3.1268 \(\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=237 \[ \frac{15 a \left (-3 a^2 b^2+2 a^4+b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d \sqrt{a^2-b^2}}+\frac{5 \cos ^3(c+d x) \left (4 a^2+a b \sin (c+d x)-b^2\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}-\frac{15 x \left (-8 a^2 b^2+8 a^4+b^4\right )}{8 b^7}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2} \]

[Out]

(-15*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(8*b^7) + (15*a*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sq
rt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^5*(3*a + b*Sin[c + d*x]))/(4*b^2*d*(a + b*Sin[c + d*x]
)^2) + (5*Cos[c + d*x]^3*(4*a^2 - b^2 + a*b*Sin[c + d*x]))/(4*b^4*d*(a + b*Sin[c + d*x])) - (15*Cos[c + d*x]*(
4*a*(2*a^2 - b^2) - b*(4*a^2 - b^2)*Sin[c + d*x]))/(8*b^6*d)

________________________________________________________________________________________

Rubi [A]  time = 0.465455, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2863, 2865, 2735, 2660, 618, 204} \[ \frac{15 a \left (-3 a^2 b^2+2 a^4+b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d \sqrt{a^2-b^2}}+\frac{5 \cos ^3(c+d x) \left (4 a^2+a b \sin (c+d x)-b^2\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}-\frac{15 x \left (-8 a^2 b^2+8 a^4+b^4\right )}{8 b^7}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

(-15*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(8*b^7) + (15*a*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sq
rt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) + (Cos[c + d*x]^5*(3*a + b*Sin[c + d*x]))/(4*b^2*d*(a + b*Sin[c + d*x]
)^2) + (5*Cos[c + d*x]^3*(4*a^2 - b^2 + a*b*Sin[c + d*x]))/(4*b^4*d*(a + b*Sin[c + d*x])) - (15*Cos[c + d*x]*(
4*a*(2*a^2 - b^2) - b*(4*a^2 - b^2)*Sin[c + d*x]))/(8*b^6*d)

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]

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))^3} \, dx &=\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}-\frac{5 \int \frac{\cos ^4(c+d x) (-2 b-6 a \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx}{8 b^2}\\ &=\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac{5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}+\frac{5 \int \frac{\cos ^2(c+d x) \left (6 a b+6 \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{8 b^4}\\ &=\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac{5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}+\frac{5 \int \frac{-6 a b \left (4 a^2-3 b^2\right )-6 \left (8 a^4-8 a^2 b^2+b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{16 b^6}\\ &=-\frac{15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac{5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}+\frac{\left (15 a \left (2 a^4-3 a^2 b^2+b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 b^7}\\ &=-\frac{15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac{5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}+\frac{\left (15 a \left (2 a^4-3 a^2 b^2+b^4\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 )}{b^7 d}\\ &=-\frac{15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac{5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}-\frac{\left (30 a \left (2 a^4-3 a^2 b^2+b^4\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 )}{b^7 d}\\ &=-\frac{15 \left (8 a^4-8 a^2 b^2+b^4\right ) x}{8 b^7}+\frac{15 a \left (2 a^4-3 a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{b^7 \sqrt{a^2-b^2} d}+\frac{\cos ^5(c+d x) (3 a+b \sin (c+d x))}{4 b^2 d (a+b \sin (c+d x))^2}+\frac{5 \cos ^3(c+d x) \left (4 a^2-b^2+a b \sin (c+d x)\right )}{4 b^4 d (a+b \sin (c+d x))}-\frac{15 \cos (c+d x) \left (4 a \left (2 a^2-b^2\right )-b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{8 b^6 d}\\ \end{align*}

Mathematica [B]  time = 8.28841, size = 1250, normalized size = 5.27 \[ \frac{\frac{18 \left (-8 (c+d x)+\frac{2 a \left (8 a^4-20 b^2 a^2+15 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac{3 b \left (4 a^4-7 b^2 a^2+2 b^4\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a b \left (4 a^2-3 b^2\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}\right )}{b^3}-\frac{10 \left (\frac{6 a b \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+\frac{\cos (c+d x) \left (a \left (2 a^2+b^2\right )+b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}\right )}{(a-b)^2 (a+b)^2}+\frac{10 \left (-8 \sin (2 (c+d x)) b^2+96 a \cos (c+d x) b+\frac{\left (112 a^6-220 b^2 a^4+115 b^4 a^2-10 b^6\right ) \cos (c+d x) b}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\frac{a \left (-16 a^4+20 b^2 a^2-5 b^4\right ) \cos (c+d x) b}{(a-b) (a+b) (a+b \sin (c+d x))^2}-24 \left (b^2-8 a^2\right ) (c+d x)-\frac{6 a \left (64 a^6-168 b^2 a^4+140 b^4 a^2-35 b^6\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}\right )}{b^5}+\frac{\frac{12 a \left (640 a^8-1920 b^2 a^6+2016 b^4 a^4-840 b^6 a^2+105 b^8\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{-3840 (c+d x) a^{10}-3840 b \cos (c+d x) a^9-7680 b (c+d x) \sin (c+d x) a^9+7680 b^2 (c+d x) a^8+1920 b^2 (c+d x) \cos (2 (c+d x)) a^8-2880 b^2 \sin (2 (c+d x)) a^8+8640 b^3 \cos (c+d x) a^7+320 b^3 \cos (3 (c+d x)) a^7+19200 b^3 (c+d x) \sin (c+d x) a^7-2976 b^4 (c+d x) a^6-4800 b^4 (c+d x) \cos (2 (c+d x)) a^6+6880 b^4 \sin (2 (c+d x)) a^6-40 b^4 \sin (4 (c+d x)) a^6-5696 b^5 \cos (c+d x) a^5-760 b^5 \cos (3 (c+d x)) a^5-8 b^5 \cos (5 (c+d x)) a^5-15552 b^5 (c+d x) \sin (c+d x) a^5-1776 b^6 (c+d x) a^4+3888 b^6 (c+d x) \cos (2 (c+d x)) a^4-5182 b^6 \sin (2 (c+d x)) a^4+88 b^6 \sin (4 (c+d x)) a^4+2 b^6 \sin (6 (c+d x)) a^4+788 b^7 \cos (c+d x) a^3+560 b^7 \cos (3 (c+d x)) a^3+16 b^7 \cos (5 (c+d x)) a^3+4224 b^7 (c+d x) \sin (c+d x) a^3+960 b^8 (c+d x) a^2-1056 b^8 (c+d x) \cos (2 (c+d x)) a^2+1221 b^8 \sin (2 (c+d x)) a^2-56 b^8 \sin (4 (c+d x)) a^2-4 b^8 \sin (6 (c+d x)) a^2+114 b^9 \cos (c+d x) a-120 b^9 \cos (3 (c+d x)) a-8 b^9 \cos (5 (c+d x)) a-192 b^9 (c+d x) \sin (c+d x) a-48 b^{10} (c+d x)+48 b^{10} (c+d x) \cos (2 (c+d x))-36 b^{10} \sin (2 (c+d x))+8 b^{10} \sin (4 (c+d x))+2 b^{10} \sin (6 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}}{b^7}}{256 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

((18*(-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2
- b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*
a^2*b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/b^3 - (10*((6*a*b*ArcTan[(b + a*Ta
n[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d
*x]))/(a + b*Sin[c + d*x])^2))/((a - b)^2*(a + b)^2) + (10*(-24*(-8*a^2 + b^2)*(c + d*x) - (6*a*(64*a^6 - 168*
a^4*b^2 + 140*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 96*a*b*C
os[c + d*x] + (a*b*(-16*a^4 + 20*a^2*b^2 - 5*b^4)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) + (b*
(112*a^6 - 220*a^4*b^2 + 115*a^2*b^4 - 10*b^6)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) - 8*b^
2*Sin[2*(c + d*x)]))/b^5 + ((12*a*(640*a^8 - 1920*a^6*b^2 + 2016*a^4*b^4 - 840*a^2*b^6 + 105*b^8)*ArcTan[(b +
a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (-3840*a^10*(c + d*x) + 7680*a^8*b^2*(c + d*x) - 297
6*a^6*b^4*(c + d*x) - 1776*a^4*b^6*(c + d*x) + 960*a^2*b^8*(c + d*x) - 48*b^10*(c + d*x) - 3840*a^9*b*Cos[c +
d*x] + 8640*a^7*b^3*Cos[c + d*x] - 5696*a^5*b^5*Cos[c + d*x] + 788*a^3*b^7*Cos[c + d*x] + 114*a*b^9*Cos[c + d*
x] + 1920*a^8*b^2*(c + d*x)*Cos[2*(c + d*x)] - 4800*a^6*b^4*(c + d*x)*Cos[2*(c + d*x)] + 3888*a^4*b^6*(c + d*x
)*Cos[2*(c + d*x)] - 1056*a^2*b^8*(c + d*x)*Cos[2*(c + d*x)] + 48*b^10*(c + d*x)*Cos[2*(c + d*x)] + 320*a^7*b^
3*Cos[3*(c + d*x)] - 760*a^5*b^5*Cos[3*(c + d*x)] + 560*a^3*b^7*Cos[3*(c + d*x)] - 120*a*b^9*Cos[3*(c + d*x)]
- 8*a^5*b^5*Cos[5*(c + d*x)] + 16*a^3*b^7*Cos[5*(c + d*x)] - 8*a*b^9*Cos[5*(c + d*x)] - 7680*a^9*b*(c + d*x)*S
in[c + d*x] + 19200*a^7*b^3*(c + d*x)*Sin[c + d*x] - 15552*a^5*b^5*(c + d*x)*Sin[c + d*x] + 4224*a^3*b^7*(c +
d*x)*Sin[c + d*x] - 192*a*b^9*(c + d*x)*Sin[c + d*x] - 2880*a^8*b^2*Sin[2*(c + d*x)] + 6880*a^6*b^4*Sin[2*(c +
d*x)] - 5182*a^4*b^6*Sin[2*(c + d*x)] + 1221*a^2*b^8*Sin[2*(c + d*x)] - 36*b^10*Sin[2*(c + d*x)] - 40*a^6*b^4
*Sin[4*(c + d*x)] + 88*a^4*b^6*Sin[4*(c + d*x)] - 56*a^2*b^8*Sin[4*(c + d*x)] + 8*b^10*Sin[4*(c + d*x)] + 2*a^
4*b^6*Sin[6*(c + d*x)] - 4*a^2*b^8*Sin[6*(c + d*x)] + 2*b^10*Sin[6*(c + d*x)])/((a^2 - b^2)^2*(a + b*Sin[c + d
*x])^2))/b^7)/(256*d)

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Maple [B]  time = 0.157, size = 1325, normalized size = 5.6 \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))^3,x)

[Out]

-6/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a^2-15/4/d/b^3*arctan(tan(1/2*d*x+1/2*c))-30/d/b^7*ar
ctan(tan(1/2*d*x+1/2*c))*a^4+30/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^2-10/d*a^5/b^6/(tan(1/2*d*x+1/2*c)^2*a+2*ta
n(1/2*d*x+1/2*c)*b+a)^2+11/d*a^3/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2-2/d/(tan(1/2*d*x+1/2*
c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2/a*tan(1/2*d*x+1/2*c)^2-4/d/b/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b
+a)^2*tan(1/2*d*x+1/2*c)-1/d/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*a+9/4/d/b^3/(1+tan(1/2*d*
x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7+1/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5-1/4/d/b^3/(1+tan(
1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3-9/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)-20/d/b^6/(1+t
an(1/2*d*x+1/2*c)^2)^4*a^3+14/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*a-60/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d
*x+1/2*c)^4*a^3+42/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^4*a+6/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*
tan(1/2*d*x+1/2*c)^3*a^2-60/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^2*a^3+38/d/b^4/(1+tan(1/2*d*x+
1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^2*a+6/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*a^2-6/d/b^5/(1+tan(1/
2*d*x+1/2*c)^2)^4*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)^4*tan(1/2*d*x+1/2*c)^6*a^3+18/d/b
^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^6*a+21/d*a/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b
+a)^2*tan(1/2*d*x+1/2*c)^2-31/d*a^4/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)
+35/d*a^2/b^3/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)+30/d*a^5/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))-45/d*a^3/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))+15/d*a/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))-9/d*a^4/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3+9/d*a^2/b^3/(t
an(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3-10/d*a^5/b^6/(tan(1/2*d*x+1/2*c)^2*a+2*
tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2-9/d*a^3/b^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2
*tan(1/2*d*x+1/2*c)^2

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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(cos(d*x+c)^6*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.91075, size = 1864, normalized size = 7.86 \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))^3,x, algorithm="fricas")

[Out]

[1/8*(4*a*b^5*cos(d*x + c)^5 - 15*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*d*x*cos(d*x + c)^2 - 10*(4*a^3*b^3 - 3*a*b^5)*
cos(d*x + c)^3 + 15*(8*a^6 - 7*a^2*b^4 + b^6)*d*x + 30*(2*a^5 + a^3*b^2 - a*b^4 - (2*a^3*b^2 - a*b^4)*cos(d*x
+ c)^2 + 2*(2*a^4*b - a^2*b^3)*sin(d*x + c))*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)) + 30*(4*a^5*b - 2*a^3*b^3 - a*b^5)*cos(d*x + c) - (2*b^6*cos(d*x + c)^5 - 5*(
2*a^2*b^4 - b^6)*cos(d*x + c)^3 - 30*(8*a^5*b - 8*a^3*b^3 + a*b^5)*d*x - 15*(12*a^4*b^2 - 11*a^2*b^4 + b^6)*co
s(d*x + c))*sin(d*x + c))/(b^9*d*cos(d*x + c)^2 - 2*a*b^8*d*sin(d*x + c) - (a^2*b^7 + b^9)*d), 1/8*(4*a*b^5*co
s(d*x + c)^5 - 15*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*d*x*cos(d*x + c)^2 - 10*(4*a^3*b^3 - 3*a*b^5)*cos(d*x + c)^3 +
15*(8*a^6 - 7*a^2*b^4 + b^6)*d*x + 60*(2*a^5 + a^3*b^2 - a*b^4 - (2*a^3*b^2 - a*b^4)*cos(d*x + c)^2 + 2*(2*a^
4*b - a^2*b^3)*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))) + 30
*(4*a^5*b - 2*a^3*b^3 - a*b^5)*cos(d*x + c) - (2*b^6*cos(d*x + c)^5 - 5*(2*a^2*b^4 - b^6)*cos(d*x + c)^3 - 30*
(8*a^5*b - 8*a^3*b^3 + a*b^5)*d*x - 15*(12*a^4*b^2 - 11*a^2*b^4 + b^6)*cos(d*x + c))*sin(d*x + c))/(b^9*d*cos(
d*x + c)^2 - 2*a*b^8*d*sin(d*x + c) - (a^2*b^7 + b^9)*d)]

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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(cos(d*x+c)**6*sin(d*x+c)/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.30803, size = 784, normalized size = 3.31 \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))^3,x, algorithm="giac")

[Out]

-1/8*(15*(8*a^4 - 8*a^2*b^2 + b^4)*(d*x + c)/b^7 - 120*(2*a^5 - 3*a^3*b^2 + a*b^4)*(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) + 8*(9*a^5*b*tan(1
/2*d*x + 1/2*c)^3 - 9*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 10*a^6*tan(1/2*d*x + 1/2*c)^2 + 9*a^4*b^2*tan(1/2*d*x +
1/2*c)^2 - 21*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 2*b^6*tan(1/2*d*x + 1/2*c)^2 + 31*a^5*b*tan(1/2*d*x + 1/2*c) -
35*a^3*b^3*tan(1/2*d*x + 1/2*c) + 4*a*b^5*tan(1/2*d*x + 1/2*c) + 10*a^6 - 11*a^4*b^2 + a^2*b^4)/((a*tan(1/2*d
*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a*b^6) + 2*(24*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 9*b^3*tan(1/2*d*
x + 1/2*c)^7 + 80*a^3*tan(1/2*d*x + 1/2*c)^6 - 72*a*b^2*tan(1/2*d*x + 1/2*c)^6 + 24*a^2*b*tan(1/2*d*x + 1/2*c)
^5 - b^3*tan(1/2*d*x + 1/2*c)^5 + 240*a^3*tan(1/2*d*x + 1/2*c)^4 - 168*a*b^2*tan(1/2*d*x + 1/2*c)^4 - 24*a^2*b
*tan(1/2*d*x + 1/2*c)^3 + b^3*tan(1/2*d*x + 1/2*c)^3 + 240*a^3*tan(1/2*d*x + 1/2*c)^2 - 152*a*b^2*tan(1/2*d*x
+ 1/2*c)^2 - 24*a^2*b*tan(1/2*d*x + 1/2*c) + 9*b^3*tan(1/2*d*x + 1/2*c) + 80*a^3 - 56*a*b^2)/((tan(1/2*d*x + 1
/2*c)^2 + 1)^4*b^6))/d