left (3 a b \left (2 a^2-3 b^2\right )+3 \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 b^4}\\ &=-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac{\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}+\frac{\int \frac{-3 a b \left (8 a^4-18 a^2 b^2+11 b^4\right )-3 \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{48 b^6}\\ &=-\frac{\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac{\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}+\frac{\left (a \left (a^2-b^2\right )^3\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^7}\\ &=-\frac{\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac{\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}+\frac{\left (2 a \left (a^2-b^2\right )^3\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 )}{b^7 d}\\ &=-\frac{\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac{\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}-\frac{\left (4 a \left (a^2-b^2\right )^3\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 )}{b^7 d}\\ &=-\frac{\left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^7}+\frac{2 a \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^7 d}-\frac{\cos ^5(c+d x) (6 a-5 b \sin (c+d x))}{30 b^2 d}+\frac{\cos ^3(c+d x) \left (8 a \left (a^2-b^2\right )-b \left (6 a^2-5 b^2\right ) \sin (c+d x)\right )}{24 b^4 d}-\frac{\cos (c+d x) \left (16 a \left (a^2-b^2\right )^2-b \left (8 a^4-14 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{16 b^6 d}\\ \end{align*}

Mathematica [A]  time = 2.27283, size = 275, normalized size = 1.21 \[ \frac{240 a^4 b^2 \sin (2 (c+d x))-480 a^2 b^4 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))-120 a b \left (-18 a^2 b^2+8 a^4+11 b^4\right ) \cos (c+d x)+20 \left (4 a^3 b^3-7 a b^5\right ) \cos (3 (c+d x))+1920 a \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )+2400 a^4 b^2 c-1800 a^2 b^4 c+2400 a^4 b^2 d x-1800 a^2 b^4 d x-960 a^6 c-960 a^6 d x-12 a b^5 \cos (5 (c+d x))+225 b^6 \sin (2 (c+d x))+45 b^6 \sin (4 (c+d x))+5 b^6 \sin (6 (c+d x))+300 b^6 c+300 b^6 d x}{960 b^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]

[Out]

(-960*a^6*c + 2400*a^4*b^2*c - 1800*a^2*b^4*c + 300*b^6*c - 960*a^6*d*x + 2400*a^4*b^2*d*x - 1800*a^2*b^4*d*x
+ 300*b^6*d*x + 1920*a*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 120*a*b*(8*a^4 - 1
8*a^2*b^2 + 11*b^4)*Cos[c + d*x] + 20*(4*a^3*b^3 - 7*a*b^5)*Cos[3*(c + d*x)] - 12*a*b^5*Cos[5*(c + d*x)] + 240
*a^4*b^2*Sin[2*(c + d*x)] - 480*a^2*b^4*Sin[2*(c + d*x)] + 225*b^6*Sin[2*(c + d*x)] - 30*a^2*b^4*Sin[4*(c + d*
x)] + 45*b^6*Sin[4*(c + d*x)] + 5*b^6*Sin[6*(c + d*x)])/(960*b^7*d)

________________________________________________________________________________________

Maple [B]  time = 0.094, size = 1551, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

5/8/d/b*arctan(tan(1/2*d*x+1/2*c))+2/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*a^4-2/d/b*a/(a^2-b^
2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))-20/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*
d*x+1/2*c)^4*a^5-9/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*a^2-2/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^
6*tan(1/2*d*x+1/2*c)^10*a^5-62/5/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a+1/d/b^5/(1+tan(1/2*d*
x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*a^4-10/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a^5+15/4/d/b/(1+
tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5-5/24/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3+11/8/d/b
/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)-2/d/b^7*arctan(tan(1/2*d*x+1/2*c))*a^6-46/15/d/b^2/(1+tan(1/2*d
*x+1/2*c)^2)^6*a+14/3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*a^3-2/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*a^5-11/8/d/b/(1+
tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11+44/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a^3+3/d
/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*a^4+5/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^4-15/4/d/b^3*arc
tan(tan(1/2*d*x+1/2*c))*a^2+140/3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6*a^3-92/3/d/b^2/(1+tan(
1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6*a-5/2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*a^2+5/24/
d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9-15/4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7-1
0/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a^5-19/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+
1/2*c)^3*a^2+6/d*a^3/b^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+22/d/b^4/(1+
tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a^3-20/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6*a^5-
18/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a-3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*
c)^9*a^4+19/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9*a^2-1/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan
(1/2*d*x+1/2*c)^11*a^4+9/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11*a^2-28/d/b^2/(1+tan(1/2*d*x+
1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a+26/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a^3+6/d/b^4/(1+tan
(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a^3-6/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a-2/d/
b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*a^4+5/2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c
)^7*a^2-6/d*a^5/b^5/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+2/d*a^7/b^7/(a^2-
b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))

________________________________________________________________________________________

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(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.84988, size = 1310, normalized size = 5.75 \begin{align*} \left [-\frac{48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \,{\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x - 120 \,{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt{-a^{2} + b^{2}} \log \left (-\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \,{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 240 \,{\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \,{\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \,{\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}, -\frac{48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \,{\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x + 240 \,{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \sin \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 240 \,{\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \,{\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \,{\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[-1/240*(48*a*b^5*cos(d*x + c)^5 - 80*(a^3*b^3 - a*b^5)*cos(d*x + c)^3 + 15*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4
- 5*b^6)*d*x - 120*(a^5 - 2*a^3*b^2 + a*b^4)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(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)) + 240*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c) - 5*(8*b^6*cos(d*x + c)^5 - 2
*(6*a^2*b^4 - 5*b^6)*cos(d*x + c)^3 + 3*(8*a^4*b^2 - 14*a^2*b^4 + 5*b^6)*cos(d*x + c))*sin(d*x + c))/(b^7*d),
-1/240*(48*a*b^5*cos(d*x + c)^5 - 80*(a^3*b^3 - a*b^5)*cos(d*x + c)^3 + 15*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 -
5*b^6)*d*x + 240*(a^5 - 2*a^3*b^2 + a*b^4)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(
d*x + c))) + 240*(a^5*b - 2*a^3*b^3 + a*b^5)*cos(d*x + c) - 5*(8*b^6*cos(d*x + c)^5 - 2*(6*a^2*b^4 - 5*b^6)*co
s(d*x + c)^3 + 3*(8*a^4*b^2 - 14*a^2*b^4 + 5*b^6)*cos(d*x + c))*sin(d*x + c))/(b^7*d)]

________________________________________________________________________________________

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(cos(d*x+c)**6*sin(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.21778, size = 992, normalized size = 4.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/240*(15*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*(d*x + c)/b^7 - 480*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6
)*(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)))/(sqrt(a^2 -
b^2)*b^7) + 2*(120*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 270*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 + 165*b^5*tan(1/2*d*x
+ 1/2*c)^11 + 240*a^5*tan(1/2*d*x + 1/2*c)^10 - 720*a^3*b^2*tan(1/2*d*x + 1/2*c)^10 + 720*a*b^4*tan(1/2*d*x +
1/2*c)^10 + 360*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 570*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 - 25*b^5*tan(1/2*d*x + 1/2*c
)^9 + 1200*a^5*tan(1/2*d*x + 1/2*c)^8 - 3120*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 2160*a*b^4*tan(1/2*d*x + 1/2*c)^
8 + 240*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 300*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 450*b^5*tan(1/2*d*x + 1/2*c)^7 + 2
400*a^5*tan(1/2*d*x + 1/2*c)^6 - 5600*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 + 3680*a*b^4*tan(1/2*d*x + 1/2*c)^6 - 240
*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 300*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 450*b^5*tan(1/2*d*x + 1/2*c)^5 + 2400*a^5
*tan(1/2*d*x + 1/2*c)^4 - 5280*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 3360*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 360*a^4*b*
tan(1/2*d*x + 1/2*c)^3 + 570*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 25*b^5*tan(1/2*d*x + 1/2*c)^3 + 1200*a^5*tan(1/2
*d*x + 1/2*c)^2 - 2640*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 1488*a*b^4*tan(1/2*d*x + 1/2*c)^2 - 120*a^4*b*tan(1/2*
d*x + 1/2*c) + 270*a^2*b^3*tan(1/2*d*x + 1/2*c) - 165*b^5*tan(1/2*d*x + 1/2*c) + 240*a^5 - 560*a^3*b^2 + 368*a
*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*b^6))/d