3.1301 \(\int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=282 \[ \frac{a \left (-20 a^2 b^2+15 a^4+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac{2 a^3 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d}-\frac{\left (6 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{24 b^3 d}+\frac{a \left (5 a^2-6 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{15 b^4 d}-\frac{\left (-10 a^2 b^2+8 a^4+b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^5 d}+\frac{x \left (-24 a^4 b^2+6 a^2 b^4+16 a^6+b^6\right )}{16 b^7}+\frac{a \sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 b d} \]

[Out]

((16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*x)/(16*b^7) - (2*a^3*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2]
)/Sqrt[a^2 - b^2]])/(b^7*d) + (a*(15*a^4 - 20*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^6*d) - ((8*a^4 - 10*a^2*b^2
+ b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^5*d) + (a*(5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^4*d) - (
(6*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x]^4)/(5*b^2*d) - (Cos[c +
d*x]*Sin[c + d*x]^5)/(6*b*d)

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Rubi [A]  time = 1.00121, antiderivative size = 282, normalized size of antiderivative = 1., 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 = {2895, 3049, 3023, 2735, 2660, 618, 204} \[ \frac{a \left (-20 a^2 b^2+15 a^4+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac{2 a^3 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{b^7 d}-\frac{\left (6 a^2-7 b^2\right ) \sin ^3(c+d x) \cos (c+d x)}{24 b^3 d}+\frac{a \left (5 a^2-6 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{15 b^4 d}-\frac{\left (-10 a^2 b^2+8 a^4+b^4\right ) \sin (c+d x) \cos (c+d x)}{16 b^5 d}+\frac{x \left (-24 a^4 b^2+6 a^2 b^4+16 a^6+b^6\right )}{16 b^7}+\frac{a \sin ^4(c+d x) \cos (c+d x)}{5 b^2 d}-\frac{\sin ^5(c+d x) \cos (c+d x)}{6 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*x)/(16*b^7) - (2*a^3*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2]
)/Sqrt[a^2 - b^2]])/(b^7*d) + (a*(15*a^4 - 20*a^2*b^2 + 3*b^4)*Cos[c + d*x])/(15*b^6*d) - ((8*a^4 - 10*a^2*b^2
+ b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*b^5*d) + (a*(5*a^2 - 6*b^2)*Cos[c + d*x]*Sin[c + d*x]^2)/(15*b^4*d) - (
(6*a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x]^3)/(24*b^3*d) + (a*Cos[c + d*x]*Sin[c + d*x]^4)/(5*b^2*d) - (Cos[c +
d*x]*Sin[c + d*x]^5)/(6*b*d)

Rule 2895

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(a*(n + 3)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d*f*(m
+ n + 3)*(m + n + 4)), x] + (-Dist[1/(b^2*(m + n + 3)*(m + n + 4)), Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x
])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*
(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x] - Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e
+ f*x])^(m + 1))/(b*d^2*f*(m + n + 4)), x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[
m, 0] || IntegersQ[2*m, 2*n]) &&  !m < -1 &&  !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]

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])))

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 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 ^4(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac{\int \frac{\sin ^3(c+d x) \left (6 \left (4 a^2-5 b^2\right )-a b \sin (c+d x)-5 \left (6 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{30 b^2}\\ &=-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac{\int \frac{\sin ^2(c+d x) \left (-15 a \left (6 a^2-7 b^2\right )+3 b \left (2 a^2-5 b^2\right ) \sin (c+d x)+24 a \left (5 a^2-6 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 b^3}\\ &=\frac{a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac{\int \frac{\sin (c+d x) \left (48 a^2 \left (5 a^2-6 b^2\right )-3 a b \left (10 a^2-9 b^2\right ) \sin (c+d x)-45 \left (8 a^4-10 a^2 b^2+b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 b^4}\\ &=-\frac{\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac{a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac{\int \frac{-45 a \left (8 a^4-10 a^2 b^2+b^4\right )+3 b \left (40 a^4-42 a^2 b^2-15 b^4\right ) \sin (c+d x)+48 a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{720 b^5}\\ &=\frac{a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac{\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac{a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac{\int \frac{-45 a b \left (8 a^4-10 a^2 b^2+b^4\right )-45 \left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{720 b^6}\\ &=\frac{\left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}+\frac{a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac{\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac{a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac{\left (a^3 \left (a^2-b^2\right )^2\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{b^7}\\ &=\frac{\left (16 a^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}+\frac{a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac{\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac{a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}-\frac{\left (2 a^3 \left (a^2-b^2\right )^2\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^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}+\frac{a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac{\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac{a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}+\frac{\left (4 a^3 \left (a^2-b^2\right )^2\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^6-24 a^4 b^2+6 a^2 b^4+b^6\right ) x}{16 b^7}-\frac{2 a^3 \left (a^2-b^2\right )^{3/2} \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{a \left (15 a^4-20 a^2 b^2+3 b^4\right ) \cos (c+d x)}{15 b^6 d}-\frac{\left (8 a^4-10 a^2 b^2+b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^5 d}+\frac{a \left (5 a^2-6 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{15 b^4 d}-\frac{\left (6 a^2-7 b^2\right ) \cos (c+d x) \sin ^3(c+d x)}{24 b^3 d}+\frac{a \cos (c+d x) \sin ^4(c+d x)}{5 b^2 d}-\frac{\cos (c+d x) \sin ^5(c+d x)}{6 b d}\\ \end{align*}

Mathematica [A]  time = 2.28009, size = 274, normalized size = 0.97 \[ \frac{-240 a^4 b^2 \sin (2 (c+d x))+240 a^2 b^4 \sin (2 (c+d x))+30 a^2 b^4 \sin (4 (c+d x))+120 a b \left (-10 a^2 b^2+8 a^4+b^4\right ) \cos (c+d x)+\left (60 a b^5-80 a^3 b^3\right ) \cos (3 (c+d x))-1920 a^3 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )-1440 a^4 b^2 c+360 a^2 b^4 c-1440 a^4 b^2 d x+360 a^2 b^4 d x+960 a^6 c+960 a^6 d x+12 a b^5 \cos (5 (c+d x))+15 b^6 \sin (2 (c+d x))-15 b^6 \sin (4 (c+d x))-5 b^6 \sin (6 (c+d x))+60 b^6 c+60 b^6 d x}{960 b^7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(960*a^6*c - 1440*a^4*b^2*c + 360*a^2*b^4*c + 60*b^6*c + 960*a^6*d*x - 1440*a^4*b^2*d*x + 360*a^2*b^4*d*x + 60
*b^6*d*x - 1920*a^3*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 120*a*b*(8*a^4 - 10*a
^2*b^2 + b^4)*Cos[c + d*x] + (-80*a^3*b^3 + 60*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)] + 240*a^2*b^4*Sin[2*(c + d*x)] + 15*b^6*Sin[2*(c + d*x)] + 30*a^2*b^4*Sin[4*(c + d*x)] - 1
5*b^6*Sin[4*(c + d*x)] - 5*b^6*Sin[6*(c + d*x)])/(960*b^7*d)

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Maple [B]  time = 0.103, size = 1501, normalized size = 5.3 \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)^4*sin(d*x+c)^3/(a+b*sin(d*x+c)),x)

[Out]

1/8/d/b*arctan(tan(1/2*d*x+1/2*c))-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+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+5/4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*a^2+2/d/
b^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^10*a^5+2/5/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*
c)^2*a-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+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^5-13/4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5+47/24/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*
tan(1/2*d*x+1/2*c)^3-1/8/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)+2/d/b^7*arctan(tan(1/2*d*x+1/2*c))*
a^6+2/5/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*a-8/3/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*a^3+2/d/b^6/(1+tan(1/2*d*x+1/2
*c)^2)^6*a^5+1/8/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11-24/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(
1/2*d*x+1/2*c)^4*a^3-3/d/b^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*a^4-3/d/b^5*arctan(tan(1/2*d*x+1/
2*c))*a^4+3/4/d/b^3*arctan(tan(1/2*d*x+1/2*c))*a^2-80/3/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+4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^6*a+1/2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x
+1/2*c)^5*a^2-47/24/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9+13/4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^6*ta
n(1/2*d*x+1/2*c)^7+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+7/4/d/b^3/(1+tan(1/2*d*x+1/2*c
)^2)^6*tan(1/2*d*x+1/2*c)^3*a^2-2/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))-12/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+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+2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a+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^4-7/4/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+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^4-5/4/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+4/d/b^2
/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^4*a-16/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^8*a^
3-4/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+2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+
1/2*c)^10*a+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^4-1/2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^6*t
an(1/2*d*x+1/2*c)^7*a^2+4/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))-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))

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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)^4*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.74563, size = 1204, normalized size = 4.27 \begin{align*} \left [\frac{48 \, a b^{5} \cos \left (d x + c\right )^{5} - 80 \, a^{3} b^{3} \cos \left (d x + c\right )^{3} + 15 \,{\left (16 \, a^{6} - 24 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} d x - 120 \,{\left (a^{5} - a^{3} b^{2}\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 - a^{3} b^{3}\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} + b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, a^{4} b^{2} - 6 \, a^{2} b^{4} - 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 \, a^{3} b^{3} \cos \left (d x + c\right )^{3} + 15 \,{\left (16 \, a^{6} - 24 \, a^{4} b^{2} + 6 \, a^{2} b^{4} + b^{6}\right )} d x + 240 \,{\left (a^{5} - a^{3} b^{2}\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 - a^{3} b^{3}\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} + b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (8 \, a^{4} b^{2} - 6 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, b^{7} d}\right ] \end{align*}

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)),x, algorithm="fricas")

[Out]

[1/240*(48*a*b^5*cos(d*x + c)^5 - 80*a^3*b^3*cos(d*x + c)^3 + 15*(16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*d*x -
120*(a^5 - a^3*b^2)*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 - a^3*b^3)*cos(d*x + c) - 5*(8*b^6*cos(d*x + c)^5 - 2*(6*a^2*b^4 + b^6)*cos(d*x + c)^3
+ 3*(8*a^4*b^2 - 6*a^2*b^4 - 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*cos(d*x + c)^3 + 15*(16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*d*x + 240*(a^5 - a^3*b^2)*sqrt(a^2 - b^2)*arc
tan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 240*(a^5*b - a^3*b^3)*cos(d*x + c) - 5*(8*b^6*cos(
d*x + c)^5 - 2*(6*a^2*b^4 + b^6)*cos(d*x + c)^3 + 3*(8*a^4*b^2 - 6*a^2*b^4 - b^6)*cos(d*x + c))*sin(d*x + c))/
(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)**4*sin(d*x+c)**3/(a+b*sin(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.18101, size = 980, normalized size = 3.48 \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)^4*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/240*(15*(16*a^6 - 24*a^4*b^2 + 6*a^2*b^4 + b^6)*(d*x + c)/b^7 - 480*(a^7 - 2*a^5*b^2 + a^3*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) + 2
*(120*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 150*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 + 15*b^5*tan(1/2*d*x + 1/2*c)^11 + 2
40*a^5*tan(1/2*d*x + 1/2*c)^10 - 480*a^3*b^2*tan(1/2*d*x + 1/2*c)^10 + 240*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 - 210*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 - 235*b^5*tan(1/2*d*x + 1/2*c)^9 + 1200*a^5
*tan(1/2*d*x + 1/2*c)^8 - 1920*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 240*a*b^4*tan(1/2*d*x + 1/2*c)^8 + 240*a^4*b*t
an(1/2*d*x + 1/2*c)^7 - 60*a^2*b^3*tan(1/2*d*x + 1/2*c)^7 + 390*b^5*tan(1/2*d*x + 1/2*c)^7 + 2400*a^5*tan(1/2*
d*x + 1/2*c)^6 - 3200*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 + 480*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 + 60*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 - 390*b^5*tan(1/2*d*x + 1/2*c)^5 + 2400*a^5*tan(1/2*d*x + 1/2
*c)^4 - 2880*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 + 480*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 + 210*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 235*b^5*tan(1/2*d*x + 1/2*c)^3 + 1200*a^5*tan(1/2*d*x + 1/2*c)^2 -
1440*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 48*a*b^4*tan(1/2*d*x + 1/2*c)^2 - 120*a^4*b*tan(1/2*d*x + 1/2*c) + 150*a
^2*b^3*tan(1/2*d*x + 1/2*c) - 15*b^5*tan(1/2*d*x + 1/2*c) + 240*a^5 - 320*a^3*b^2 + 48*a*b^4)/((tan(1/2*d*x +
1/2*c)^2 + 1)^6*b^6))/d