3.1127 \(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=307 \[ -\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {3 a \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^3 d}+\frac {6 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d \sqrt {a^2-b^2}}-\frac {3 a x \left (8 a^4-8 a^2 b^2+b^4\right )}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d} \]

[Out]

-3/4*a*(8*a^4-8*a^2*b^2+b^4)*x/b^7-1/5*(30*a^4-25*a^2*b^2+b^4)*cos(d*x+c)/b^6/d+3/4*a*(4*a^2-3*b^2)*cos(d*x+c)
*sin(d*x+c)/b^5/d-1/5*(10*a^2-7*b^2)*cos(d*x+c)*sin(d*x+c)^2/b^4/d+1/2*(3*a^2-2*b^2)*cos(d*x+c)*sin(d*x+c)^3/a
/b^3/d-1/5*cos(d*x+c)*sin(d*x+c)^4/b^2/d-(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/a/b^2/d/(a+b*sin(d*x+c))+6*a^2*(2*a
^4-3*a^2*b^2+b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^7/d/(a^2-b^2)^(1/2)

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Rubi [A]  time = 1.02, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2892, 3049, 3023, 2735, 2660, 618, 204} \[ -\frac {\left (-25 a^2 b^2+30 a^4+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {6 a^2 \left (-3 a^2 b^2+2 a^4+b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^7 d \sqrt {a^2-b^2}}-\frac {\left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\left (3 a^2-2 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{2 a b^3 d}-\frac {\left (10 a^2-7 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{5 b^4 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 b^5 d}-\frac {3 a x \left (-8 a^2 b^2+8 a^4+b^4\right )}{4 b^7}-\frac {\sin ^4(c+d x) \cos (c+d x)}{5 b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]

[Out]

(-3*a*(8*a^4 - 8*a^2*b^2 + b^4)*x)/(4*b^7) + (6*a^2*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/
Sqrt[a^2 - b^2]])/(b^7*Sqrt[a^2 - b^2]*d) - ((30*a^4 - 25*a^2*b^2 + b^4)*Cos[c + d*x])/(5*b^6*d) + (3*a*(4*a^2
- 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(4*b^5*d) - ((10*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(5*b^4*d) + ((
3*a^2 - 2*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(2*a*b^3*d) - (Cos[c + d*x]*Sin[c + d*x]^4)/(5*b^2*d) - ((a^2 - b^
2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*(a + b*Sin[c + d*x]))

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 2892

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[((a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*b^2*d*
f*(m + 1)), x] + (-Dist[1/(a*b^2*(m + 1)*(m + n + 4)), Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n*Sim
p[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m
+ n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] - Simp[(Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e +
f*x])^(n + 1))/(b^2*d*f*(m + n + 4)), x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2
*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && NeQ[m + n + 4, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rubi steps

\begin {align*} \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^3(c+d x) \left (3 \left (8 a^2-5 b^2\right )-a b \sin (c+d x)-10 \left (3 a^2-2 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{5 a b^2}\\ &=\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin ^2(c+d x) \left (-30 a \left (3 a^2-2 b^2\right )+6 a^2 b \sin (c+d x)+12 a \left (10 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{20 a b^3}\\ &=-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\sin (c+d x) \left (24 a^2 \left (10 a^2-7 b^2\right )-6 a b \left (5 a^2-2 b^2\right ) \sin (c+d x)-90 a^2 \left (4 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60 a b^4}\\ &=\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {-90 a^3 \left (4 a^2-3 b^2\right )+6 a^2 b \left (20 a^2-11 b^2\right ) \sin (c+d x)+24 a \left (30 a^4-25 a^2 b^2+b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{120 a b^5}\\ &=-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {-90 a^3 b \left (4 a^2-3 b^2\right )-90 a^2 \left (8 a^4-8 a^2 b^2+b^4\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{120 a b^6}\\ &=-\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\left (3 a^2 \left (2 a^4-3 a^2 b^2+b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^7}\\ &=-\frac {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}+\frac {\left (6 a^2 \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 {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}-\frac {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}-\frac {\left (12 a^2 \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 {3 a \left (8 a^4-8 a^2 b^2+b^4\right ) x}{4 b^7}+\frac {6 a^2 \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 {\left (30 a^4-25 a^2 b^2+b^4\right ) \cos (c+d x)}{5 b^6 d}+\frac {3 a \left (4 a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 b^5 d}-\frac {\left (10 a^2-7 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{5 b^4 d}+\frac {\left (3 a^2-2 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{2 a b^3 d}-\frac {\cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac {\left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 4.57, size = 378, normalized size = 1.23 \[ \frac {\frac {960 a^2 \left (2 a^4-3 a^2 b^2+b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {960 a^6 c+960 a^6 d x+960 a^5 b c \sin (c+d x)+960 a^5 b d x \sin (c+d x)+240 a^4 b^2 \sin (2 (c+d x))-960 a^4 b^2 c-960 a^4 b^2 d x-960 a^3 b^3 c \sin (c+d x)-960 a^3 b^3 d x \sin (c+d x)+5 \left (8 a^3 b^3-5 a b^5\right ) \cos (3 (c+d x))-200 a^2 b^4 \sin (2 (c+d x))-10 a^2 b^4 \sin (4 (c+d x))+120 a^2 b^4 c+120 a^2 b^4 d x+60 a b \left (16 a^4-14 a^2 b^2+b^4\right ) \cos (c+d x)+120 a b^5 c \sin (c+d x)+120 a b^5 d x \sin (c+d x)-3 a b^5 \cos (5 (c+d x))+5 b^6 \sin (2 (c+d x))+4 b^6 \sin (4 (c+d x))+b^6 \sin (6 (c+d x))}{a+b \sin (c+d x)}}{160 b^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^2,x]

[Out]

((960*a^2*(2*a^4 - 3*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (960*a
^6*c - 960*a^4*b^2*c + 120*a^2*b^4*c + 960*a^6*d*x - 960*a^4*b^2*d*x + 120*a^2*b^4*d*x + 60*a*b*(16*a^4 - 14*a
^2*b^2 + b^4)*Cos[c + d*x] + 5*(8*a^3*b^3 - 5*a*b^5)*Cos[3*(c + d*x)] - 3*a*b^5*Cos[5*(c + d*x)] + 960*a^5*b*c
*Sin[c + d*x] - 960*a^3*b^3*c*Sin[c + d*x] + 120*a*b^5*c*Sin[c + d*x] + 960*a^5*b*d*x*Sin[c + d*x] - 960*a^3*b
^3*d*x*Sin[c + d*x] + 120*a*b^5*d*x*Sin[c + d*x] + 240*a^4*b^2*Sin[2*(c + d*x)] - 200*a^2*b^4*Sin[2*(c + d*x)]
+ 5*b^6*Sin[2*(c + d*x)] - 10*a^2*b^4*Sin[4*(c + d*x)] + 4*b^6*Sin[4*(c + d*x)] + b^6*Sin[6*(c + d*x)])/(a +
b*Sin[c + d*x]))/(160*b^7*d)

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fricas [A]  time = 0.83, size = 649, normalized size = 2.11 \[ \left [\frac {6 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{6} - 8 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x - 30 \, {\left (2 \, a^{5} - a^{3} b^{2} + {\left (2 \, a^{4} b - a^{2} b^{3}\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 (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - {\left (4 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, a^{2} b^{4} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} d x + 15 \, {\left (4 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}}, \frac {6 \, a b^{5} \cos \left (d x + c\right )^{5} - 5 \, {\left (4 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{6} - 8 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x - 60 \, {\left (2 \, a^{5} - a^{3} b^{2} + {\left (2 \, a^{4} b - a^{2} b^{3}\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 (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - {\left (4 \, b^{6} \cos \left (d x + c\right )^{5} - 10 \, a^{2} b^{4} \cos \left (d x + c\right )^{3} + 15 \, {\left (8 \, a^{5} b - 8 \, a^{3} b^{3} + a b^{5}\right )} d x + 15 \, {\left (4 \, a^{4} b^{2} - 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20 \, {\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)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[1/20*(6*a*b^5*cos(d*x + c)^5 - 5*(4*a^3*b^3 - a*b^5)*cos(d*x + c)^3 - 15*(8*a^6 - 8*a^4*b^2 + a^2*b^4)*d*x -
30*(2*a^5 - a^3*b^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)) - 15*(8*a^5*b - 8*a^3*b^3 + a*b^5)*cos(d*x + c) - (4*b^6*cos(d*x +
c)^5 - 10*a^2*b^4*cos(d*x + c)^3 + 15*(8*a^5*b - 8*a^3*b^3 + a*b^5)*d*x + 15*(4*a^4*b^2 - 3*a^2*b^4)*cos(d*x
+ c))*sin(d*x + c))/(b^8*d*sin(d*x + c) + a*b^7*d), 1/20*(6*a*b^5*cos(d*x + c)^5 - 5*(4*a^3*b^3 - a*b^5)*cos(d
*x + c)^3 - 15*(8*a^6 - 8*a^4*b^2 + a^2*b^4)*d*x - 60*(2*a^5 - a^3*b^2 + (2*a^4*b - a^2*b^3)*sin(d*x + c))*sqr
t(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 15*(8*a^5*b - 8*a^3*b^3 + a*b^5)*c
os(d*x + c) - (4*b^6*cos(d*x + c)^5 - 10*a^2*b^4*cos(d*x + c)^3 + 15*(8*a^5*b - 8*a^3*b^3 + a*b^5)*d*x + 15*(4
*a^4*b^2 - 3*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/(b^8*d*sin(d*x + c) + a*b^7*d)]

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giac [A]  time = 0.23, size = 536, normalized size = 1.75 \[ -\frac {\frac {15 \, {\left (8 \, a^{5} - 8 \, a^{3} b^{2} + a b^{4}\right )} {\left (d x + c\right )}}{b^{7}} - \frac {120 \, {\left (2 \, a^{6} - 3 \, a^{4} b^{2} + a^{2} b^{4}\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 {40 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} - a^{3} b^{2}\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 (40 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 25 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 100 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 120 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 20 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 80 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 10 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 400 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 600 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 440 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 80 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 400 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 280 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 40 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 100 \, a^{4} - 80 \, a^{2} b^{2} + 4 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5} b^{6}}}{20 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/20*(15*(8*a^5 - 8*a^3*b^2 + a*b^4)*(d*x + c)/b^7 - 120*(2*a^6 - 3*a^4*b^2 + a^2*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) + 40*(a^4*b*t
an(1/2*d*x + 1/2*c) - a^2*b^3*tan(1/2*d*x + 1/2*c) + a^5 - a^3*b^2)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d
*x + 1/2*c) + a)*b^6) + 2*(40*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 25*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 100*a^4*tan(1/2
*d*x + 1/2*c)^8 - 120*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 20*b^4*tan(1/2*d*x + 1/2*c)^8 + 80*a^3*b*tan(1/2*d*x +
1/2*c)^7 - 10*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 400*a^4*tan(1/2*d*x + 1/2*c)^6 - 360*a^2*b^2*tan(1/2*d*x + 1/2*c)
^6 + 600*a^4*tan(1/2*d*x + 1/2*c)^4 - 440*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 40*b^4*tan(1/2*d*x + 1/2*c)^4 - 80*
a^3*b*tan(1/2*d*x + 1/2*c)^3 + 10*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 400*a^4*tan(1/2*d*x + 1/2*c)^2 - 280*a^2*b^2*
tan(1/2*d*x + 1/2*c)^2 - 40*a^3*b*tan(1/2*d*x + 1/2*c) + 25*a*b^3*tan(1/2*d*x + 1/2*c) + 100*a^4 - 80*a^2*b^2
+ 4*b^4)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*b^6))/d

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maple [B]  time = 0.58, size = 1119, normalized size = 3.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^2,x)

[Out]

-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+2/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)+1/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7*a+12/d/b^5*arctan(tan(
1/2*d*x+1/2*c))*a^3-3/2/d/b^3*arctan(tan(1/2*d*x+1/2*c))*a+2/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)-10/d/b^6/(1+tan(1/2*d*x+1/2*c)^2)^5*a^4+8/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*a^2-12/d/b^7*arctan(tan(
1/2*d*x+1/2*c))*a^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)-2/d/b^2/(1+tan(1/2*d*x+1/2*c
)^2)^5*tan(1/2*d*x+1/2*c)^8-4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4-2/5/d/b^2/(1+tan(1/2*d*x+1
/2*c)^2)^5-40/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+28/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*ta
n(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^3-5/2/d/b^3/(1+tan(1/2*d*x+1/2*
c)^2)^5*tan(1/2*d*x+1/2*c)*a-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)+
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))-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+5/2/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+ta
n(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^8*a^4+12/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^8*a^2-40
/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+36/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+44/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^3-1/d/b^3/(1+tan(1/2*d*x+1/2*c
)^2)^5*tan(1/2*d*x+1/2*c)^3*a-18/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))+6/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))

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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)^4*sin(d*x+c)^3/(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 = 12.32, size = 2390, normalized size = 7.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^2,x)

[Out]

- ((2*(a*b^4 + 30*a^5 - 25*a^3*b^2))/(5*b^6) - (3*tan(c/2 + (d*x)/2)^10*(a*b^4 - 4*a^5 + 2*a^3*b^2))/b^6 + (6*
tan(c/2 + (d*x)/2)^4*(a*b^4 + 20*a^5 - 18*a^3*b^2))/b^6 + (4*tan(c/2 + (d*x)/2)^6*(a*b^4 + 30*a^5 - 25*a^3*b^2
))/b^6 + (3*tan(c/2 + (d*x)/2)^2*(9*a*b^4 + 100*a^5 - 90*a^3*b^2))/(5*b^6) + (tan(c/2 + (d*x)/2)*(180*a^4 + 8*
b^4 - 155*a^2*b^2))/(10*b^5) + (3*tan(c/2 + (d*x)/2)^11*(4*a^4 - 3*a^2*b^2))/(2*b^5) + (6*tan(c/2 + (d*x)/2)^8
*(10*a^5 - 7*a^3*b^2))/b^6 + (3*tan(c/2 + (d*x)/2)^7*(36*a^4 - 31*a^2*b^2))/b^5 + (tan(c/2 + (d*x)/2)^3*(156*a
^4 - 125*a^2*b^2))/(2*b^5) + (tan(c/2 + (d*x)/2)^9*(84*a^4 + 8*b^4 - 75*a^2*b^2))/(2*b^5) + (tan(c/2 + (d*x)/2
)^5*(132*a^4 + 8*b^4 - 107*a^2*b^2))/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*t
an(c/2 + (d*x)/2)^9 + 2*b*tan(c/2 + (d*x)/2)^11)) - (2*atanh((108*a^6*(b^2 - a^2)^(1/2))/(108*a^6*b - (324*a^8
)/b + (216*a^10)/b^3 - 648*a^7*tan(c/2 + (d*x)/2) + 216*a^5*b^2*tan(c/2 + (d*x)/2) + (432*a^9*tan(c/2 + (d*x)/
2))/b^2) - (216*a^8*(b^2 - a^2)^(1/2))/(108*a^6*b^3 - 324*a^8*b + (216*a^10)/b + 432*a^9*tan(c/2 + (d*x)/2) +
216*a^5*b^4*tan(c/2 + (d*x)/2) - 648*a^7*b^2*tan(c/2 + (d*x)/2)) + (216*a^5*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/
2))/(108*a^6 - (324*a^8)/b^2 + (216*a^10)/b^4 + 216*a^5*b*tan(c/2 + (d*x)/2) - (648*a^7*tan(c/2 + (d*x)/2))/b
+ (432*a^9*tan(c/2 + (d*x)/2))/b^3) - (540*a^7*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(108*a^6*b^2 - 324*a^8 +
(216*a^10)/b^2 - 648*a^7*b*tan(c/2 + (d*x)/2) + 216*a^5*b^3*tan(c/2 + (d*x)/2) + (432*a^9*tan(c/2 + (d*x)/2))/
b) + (216*a^9*tan(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))/(216*a^10 + 108*a^6*b^4 - 324*a^8*b^2 + 432*a^9*b*tan(c/2
+ (d*x)/2) + 216*a^5*b^5*tan(c/2 + (d*x)/2) - 648*a^7*b^3*tan(c/2 + (d*x)/2)))*(6*a^4*(b^2 - a^2)^(1/2) - 3*a^
2*b^2*(b^2 - a^2)^(1/2)))/(b^7*d) - (3*a*atan(((3*a*(8*a^4 + b^4 - 8*a^2*b^2)*((2*(9*a^4*b^14 - 144*a^6*b^12 +
720*a^8*b^10 - 1152*a^10*b^8 + 576*a^12*b^6))/b^17 + (2*tan(c/2 + (d*x)/2)*(18*a^3*b^16 - 441*a^5*b^14 + 2448
*a^7*b^12 - 4896*a^9*b^10 + 4032*a^11*b^8 - 1152*a^13*b^6))/b^18 - (a*((2*(12*a^2*b^18 - 60*a^4*b^16 + 48*a^6*
b^14))/b^17 + (2*tan(c/2 + (d*x)/2)*(96*a^3*b^18 - 288*a^5*b^16 + 192*a^7*b^14))/b^18 - (a*(32*a^2*b^3 + (2*ta
n(c/2 + (d*x)/2)*(48*a*b^22 - 32*a^3*b^20))/b^18)*(8*a^4 + b^4 - 8*a^2*b^2)*3i)/(4*b^7))*(8*a^4 + b^4 - 8*a^2*
b^2)*3i)/(4*b^7)))/(4*b^7) + (3*a*(8*a^4 + b^4 - 8*a^2*b^2)*((2*(9*a^4*b^14 - 144*a^6*b^12 + 720*a^8*b^10 - 11
52*a^10*b^8 + 576*a^12*b^6))/b^17 + (2*tan(c/2 + (d*x)/2)*(18*a^3*b^16 - 441*a^5*b^14 + 2448*a^7*b^12 - 4896*a
^9*b^10 + 4032*a^11*b^8 - 1152*a^13*b^6))/b^18 + (a*((2*(12*a^2*b^18 - 60*a^4*b^16 + 48*a^6*b^14))/b^17 + (2*t
an(c/2 + (d*x)/2)*(96*a^3*b^18 - 288*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)*(8*a^4 + b^4 - 8*a^2*b^2)*3i)/(4*b^7))*(8*a^4 + b^4 - 8*a^2*b^2)*3i)/(4*b^7)))
/(4*b^7))/((4*(1728*a^16 - 81*a^6*b^10 + 999*a^8*b^8 - 3942*a^10*b^6 + 6912*a^12*b^4 - 5616*a^14*b^2))/b^17 +
(4*tan(c/2 + (d*x)/2)*(6912*a^17 + 54*a^5*b^12 - 1026*a^7*b^10 + 7020*a^9*b^8 - 21600*a^11*b^6 + 32832*a^13*b^
4 - 24192*a^15*b^2))/b^18 - (a*(8*a^4 + b^4 - 8*a^2*b^2)*((2*(9*a^4*b^14 - 144*a^6*b^12 + 720*a^8*b^10 - 1152*
a^10*b^8 + 576*a^12*b^6))/b^17 + (2*tan(c/2 + (d*x)/2)*(18*a^3*b^16 - 441*a^5*b^14 + 2448*a^7*b^12 - 4896*a^9*
b^10 + 4032*a^11*b^8 - 1152*a^13*b^6))/b^18 - (a*((2*(12*a^2*b^18 - 60*a^4*b^16 + 48*a^6*b^14))/b^17 + (2*tan(
c/2 + (d*x)/2)*(96*a^3*b^18 - 288*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)*(8*a^4 + b^4 - 8*a^2*b^2)*3i)/(4*b^7))*(8*a^4 + b^4 - 8*a^2*b^2)*3i)/(4*b^7))*3i)
/(4*b^7) + (a*(8*a^4 + b^4 - 8*a^2*b^2)*((2*(9*a^4*b^14 - 144*a^6*b^12 + 720*a^8*b^10 - 1152*a^10*b^8 + 576*a^
12*b^6))/b^17 + (2*tan(c/2 + (d*x)/2)*(18*a^3*b^16 - 441*a^5*b^14 + 2448*a^7*b^12 - 4896*a^9*b^10 + 4032*a^11*
b^8 - 1152*a^13*b^6))/b^18 + (a*((2*(12*a^2*b^18 - 60*a^4*b^16 + 48*a^6*b^14))/b^17 + (2*tan(c/2 + (d*x)/2)*(9
6*a^3*b^18 - 288*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)*(8*a^4 + b^4 - 8*a^2*b^2)*3i)/(4*b^7))*(8*a^4 + b^4 - 8*a^2*b^2)*3i)/(4*b^7))*3i)/(4*b^7)))*(8*a^4
+ b^4 - 8*a^2*b^2))/(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)**4*sin(d*x+c)**3/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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