3.1258 \(\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=231 \[ -\frac{2 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac{\cos (c+d x) \left (4 \left (-7 a^2 b^2+6 a^4+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}+\frac{a x \left (-40 a^2 b^2+24 a^4+15 b^4\right )}{4 b^7}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))} \]

[Out]

(a*(24*a^4 - 40*a^2*b^2 + 15*b^4)*x)/(4*b^7) - (2*(a^2 - b^2)^(3/2)*(6*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/
2])/Sqrt[a^2 - b^2]])/(b^7*d) + (Cos[c + d*x]^5*(6*a + b*Sin[c + d*x]))/(5*b^2*d*(a + b*Sin[c + d*x])) - (Cos[
c + d*x]^3*(2*(6*a^2 - b^2) - 9*a*b*Sin[c + d*x]))/(6*b^4*d) + (Cos[c + d*x]*(4*(6*a^4 - 7*a^2*b^2 + b^4) - a*
b*(12*a^2 - 11*b^2)*Sin[c + d*x]))/(4*b^6*d)

________________________________________________________________________________________

Rubi [A]  time = 0.512611, antiderivative size = 231, 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} \[ -\frac{2 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac{\cos (c+d x) \left (4 \left (-7 a^2 b^2+6 a^4+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}+\frac{a x \left (-40 a^2 b^2+24 a^4+15 b^4\right )}{4 b^7}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]

[Out]

(a*(24*a^4 - 40*a^2*b^2 + 15*b^4)*x)/(4*b^7) - (2*(a^2 - b^2)^(3/2)*(6*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/
2])/Sqrt[a^2 - b^2]])/(b^7*d) + (Cos[c + d*x]^5*(6*a + b*Sin[c + d*x]))/(5*b^2*d*(a + b*Sin[c + d*x])) - (Cos[
c + d*x]^3*(2*(6*a^2 - b^2) - 9*a*b*Sin[c + d*x]))/(6*b^4*d) + (Cos[c + d*x]*(4*(6*a^4 - 7*a^2*b^2 + b^4) - a*
b*(12*a^2 - 11*b^2)*Sin[c + d*x]))/(4*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))^2} \, dx &=\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac{\int \frac{\cos ^4(c+d x) (-b-6 a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{b^2}\\ &=\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}-\frac{\int \frac{\cos ^2(c+d x) \left (2 b \left (3 a^2-2 b^2\right )+2 a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b^4}\\ &=\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac{\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}-\frac{\int \frac{-2 b \left (12 a^4-17 a^2 b^2+4 b^4\right )-2 a \left (24 a^4-40 a^2 b^2+15 b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{8 b^6}\\ &=\frac{a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac{\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}-\frac{\left (\left (a^2-b^2\right )^2 \left (6 a^2-b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^7}\\ &=\frac{a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac{\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}-\frac{\left (2 \left (a^2-b^2\right )^2 \left (6 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 )}{b^7 d}\\ &=\frac{a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac{\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}+\frac{\left (4 \left (a^2-b^2\right )^2 \left (6 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 )}{b^7 d}\\ &=\frac{a \left (24 a^4-40 a^2 b^2+15 b^4\right ) x}{4 b^7}-\frac{2 \left (a^2-b^2\right )^{3/2} \left (6 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 )}{b^7 d}+\frac{\cos ^5(c+d x) (6 a+b \sin (c+d x))}{5 b^2 d (a+b \sin (c+d x))}-\frac{\cos ^3(c+d x) \left (2 \left (6 a^2-b^2\right )-9 a b \sin (c+d x)\right )}{6 b^4 d}+\frac{\cos (c+d x) \left (4 \left (6 a^4-7 a^2 b^2+b^4\right )-a b \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{4 b^6 d}\\ \end{align*}

Mathematica [A]  time = 3.99716, size = 371, normalized size = 1.61 \[ \frac{\frac{720 a^4 b^2 \sin (2 (c+d x))-4800 a^3 b^3 c \sin (c+d x)-4800 a^3 b^3 d x \sin (c+d x)-1080 a^2 b^4 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))+60 a b \left (-74 a^2 b^2+48 a^4+23 b^4\right ) \cos (c+d x)+5 \left (24 a^3 b^3-31 a b^5\right ) \cos (3 (c+d x))-4800 a^4 b^2 c+1800 a^2 b^4 c-4800 a^4 b^2 d x+1800 a^2 b^4 d x+2880 a^5 b c \sin (c+d x)+2880 a^5 b d x \sin (c+d x)+2880 a^6 c+2880 a^6 d x+1800 a b^5 c \sin (c+d x)+1800 a b^5 d x \sin (c+d x)-9 a b^5 \cos (5 (c+d x))+295 b^6 \sin (2 (c+d x))+32 b^6 \sin (4 (c+d x))+3 b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}-960 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{480 b^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^6*Sin[c + d*x])/(a + b*Sin[c + d*x])^2,x]

[Out]

(-960*(a^2 - b^2)^(3/2)*(6*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + (2880*a^6*c - 4800*a^
4*b^2*c + 1800*a^2*b^4*c + 2880*a^6*d*x - 4800*a^4*b^2*d*x + 1800*a^2*b^4*d*x + 60*a*b*(48*a^4 - 74*a^2*b^2 +
23*b^4)*Cos[c + d*x] + 5*(24*a^3*b^3 - 31*a*b^5)*Cos[3*(c + d*x)] - 9*a*b^5*Cos[5*(c + d*x)] + 2880*a^5*b*c*Si
n[c + d*x] - 4800*a^3*b^3*c*Sin[c + d*x] + 1800*a*b^5*c*Sin[c + d*x] + 2880*a^5*b*d*x*Sin[c + d*x] - 4800*a^3*
b^3*d*x*Sin[c + d*x] + 1800*a*b^5*d*x*Sin[c + d*x] + 720*a^4*b^2*Sin[2*(c + d*x)] - 1080*a^2*b^4*Sin[2*(c + d*
x)] + 295*b^6*Sin[2*(c + d*x)] - 30*a^2*b^4*Sin[4*(c + d*x)] + 32*b^6*Sin[4*(c + d*x)] + 3*b^6*Sin[6*(c + d*x)
])/(a + b*Sin[c + d*x]))/(480*b^7*d)

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Maple [B]  time = 0.134, size = 1321, normalized size = 5.7 \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))^2,x)

[Out]

46/15/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5+2/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)-4/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)-20/d/b^5*arctan(tan(1/2*d*x+1/2*c))*a^3+15/2/d/b^3*arctan
(tan(1/2*d*x+1/2*c))*a+12/d/b^7*arctan(tan(1/2*d*x+1/2*c))*a^5+10/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^5*a^4-14/d/b^
4/(1+tan(1/2*d*x+1/2*c)^2)^5*a^2+2/d/b^2/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*a-16/d*a^2/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/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))+2/d/b/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)*tan(1/2*d*x+1/2*
c)+6/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^8+12/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2
*c)^6+56/3/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4+28/3/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2
*d*x+1/2*c)^2+10/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^8*a^4-18/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5
*tan(1/2*d*x+1/2*c)^8*a^2+2/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)*tan(1/2*d*x+1/2*c)-4/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)*tan(1/2*d*x+1/2*c)-12/d*a^6/b^7/(a^2-b^2)^(1/2)*arc
tan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))+26/d*a^4/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))+8/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7*a^3-5/d/b^3/(1+tan(1/2*d
*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7*a+40/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6*a^4-60/d/b^4/(1
+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^6*a^2+60/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4*a^4
-80/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4*a^2-8/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1
/2*c)^3*a^3+5/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^3*a+40/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(
1/2*d*x+1/2*c)^2*a^4-52/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2*a^2-4/d/b^5/(1+tan(1/2*d*x+1/2*c
)^2)^5*tan(1/2*d*x+1/2*c)*a^3+9/2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)*a+4/d/b^5/(1+tan(1/2*d*x
+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9*a^3-9/2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9*a

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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))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.26105, size = 1619, normalized size = 7.01 \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))^2,x, algorithm="fricas")

[Out]

[-1/60*(18*a*b^5*cos(d*x + c)^5 - 5*(12*a^3*b^3 - 11*a*b^5)*cos(d*x + c)^3 - 15*(24*a^6 - 40*a^4*b^2 + 15*a^2*
b^4)*d*x - 30*(6*a^5 - 7*a^3*b^2 + a*b^4 + (6*a^4*b - 7*a^2*b^3 + b^5)*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)) - 15*(24*a^5*b - 40*a^3*b^3 + 15*a*b^
5)*cos(d*x + c) - (12*b^6*cos(d*x + c)^5 - 10*(3*a^2*b^4 - 2*b^6)*cos(d*x + c)^3 + 15*(24*a^5*b - 40*a^3*b^3 +
15*a*b^5)*d*x + 15*(12*a^4*b^2 - 17*a^2*b^4 + 4*b^6)*cos(d*x + c))*sin(d*x + c))/(b^8*d*sin(d*x + c) + a*b^7*
d), -1/60*(18*a*b^5*cos(d*x + c)^5 - 5*(12*a^3*b^3 - 11*a*b^5)*cos(d*x + c)^3 - 15*(24*a^6 - 40*a^4*b^2 + 15*a
^2*b^4)*d*x - 60*(6*a^5 - 7*a^3*b^2 + a*b^4 + (6*a^4*b - 7*a^2*b^3 + b^5)*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))) - 15*(24*a^5*b - 40*a^3*b^3 + 15*a*b^5)*cos(d*x + c) -
(12*b^6*cos(d*x + c)^5 - 10*(3*a^2*b^4 - 2*b^6)*cos(d*x + c)^3 + 15*(24*a^5*b - 40*a^3*b^3 + 15*a*b^5)*d*x + 1
5*(12*a^4*b^2 - 17*a^2*b^4 + 4*b^6)*cos(d*x + c))*sin(d*x + c))/(b^8*d*sin(d*x + c) + a*b^7*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))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.24293, size = 801, normalized size = 3.47 \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))^2,x, algorithm="giac")

[Out]

1/60*(15*(24*a^5 - 40*a^3*b^2 + 15*a*b^4)*(d*x + c)/b^7 - 120*(6*a^6 - 13*a^4*b^2 + 8*a^2*b^4 - 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)
+ 120*(a^4*b*tan(1/2*d*x + 1/2*c) - 2*a^2*b^3*tan(1/2*d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/2*c) + a^5 - 2*a^3*b^
2 + a*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*b^6) + 2*(120*a^3*b*tan(1/2*d*x + 1/2*c)
^9 - 135*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 300*a^4*tan(1/2*d*x + 1/2*c)^8 - 540*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 +
180*b^4*tan(1/2*d*x + 1/2*c)^8 + 240*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 150*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 1200*a^
4*tan(1/2*d*x + 1/2*c)^6 - 1800*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 + 360*b^4*tan(1/2*d*x + 1/2*c)^6 + 1800*a^4*tan
(1/2*d*x + 1/2*c)^4 - 2400*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 560*b^4*tan(1/2*d*x + 1/2*c)^4 - 240*a^3*b*tan(1/2
*d*x + 1/2*c)^3 + 150*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 1200*a^4*tan(1/2*d*x + 1/2*c)^2 - 1560*a^2*b^2*tan(1/2*d*
x + 1/2*c)^2 + 280*b^4*tan(1/2*d*x + 1/2*c)^2 - 120*a^3*b*tan(1/2*d*x + 1/2*c) + 135*a*b^3*tan(1/2*d*x + 1/2*c
) + 300*a^4 - 420*a^2*b^2 + 92*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*b^6))/d