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

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

[Out]

-1/16*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*x/b^7+2*a*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)
^(1/2))/b^7/d-1/30*cos(d*x+c)^5*(6*a-5*b*sin(d*x+c))/b^2/d+1/24*cos(d*x+c)^3*(8*a*(a^2-b^2)-b*(6*a^2-5*b^2)*si
n(d*x+c))/b^4/d-1/16*cos(d*x+c)*(16*a*(a^2-b^2)^2-b*(8*a^4-14*a^2*b^2+5*b^4)*sin(d*x+c))/b^6/d

________________________________________________________________________________________

Rubi [A]  time = 0.52, antiderivative size = 228, normalized size of antiderivative = 1.00, 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 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 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 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]

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 = 2.26, size = 275, normalized size = 1.21 \[ \frac {-960 a^6 c-960 a^6 d x+240 a^4 b^2 \sin (2 (c+d x))+2400 a^4 b^2 c+2400 a^4 b^2 d x+20 \left (4 a^3 b^3-7 a b^5\right ) \cos (3 (c+d x))-480 a^2 b^4 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))-1800 a^2 b^4 c-1800 a^2 b^4 d x+1920 a \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )-120 a b \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x)-12 a b^5 \cos (5 (c+d x))+225 b^6 \sin (2 (c+d x))+45 b^6 \sin (4 (c+d x))+5 b^6 \sin (6 (c+d x))+300 b^6 c+300 b^6 d x}{960 b^7 d} \]

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)

________________________________________________________________________________________

fricas [A]  time = 1.03, size = 570, normalized size = 2.50 \[ \left [-\frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x - 120 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 240 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}, -\frac {48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (16 \, a^{6} - 40 \, a^{4} b^{2} + 30 \, a^{2} b^{4} - 5 \, b^{6}\right )} d x + 240 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + 240 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 5 \, {\left (8 \, b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{4} b^{2} - 14 \, a^{2} b^{4} + 5 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}\right ] \]

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)]

________________________________________________________________________________________

giac [B]  time = 0.18, size = 735, normalized size = 3.22 \[ \text {result too large to display} \]

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

________________________________________________________________________________________

maple [B]  time = 0.26, size = 1551, normalized size = 6.80 \[ \text {result too large to display} \]

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]

-15/4/d/b^3*arctan(tan(1/2*d*x+1/2*c))*a^2-5/2/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+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^4-2/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^1
0*a^5+5/8/d/b*arctan(tan(1/2*d*x+1/2*c))-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+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^3-20/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+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+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+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^3-18/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a+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))-2/d*a/b/(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))+5/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^4-2/d/b^7*arctan(tan(1/2*d*x+1/2*c))*a
^6-2/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+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+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)+14/3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*a^3-46/15/d/b^2/(1+
tan(1/2*d*x+1/2*c)^2)^6*a-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^2-10/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+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^4-3/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
^2-2/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^2-20/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+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^3-62/5/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^2*a-10/d/b^6/(1+tan(1/2*d*
x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a^5-28/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a-6/d/b^2/(1+t
an(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a+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+
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+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))-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))-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

________________________________________________________________________________________

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

________________________________________________________________________________________

mupad [B]  time = 14.38, size = 3683, normalized size = 16.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*(23*a*b^4 + 15*a^5 - 35*a^3*b^2))/(15*b^6) + (4*tan(c/2 + (d*x)/2)^4*(7*a*b^4 + 5*a^5 - 11*a^3*b^2))/b^6
+ (2*tan(c/2 + (d*x)/2)^8*(9*a*b^4 + 5*a^5 - 13*a^3*b^2))/b^6 + (4*tan(c/2 + (d*x)/2)^6*(23*a*b^4 + 15*a^5 -
35*a^3*b^2))/(3*b^6) + (2*tan(c/2 + (d*x)/2)^2*(31*a*b^4 + 25*a^5 - 55*a^3*b^2))/(5*b^6) - (tan(c/2 + (d*x)/2)
*(8*a^4 + 11*b^4 - 18*a^2*b^2))/(8*b^5) - (tan(c/2 + (d*x)/2)^5*(8*a^4 + 15*b^4 - 10*a^2*b^2))/(4*b^5) + (tan(
c/2 + (d*x)/2)^7*(8*a^4 + 15*b^4 - 10*a^2*b^2))/(4*b^5) + (tan(c/2 + (d*x)/2)^11*(8*a^4 + 11*b^4 - 18*a^2*b^2)
)/(8*b^5) + (tan(c/2 + (d*x)/2)^3*(5*b^4 - 72*a^4 + 114*a^2*b^2))/(24*b^5) - (tan(c/2 + (d*x)/2)^9*(5*b^4 - 72
*a^4 + 114*a^2*b^2))/(24*b^5) + (2*tan(c/2 + (d*x)/2)^10*(3*a*b^4 + a^5 - 3*a^3*b^2))/b^6)/(d*(6*tan(c/2 + (d*
x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*x)/2)^8 + 6*tan(c/2 + (d*x)/2)^1
0 + tan(c/2 + (d*x)/2)^12 + 1)) - (atan((((((25*a^2*b^18)/8 - (75*a^4*b^16)/2 + (325*a^6*b^14)/2 - 320*a^8*b^1
2 + 320*a^10*b^10 - 160*a^12*b^8 + 32*a^14*b^6)/b^17 - (((10*a*b^20 - 38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*b^14)
/b^17 - ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18))*(a^6*16i - b^6*5i + a^2*b^4*
30i - a^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d*x)/2)*(512*a^2*b^20 - 1536*a^4*b^18 + 1536*a^6*b^16 - 512*a^8*b^1
4))/(8*b^18))*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d*x)/2)*(50*a*b^20 - 881*
a^3*b^18 + 4436*a^5*b^16 - 10260*a^7*b^14 + 12800*a^9*b^12 - 8960*a^11*b^10 + 3328*a^13*b^8 - 512*a^15*b^6))/(
8*b^18))*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i)*1i)/(16*b^7) + ((((25*a^2*b^18)/8 - (75*a^4*b^16)/2 +
(325*a^6*b^14)/2 - 320*a^8*b^12 + 320*a^10*b^10 - 160*a^12*b^8 + 32*a^14*b^6)/b^17 + (((10*a*b^20 - 38*a^3*b^1
8 + 44*a^5*b^16 - 16*a^7*b^14)/b^17 + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18)
)*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d*x)/2)*(512*a^2*b^20 - 1536*a^4*b^18
+ 1536*a^6*b^16 - 512*a^8*b^14))/(8*b^18))*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (tan(c/
2 + (d*x)/2)*(50*a*b^20 - 881*a^3*b^18 + 4436*a^5*b^16 - 10260*a^7*b^14 + 12800*a^9*b^12 - 8960*a^11*b^10 + 33
28*a^13*b^8 - 512*a^15*b^6))/(8*b^18))*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i)*1i)/(16*b^7))/((32*a^19
+ (55*a^3*b^16)/4 - (585*a^5*b^14)/4 + (2445*a^7*b^12)/4 - (5511*a^9*b^10)/4 + 1874*a^11*b^8 - 1595*a^13*b^6 +
836*a^15*b^4 - 248*a^17*b^2)/b^17 + (tan(c/2 + (d*x)/2)*(512*a^20 - 50*a^2*b^18 + 750*a^4*b^16 - 4550*a^6*b^1
4 + 14770*a^8*b^12 - 28880*a^10*b^10 + 35880*a^12*b^8 - 28672*a^14*b^6 + 14336*a^16*b^4 - 4096*a^18*b^2))/(4*b
^18) + ((((25*a^2*b^18)/8 - (75*a^4*b^16)/2 + (325*a^6*b^14)/2 - 320*a^8*b^12 + 320*a^10*b^10 - 160*a^12*b^8 +
32*a^14*b^6)/b^17 - (((10*a*b^20 - 38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*b^14)/b^17 - ((32*a^2*b^3 + (tan(c/2 +
(d*x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18))*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (ta
n(c/2 + (d*x)/2)*(512*a^2*b^20 - 1536*a^4*b^18 + 1536*a^6*b^16 - 512*a^8*b^14))/(8*b^18))*(a^6*16i - b^6*5i +
a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d*x)/2)*(50*a*b^20 - 881*a^3*b^18 + 4436*a^5*b^16 - 10260*a
^7*b^14 + 12800*a^9*b^12 - 8960*a^11*b^10 + 3328*a^13*b^8 - 512*a^15*b^6))/(8*b^18))*(a^6*16i - b^6*5i + a^2*b
^4*30i - a^4*b^2*40i))/(16*b^7) - ((((25*a^2*b^18)/8 - (75*a^4*b^16)/2 + (325*a^6*b^14)/2 - 320*a^8*b^12 + 320
*a^10*b^10 - 160*a^12*b^8 + 32*a^14*b^6)/b^17 + (((10*a*b^20 - 38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*b^14)/b^17 +
((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18))*(a^6*16i - b^6*5i + a^2*b^4*30i - a
^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d*x)/2)*(512*a^2*b^20 - 1536*a^4*b^18 + 1536*a^6*b^16 - 512*a^8*b^14))/(8*
b^18))*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7) + (tan(c/2 + (d*x)/2)*(50*a*b^20 - 881*a^3*b^1
8 + 4436*a^5*b^16 - 10260*a^7*b^14 + 12800*a^9*b^12 - 8960*a^11*b^10 + 3328*a^13*b^8 - 512*a^15*b^6))/(8*b^18)
)*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i))/(16*b^7)))*(a^6*16i - b^6*5i + a^2*b^4*30i - a^4*b^2*40i)*1i
)/(8*b^7*d) - (a*atan(((a*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^2*b^18)/8 - (75*a^4*b^16)/2 + (325*a^6*b^14)/2
- 320*a^8*b^12 + 320*a^10*b^10 - 160*a^12*b^8 + 32*a^14*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(50*a*b^20 - 881*a^3*b
^18 + 4436*a^5*b^16 - 10260*a^7*b^14 + 12800*a^9*b^12 - 8960*a^11*b^10 + 3328*a^13*b^8 - 512*a^15*b^6))/(8*b^1
8) + (a*(-(a + b)^5*(a - b)^5)^(1/2)*((10*a*b^20 - 38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*b^14)/b^17 + (tan(c/2 +
(d*x)/2)*(512*a^2*b^20 - 1536*a^4*b^18 + 1536*a^6*b^16 - 512*a^8*b^14))/(8*b^18) + (a*(-(a + b)^5*(a - b)^5)^(
1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18)))/b^7))/b^7)*1i)/b^7 + (a*(-(a +
b)^5*(a - b)^5)^(1/2)*(((25*a^2*b^18)/8 - (75*a^4*b^16)/2 + (325*a^6*b^14)/2 - 320*a^8*b^12 + 320*a^10*b^10 -
160*a^12*b^8 + 32*a^14*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(50*a*b^20 - 881*a^3*b^18 + 4436*a^5*b^16 - 10260*a^7*b
^14 + 12800*a^9*b^12 - 8960*a^11*b^10 + 3328*a^13*b^8 - 512*a^15*b^6))/(8*b^18) - (a*(-(a + b)^5*(a - b)^5)^(1
/2)*((10*a*b^20 - 38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*b^14)/b^17 + (tan(c/2 + (d*x)/2)*(512*a^2*b^20 - 1536*a^4
*b^18 + 1536*a^6*b^16 - 512*a^8*b^14))/(8*b^18) - (a*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*
x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18)))/b^7))/b^7)*1i)/b^7)/((32*a^19 + (55*a^3*b^16)/4 - (585*a^5*b^14)
/4 + (2445*a^7*b^12)/4 - (5511*a^9*b^10)/4 + 1874*a^11*b^8 - 1595*a^13*b^6 + 836*a^15*b^4 - 248*a^17*b^2)/b^17
+ (tan(c/2 + (d*x)/2)*(512*a^20 - 50*a^2*b^18 + 750*a^4*b^16 - 4550*a^6*b^14 + 14770*a^8*b^12 - 28880*a^10*b^
10 + 35880*a^12*b^8 - 28672*a^14*b^6 + 14336*a^16*b^4 - 4096*a^18*b^2))/(4*b^18) - (a*(-(a + b)^5*(a - b)^5)^(
1/2)*(((25*a^2*b^18)/8 - (75*a^4*b^16)/2 + (325*a^6*b^14)/2 - 320*a^8*b^12 + 320*a^10*b^10 - 160*a^12*b^8 + 32
*a^14*b^6)/b^17 + (tan(c/2 + (d*x)/2)*(50*a*b^20 - 881*a^3*b^18 + 4436*a^5*b^16 - 10260*a^7*b^14 + 12800*a^9*b
^12 - 8960*a^11*b^10 + 3328*a^13*b^8 - 512*a^15*b^6))/(8*b^18) + (a*(-(a + b)^5*(a - b)^5)^(1/2)*((10*a*b^20 -
38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*b^14)/b^17 + (tan(c/2 + (d*x)/2)*(512*a^2*b^20 - 1536*a^4*b^18 + 1536*a^6*
b^16 - 512*a^8*b^14))/(8*b^18) + (a*(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^22
- 512*a^3*b^20))/(8*b^18)))/b^7))/b^7))/b^7 + (a*(-(a + b)^5*(a - b)^5)^(1/2)*(((25*a^2*b^18)/8 - (75*a^4*b^1
6)/2 + (325*a^6*b^14)/2 - 320*a^8*b^12 + 320*a^10*b^10 - 160*a^12*b^8 + 32*a^14*b^6)/b^17 + (tan(c/2 + (d*x)/2
)*(50*a*b^20 - 881*a^3*b^18 + 4436*a^5*b^16 - 10260*a^7*b^14 + 12800*a^9*b^12 - 8960*a^11*b^10 + 3328*a^13*b^8
- 512*a^15*b^6))/(8*b^18) - (a*(-(a + b)^5*(a - b)^5)^(1/2)*((10*a*b^20 - 38*a^3*b^18 + 44*a^5*b^16 - 16*a^7*
b^14)/b^17 + (tan(c/2 + (d*x)/2)*(512*a^2*b^20 - 1536*a^4*b^18 + 1536*a^6*b^16 - 512*a^8*b^14))/(8*b^18) - (a*
(-(a + b)^5*(a - b)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(768*a*b^22 - 512*a^3*b^20))/(8*b^18)))/b^7))/b
^7))/b^7))*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(b^7*d)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________