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

[Out]

1/4*a*(24*a^4-40*a^2*b^2+15*b^4)*x/b^7-2*(a^2-b^2)^(3/2)*(6*a^2-b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)
^(1/2))/b^7/d+1/5*cos(d*x+c)^5*(6*a+b*sin(d*x+c))/b^2/d/(a+b*sin(d*x+c))-1/6*cos(d*x+c)^3*(12*a^2-2*b^2-9*a*b*
sin(d*x+c))/b^4/d+1/4*cos(d*x+c)*(24*a^4-28*a^2*b^2+4*b^4-a*b*(12*a^2-11*b^2)*sin(d*x+c))/b^6/d

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Rubi [A]  time = 0.51, antiderivative size = 231, normalized size of antiderivative = 1.00, 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 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 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]

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*}

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Mathematica [A]  time = 4.35, size = 371, normalized size = 1.61 \[ \frac {\frac {2880 a^6 c+2880 a^6 d x+2880 a^5 b c \sin (c+d x)+2880 a^5 b d x \sin (c+d x)+720 a^4 b^2 \sin (2 (c+d x))-4800 a^4 b^2 c-4800 a^4 b^2 d x-4800 a^3 b^3 c \sin (c+d x)-4800 a^3 b^3 d x \sin (c+d x)+5 \left (24 a^3 b^3-31 a b^5\right ) \cos (3 (c+d x))-1080 a^2 b^4 \sin (2 (c+d x))-30 a^2 b^4 \sin (4 (c+d x))+1800 a^2 b^4 c+1800 a^2 b^4 d x+60 a b \left (48 a^4-74 a^2 b^2+23 b^4\right ) \cos (c+d x)+1800 a b^5 c \sin (c+d x)+1800 a b^5 d x \sin (c+d x)-9 a b^5 \cos (5 (c+d x))+295 b^6 \sin (2 (c+d x))+32 b^6 \sin (4 (c+d x))+3 b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}-960 \left (a^2-b^2\right )^{3/2} \left (6 a^2-b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{480 b^7 d} \]

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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fricas [A]  time = 1.24, size = 697, normalized size = 3.02 \[ \left [-\frac {18 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (12 \, a^{3} b^{3} - 11 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (24 \, a^{6} - 40 \, a^{4} b^{2} + 15 \, a^{2} b^{4}\right )} d x - 30 \, {\left (6 \, a^{5} - 7 \, a^{3} b^{2} + a b^{4} + {\left (6 \, a^{4} b - 7 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )\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 ) - 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right ) - {\left (12 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} d x + 15 \, {\left (12 \, a^{4} b^{2} - 17 \, a^{2} b^{4} + 4 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}}, -\frac {18 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (12 \, a^{3} b^{3} - 11 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (24 \, a^{6} - 40 \, a^{4} b^{2} + 15 \, a^{2} b^{4}\right )} d x - 60 \, {\left (6 \, a^{5} - 7 \, a^{3} b^{2} + a b^{4} + {\left (6 \, a^{4} b - 7 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )\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 ) - 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} \cos \left (d x + c\right ) - {\left (12 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, {\left (3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (24 \, a^{5} b - 40 \, a^{3} b^{3} + 15 \, a b^{5}\right )} d x + 15 \, {\left (12 \, a^{4} b^{2} - 17 \, a^{2} b^{4} + 4 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{60 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}}\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))^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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giac [B]  time = 0.23, size = 593, normalized size = 2.57 \[ \frac {\frac {15 \, {\left (24 \, a^{5} - 40 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{b^{7}} - \frac {120 \, {\left (6 \, a^{6} - 13 \, a^{4} b^{2} + 8 \, a^{2} b^{4} - b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{7}} + \frac {120 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} b^{6}} + \frac {2 \, {\left (120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 300 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 540 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 180 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1200 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1800 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1800 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2400 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 560 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1200 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1560 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 280 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 300 \, a^{4} - 420 \, a^{2} b^{2} + 92 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} b^{6}}}{60 \, d} \]

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

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maple [B]  time = 0.47, size = 1321, normalized size = 5.72 \[ \text {result too large to display} \]

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))^2,x)

[Out]

-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^3-20/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+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)+12/d/b^7*arctan(tan(1/2*d*x+1/2*c))*a^5+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+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))+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*a^4-14/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*a^2+2/d/b/(ta
n(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)+46/15/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5-9/2/d
/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9*a+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+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+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))-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*a^4/b^5/(tan(1/2*d*x+1/2*c)^2*a+2*t
an(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)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))

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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)^6*sin(d*x+c)/(a+b*sin(d*x+c))^2,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 = 14.53, size = 3135, normalized size = 13.57 \[ \text {result too large to display} \]

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))^2,x)

[Out]

((2*(38*a*b^4 + 90*a^5 - 135*a^3*b^2))/(15*b^6) - (tan(c/2 + (d*x)/2)^10*(a*b^4 - 12*a^5 + 14*a^3*b^2))/b^6 +
(2*tan(c/2 + (d*x)/2)^8*(9*a*b^4 + 30*a^5 - 41*a^3*b^2))/b^6 + (2*tan(c/2 + (d*x)/2)^4*(29*a*b^4 + 60*a^5 - 94
*a^3*b^2))/b^6 + (4*tan(c/2 + (d*x)/2)^6*(38*a*b^4 + 90*a^5 - 135*a^3*b^2))/(3*b^6) + (tan(c/2 + (d*x)/2)^2*(1
57*a*b^4 + 300*a^5 - 470*a^3*b^2))/(5*b^6) + (tan(c/2 + (d*x)/2)*(540*a^4 + 244*b^4 - 825*a^2*b^2))/(30*b^5) +
(tan(c/2 + (d*x)/2)^11*(12*a^4 + 4*b^4 - 17*a^2*b^2))/(2*b^5) + (tan(c/2 + (d*x)/2)^9*(84*a^4 + 44*b^4 - 131*
a^2*b^2))/(2*b^5) + (tan(c/2 + (d*x)/2)^7*(108*a^4 + 44*b^4 - 165*a^2*b^2))/b^5 + (tan(c/2 + (d*x)/2)^5*(396*a
^4 + 172*b^4 - 585*a^2*b^2))/(3*b^5) + (tan(c/2 + (d*x)/2)^3*(468*a^4 + 172*b^4 - 687*a^2*b^2))/(6*b^5))/(d*(a
+ 2*b*tan(c/2 + (d*x)/2) + 6*a*tan(c/2 + (d*x)/2)^2 + 15*a*tan(c/2 + (d*x)/2)^4 + 20*a*tan(c/2 + (d*x)/2)^6 +
15*a*tan(c/2 + (d*x)/2)^8 + 6*a*tan(c/2 + (d*x)/2)^10 + a*tan(c/2 + (d*x)/2)^12 + 10*b*tan(c/2 + (d*x)/2)^3 +
20*b*tan(c/2 + (d*x)/2)^5 + 20*b*tan(c/2 + (d*x)/2)^7 + 10*b*tan(c/2 + (d*x)/2)^9 + 2*b*tan(c/2 + (d*x)/2)^11
)) + (a*atan(((a*((2*(225*a^4*b^14 - 1200*a^6*b^12 + 2320*a^8*b^10 - 1920*a^10*b^8 + 576*a^12*b^6))/b^17 - (2*
tan(c/2 + (d*x)/2)*(16*a*b^18 - 706*a^3*b^16 + 4065*a^5*b^14 - 9360*a^7*b^12 + 10400*a^9*b^10 - 5568*a^11*b^8
+ 1152*a^13*b^6))/b^18 + (a*(24*a^4 + 15*b^4 - 40*a^2*b^2)*((2*tan(c/2 + (d*x)/2)*(32*a*b^20 - 256*a^3*b^18 +
416*a^5*b^16 - 192*a^7*b^14))/b^18 - (2*(44*a^2*b^18 - 92*a^4*b^16 + 48*a^6*b^14))/b^17 + (a*(32*a^2*b^3 + (2*
tan(c/2 + (d*x)/2)*(48*a*b^22 - 32*a^3*b^20))/b^18)*(24*a^4 + 15*b^4 - 40*a^2*b^2)*1i)/(4*b^7))*1i)/(4*b^7))*(
24*a^4 + 15*b^4 - 40*a^2*b^2))/(4*b^7) + (a*((2*(225*a^4*b^14 - 1200*a^6*b^12 + 2320*a^8*b^10 - 1920*a^10*b^8
+ 576*a^12*b^6))/b^17 - (2*tan(c/2 + (d*x)/2)*(16*a*b^18 - 706*a^3*b^16 + 4065*a^5*b^14 - 9360*a^7*b^12 + 1040
0*a^9*b^10 - 5568*a^11*b^8 + 1152*a^13*b^6))/b^18 + (a*(24*a^4 + 15*b^4 - 40*a^2*b^2)*((2*(44*a^2*b^18 - 92*a^
4*b^16 + 48*a^6*b^14))/b^17 - (2*tan(c/2 + (d*x)/2)*(32*a*b^20 - 256*a^3*b^18 + 416*a^5*b^16 - 192*a^7*b^14))/
b^18 + (a*(32*a^2*b^3 + (2*tan(c/2 + (d*x)/2)*(48*a*b^22 - 32*a^3*b^20))/b^18)*(24*a^4 + 15*b^4 - 40*a^2*b^2)*
1i)/(4*b^7))*1i)/(4*b^7))*(24*a^4 + 15*b^4 - 40*a^2*b^2))/(4*b^7))/((4*(1728*a^16 - 60*a^2*b^14 + 895*a^4*b^12
- 5056*a^6*b^10 + 14291*a^8*b^8 - 22310*a^10*b^6 + 19584*a^12*b^4 - 9072*a^14*b^2))/b^17 + (4*tan(c/2 + (d*x)
/2)*(6912*a^17 - 450*a^3*b^14 + 6000*a^5*b^12 - 29690*a^7*b^10 + 74860*a^9*b^8 - 106592*a^11*b^6 + 86976*a^13*
b^4 - 38016*a^15*b^2))/b^18 - (a*((2*(225*a^4*b^14 - 1200*a^6*b^12 + 2320*a^8*b^10 - 1920*a^10*b^8 + 576*a^12*
b^6))/b^17 - (2*tan(c/2 + (d*x)/2)*(16*a*b^18 - 706*a^3*b^16 + 4065*a^5*b^14 - 9360*a^7*b^12 + 10400*a^9*b^10
- 5568*a^11*b^8 + 1152*a^13*b^6))/b^18 + (a*(24*a^4 + 15*b^4 - 40*a^2*b^2)*((2*tan(c/2 + (d*x)/2)*(32*a*b^20 -
256*a^3*b^18 + 416*a^5*b^16 - 192*a^7*b^14))/b^18 - (2*(44*a^2*b^18 - 92*a^4*b^16 + 48*a^6*b^14))/b^17 + (a*(
32*a^2*b^3 + (2*tan(c/2 + (d*x)/2)*(48*a*b^22 - 32*a^3*b^20))/b^18)*(24*a^4 + 15*b^4 - 40*a^2*b^2)*1i)/(4*b^7)
)*1i)/(4*b^7))*(24*a^4 + 15*b^4 - 40*a^2*b^2)*1i)/(4*b^7) + (a*((2*(225*a^4*b^14 - 1200*a^6*b^12 + 2320*a^8*b^
10 - 1920*a^10*b^8 + 576*a^12*b^6))/b^17 - (2*tan(c/2 + (d*x)/2)*(16*a*b^18 - 706*a^3*b^16 + 4065*a^5*b^14 - 9
360*a^7*b^12 + 10400*a^9*b^10 - 5568*a^11*b^8 + 1152*a^13*b^6))/b^18 + (a*(24*a^4 + 15*b^4 - 40*a^2*b^2)*((2*(
44*a^2*b^18 - 92*a^4*b^16 + 48*a^6*b^14))/b^17 - (2*tan(c/2 + (d*x)/2)*(32*a*b^20 - 256*a^3*b^18 + 416*a^5*b^1
6 - 192*a^7*b^14))/b^18 + (a*(32*a^2*b^3 + (2*tan(c/2 + (d*x)/2)*(48*a*b^22 - 32*a^3*b^20))/b^18)*(24*a^4 + 15
*b^4 - 40*a^2*b^2)*1i)/(4*b^7))*1i)/(4*b^7))*(24*a^4 + 15*b^4 - 40*a^2*b^2)*1i)/(4*b^7)))*(24*a^4 + 15*b^4 - 4
0*a^2*b^2))/(2*b^7*d) - (2*atanh((64*a^2*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(64*a^2*b^3 - 572*a^4*b +
(1312*a^6)/b - (1164*a^8)/b^3 + (360*a^10)/b^5 + 2624*a^5*tan(c/2 + (d*x)/2) + 128*a*b^4*tan(c/2 + (d*x)/2) -
1144*a^3*b^2*tan(c/2 + (d*x)/2) - (2328*a^7*tan(c/2 + (d*x)/2))/b^2 + (720*a^9*tan(c/2 + (d*x)/2))/b^4) - (444
*a^4*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(1312*a^6*b + 64*a^2*b^5 - 572*a^4*b^3 - (1164*a^8)/b + (360*a
^10)/b^3 - 2328*a^7*tan(c/2 + (d*x)/2) + 128*a*b^6*tan(c/2 + (d*x)/2) - 1144*a^3*b^4*tan(c/2 + (d*x)/2) + 2624
*a^5*b^2*tan(c/2 + (d*x)/2) + (720*a^9*tan(c/2 + (d*x)/2))/b^2) + (360*a^6*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)
^(1/2))/(64*a^2*b^7 - 1164*a^8*b - 572*a^4*b^5 + 1312*a^6*b^3 + (360*a^10)/b + 720*a^9*tan(c/2 + (d*x)/2) + 12
8*a*b^8*tan(c/2 + (d*x)/2) - 1144*a^3*b^6*tan(c/2 + (d*x)/2) + 2624*a^5*b^4*tan(c/2 + (d*x)/2) - 2328*a^7*b^2*
tan(c/2 + (d*x)/2)) + (128*a*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(64*a^2*b^2 - 572*a
^4 + (1312*a^6)/b^2 - (1164*a^8)/b^4 + (360*a^10)/b^6 + 128*a*b^3*tan(c/2 + (d*x)/2) - 1144*a^3*b*tan(c/2 + (d
*x)/2) + (2624*a^5*tan(c/2 + (d*x)/2))/b - (2328*a^7*tan(c/2 + (d*x)/2))/b^3 + (720*a^9*tan(c/2 + (d*x)/2))/b^
5) - (952*a^3*tan(c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(1312*a^6 + 64*a^2*b^4 - 572*a^4*b
^2 - (1164*a^8)/b^2 + (360*a^10)/b^4 + 128*a*b^5*tan(c/2 + (d*x)/2) + 2624*a^5*b*tan(c/2 + (d*x)/2) - 1144*a^3
*b^3*tan(c/2 + (d*x)/2) - (2328*a^7*tan(c/2 + (d*x)/2))/b + (720*a^9*tan(c/2 + (d*x)/2))/b^3) + (1164*a^5*tan(
c/2 + (d*x)/2)*(b^6 - a^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(64*a^2*b^6 - 1164*a^8 - 572*a^4*b^4 + 1312*a^6*b^2
+ (360*a^10)/b^2 + 128*a*b^7*tan(c/2 + (d*x)/2) - 2328*a^7*b*tan(c/2 + (d*x)/2) - 1144*a^3*b^5*tan(c/2 + (d*x)
/2) + 2624*a^5*b^3*tan(c/2 + (d*x)/2) + (720*a^9*tan(c/2 + (d*x)/2))/b) - (360*a^7*tan(c/2 + (d*x)/2)*(b^6 - a
^6 - 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(360*a^10 + 64*a^2*b^8 - 572*a^4*b^6 + 1312*a^6*b^4 - 1164*a^8*b^2 + 128*a*
b^9*tan(c/2 + (d*x)/2) + 720*a^9*b*tan(c/2 + (d*x)/2) - 1144*a^3*b^7*tan(c/2 + (d*x)/2) + 2624*a^5*b^5*tan(c/2
+ (d*x)/2) - 2328*a^7*b^3*tan(c/2 + (d*x)/2)))*(6*a^2 - b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(b^7*d)

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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)**6*sin(d*x+c)/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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