3.7.0 ∫ (3+b sin(e+fx))3 _________________________

Integrand size = 25, antiderivative size = 314 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=-\frac {(3 c-b d) \left (30 b c d-b^2 \left (3 c^2+2 d^2\right )-9 \left (2 c^2+3 d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-3 d)^2 \cos (e+f x) (3+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-3 d)^2 \left (2 b c^2+15 c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {(b c-3 d) \left (15 b c d \left (c^2-7 d^2\right )+9 d^2 \left (11 c^2+4 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))} \]

-(a*c-b*d)*(10*a*b*c*d-b^2*(3*c^2+2*d^2)-a^2*(2*c^2+3*d^2))*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(

c^2-d^2)^(7/2)/f+1/3*(-a*d+b*c)^2*cos(f*x+e)*(a+b*sin(f*x+e))/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^3+1/6*(-a*d+b*c)^

2*(5*a*c*d+2*b*c^2-7*b*d^2)*cos(f*x+e)/d^2/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^2-1/6*(-a*d+b*c)*(5*a*b*c*d*(c^2-7*d

^2)+a^2*d^2*(11*c^2+4*d^2)+b^2*(2*c^4-5*c^2*d^2+18*d^4))*cos(f*x+e)/d^2/(c^2-d^2)^3/f/(c+d*sin(f*x+e))

Rubi [A] (veriﬁed)

Time = 0.50 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2871, 3100, 2833, 12, 2739, 632, 210} \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=-\frac {(a c-b d) \left (-\left (a^2 \left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{7/2}}-\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )+5 a b c d \left (c^2-7 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) (b c-a d) \cos (e+f x)}{6 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))}+\frac {\left (5 a c d+2 b c^2-7 b d^2\right ) (b c-a d)^2 \cos (e+f x)}{6 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3} \]

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

-(((a*c - b*d)*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c

^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x]*(a + b*Sin[e + f*x]))/(3*d*(c^2 - d^2)*f*(c +

d*Sin[e + f*x])^3) + ((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*Cos[e + f*x])/(6*d^2*(c^2 - d^2)^2*f*(c + d

*Sin[e + f*x])^2) - ((b*c - a*d)*(5*a*b*c*d*(c^2 - 7*d^2) + a^2*d^2*(11*c^2 + 4*d^2) + b^2*(2*c^4 - 5*c^2*d^2

+ 18*d^4))*Cos[e + f*x])/(6*d^2*(c^2 - d^2)^3*f*(c + d*Sin[e + f*x]))

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])

], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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]

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] &&

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(

b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b

^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x]

/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si

mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/

(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e

+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +

c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +

d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f

_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m

+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +

a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,

e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {\int \frac {b^3 c^2-3 a^3 c d-5 a b^2 c d+7 a^2 b d^2-\left (7 a^2 b c d+3 b^3 c d-2 a^3 d^2+a b^2 \left (c^2-9 d^2\right )\right ) \sin (e+f x)-b \left (2 a b c d-a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 d \left (c^2-d^2\right )} \\ & = \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (2 b c^2+5 a c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\int \frac {-2 d \left (15 a^2 b c d^2-a^3 d \left (3 c^2+2 d^2\right )-b^3 \left (c^3-6 c d^2\right )-a b^2 \left (6 c^2 d+9 d^3\right )\right )-\left (5 a^3 c d^3-3 a b^2 c d \left (c^2-6 d^2\right )-3 a^2 b d^2 \left (2 c^2+3 d^2\right )-b^3 \left (2 c^4-3 c^2 d^2+6 d^4\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 d^2 \left (c^2-d^2\right )^2} \\ & = \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (2 b c^2+5 a c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {(b c-a d) \left (5 a b c d \left (c^2-7 d^2\right )+a^2 d^2 \left (11 c^2+4 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac {\int -\frac {3 d^2 (a c-b d) \left (2 a^2 c^2+3 b^2 c^2-10 a b c d+3 a^2 d^2+2 b^2 d^2\right )}{c+d \sin (e+f x)} \, dx}{6 d^2 \left (c^2-d^2\right )^3} \\ & = \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (2 b c^2+5 a c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {(b c-a d) \left (5 a b c d \left (c^2-7 d^2\right )+a^2 d^2 \left (11 c^2+4 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac {\left ((a c-b d) \left (10 a b c d-b^2 \left (3 c^2+2 d^2\right )-a^2 \left (2 c^2+3 d^2\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 \left (c^2-d^2\right )^3} \\ & = \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (2 b c^2+5 a c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {(b c-a d) \left (5 a b c d \left (c^2-7 d^2\right )+a^2 d^2 \left (11 c^2+4 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac {\left ((a c-b d) \left (10 a b c d-b^2 \left (3 c^2+2 d^2\right )-a^2 \left (2 c^2+3 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (c^2-d^2\right )^3 f} \\ & = \frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (2 b c^2+5 a c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {(b c-a d) \left (5 a b c d \left (c^2-7 d^2\right )+a^2 d^2 \left (11 c^2+4 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}+\frac {\left (2 (a c-b d) \left (10 a b c d-b^2 \left (3 c^2+2 d^2\right )-a^2 \left (2 c^2+3 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (c^2-d^2\right )^3 f} \\ & = -\frac {(a c-b d) \left (10 a b c d-b^2 \left (3 c^2+2 d^2\right )-a^2 \left (2 c^2+3 d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {(b c-a d)^2 \left (2 b c^2+5 a c d-7 b d^2\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {(b c-a d) \left (5 a b c d \left (c^2-7 d^2\right )+a^2 d^2 \left (11 c^2+4 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (e+f x)}{6 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 4.86 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.01 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\frac {\frac {12 \left (9 \left (6+b^2\right ) c^3-3 b \left (36+b^2\right ) c^2 d+9 \left (9+4 b^2\right ) c d^2-b \left (27+2 b^2\right ) d^3\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+\frac {(b c-3 d) \cos (e+f x) \left (-324 c^4-10 b^2 c^4+195 b c^3 d-9 c^2 d^2-17 b^2 c^2 d^2+75 b c d^3-72 d^4-18 b^2 d^4+\left (99 c^2 d^2+36 d^4+15 b c d \left (c^2-7 d^2\right )+b^2 \left (2 c^4-5 c^2 d^2+18 d^4\right )\right ) \cos (2 (e+f x))-6 \left (9 c d \left (9 c^2+d^2\right )+b^2 c d \left (c^2+9 d^2\right )+b \left (9 c^4-78 c^2 d^2+9 d^4\right )\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^3 (c+d \sin (e+f x))^3}}{12 f} \]

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

((12*(9*(6 + b^2)*c^3 - 3*b*(36 + b^2)*c^2*d + 9*(9 + 4*b^2)*c*d^2 - b*(27 + 2*b^2)*d^3)*ArcTan[(d + c*Tan[(e

+ f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(7/2) + ((b*c - 3*d)*Cos[e + f*x]*(-324*c^4 - 10*b^2*c^4 + 195*b*c^3*

d - 9*c^2*d^2 - 17*b^2*c^2*d^2 + 75*b*c*d^3 - 72*d^4 - 18*b^2*d^4 + (99*c^2*d^2 + 36*d^4 + 15*b*c*d*(c^2 - 7*d

^2) + b^2*(2*c^4 - 5*c^2*d^2 + 18*d^4))*Cos[2*(e + f*x)] - 6*(9*c*d*(9*c^2 + d^2) + b^2*c*d*(c^2 + 9*d^2) + b*

(9*c^4 - 78*c^2*d^2 + 9*d^4))*Sin[e + f*x]))/((c^2 - d^2)^3*(c + d*Sin[e + f*x])^3))/(12*f)

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1153\) vs. \(2(314)=628\).

Time = 2.76 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.68


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1154\)
default	\(\text {Expression too large to display}\)	\(1154\)
risch	\(\text {Expression too large to display}\)	\(2740\)

int((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

1/f*(2*(1/2*(9*a^3*c^4*d^2-6*a^3*c^2*d^4+2*a^3*d^6-12*a^2*b*c^5*d-3*a^2*b*c^3*d^3+3*a*b^2*c^6+12*a*b^2*c^4*d^2

-3*b^3*c^5*d-2*b^3*c^3*d^3)/c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5+1/2*(6*a^3*c^6*d+27*a^3*c^4*d

^3-12*a^3*c^2*d^5+4*a^3*d^7-6*a^2*b*c^7-42*a^2*b*c^5*d^2-33*a^2*b*c^3*d^4+6*a^2*b*c*d^6+15*a*b^2*c^6*d+60*a*b^

2*c^4*d^3-15*b^3*c^5*d^2-10*b^3*c^3*d^4)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/c^2*tan(1/2*f*x+1/2*e)^4+1/3/c^3*d*(54*

a^3*c^6*d+21*a^3*c^4*d^3-4*a^3*c^2*d^5+4*a^3*d^7-54*a^2*b*c^7-126*a^2*b*c^5*d^2-51*a^2*b*c^3*d^4+6*a^2*b*c*d^6

+117*a*b^2*c^6*d+96*a*b^2*c^4*d^3+12*a*b^2*c^2*d^5-12*b^3*c^7-41*b^3*c^5*d^2-22*b^3*c^3*d^4)/(c^6-3*c^4*d^2+3*

c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^3+(6*a^3*c^6*d+20*a^3*c^4*d^3-3*a^3*c^2*d^5+2*a^3*d^7-6*a^2*b*c^7-30*a^2*b*c^5

*d^2-42*a^2*b*c^3*d^4+3*a^2*b*c*d^6+12*a*b^2*c^6*d+51*a*b^2*c^4*d^3+12*a*b^2*c^2*d^5-2*b^3*c^7-6*b^3*c^5*d^2-1

7*b^3*c^3*d^4)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/c^2*tan(1/2*f*x+1/2*e)^2+1/2*(27*a^3*c^4*d^2-4*a^3*c^2*d^4+2*a^3*

d^6-24*a^2*b*c^5*d-57*a^2*b*c^3*d^3+6*a^2*b*c*d^5-3*a*b^2*c^6+66*a*b^2*c^4*d^2+12*a*b^2*c^2*d^4-5*b^3*c^5*d-20

*b^3*c^3*d^3)/c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)+1/6*(18*a^3*c^4*d-5*a^3*c^2*d^3+2*a^3*d^5-18*

a^2*b*c^5-30*a^2*b*c^3*d^2+3*a^2*b*c*d^4+39*a*b^2*c^4*d+6*a*b^2*c^2*d^3-4*b^3*c^5-11*b^3*c^3*d^2)/(c^6-3*c^4*d

^2+3*c^2*d^4-d^6))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)^3+(2*a^3*c^3+3*a^3*c*d^2-12*a^2*b*c^2*d-3

*a^2*b*d^3+3*a*b^2*c^3+12*a*b^2*c*d^2-3*b^3*c^2*d-2*b^3*d^3)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/(c^2-d^2)^(1/2)*arc

tan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2)))

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1026 vs. \(2 (314) = 628\).

Time = 0.40 (sec) , antiderivative size = 2136, normalized size of antiderivative = 6.80 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]

integrate((a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^4,x, algorithm="fricas")

[-1/12*(2*(2*b^3*c^7 + 3*a*b^2*c^6*d + (6*a^2*b - 7*b^3)*c^5*d^2 - 11*(a^3 + 3*a*b^2)*c^4*d^3 + (33*a^2*b + 23

*b^3)*c^3*d^4 + (7*a^3 + 12*a*b^2)*c^2*d^5 - 3*(13*a^2*b + 6*b^3)*c*d^6 + 2*(2*a^3 + 9*a*b^2)*d^7)*cos(f*x + e

)^3 - 6*(3*a*b^2*c^7 + 3*a^2*b*d^7 + (6*a^2*b + b^3)*c^6*d - 3*(3*a^3 + 10*a*b^2)*c^5*d^2 + (21*a^2*b + 8*b^3)

*c^4*d^3 + (8*a^3 + 21*a*b^2)*c^3*d^4 - 3*(10*a^2*b + 3*b^3)*c^2*d^5 + (a^3 + 6*a*b^2)*c*d^6)*cos(f*x + e)*sin

(f*x + e) - 3*((2*a^3 + 3*a*b^2)*c^6 - 3*(4*a^2*b + b^3)*c^5*d + 3*(3*a^3 + 7*a*b^2)*c^4*d^2 - (39*a^2*b + 11*

b^3)*c^3*d^3 + 9*(a^3 + 4*a*b^2)*c^2*d^4 - 3*(3*a^2*b + 2*b^3)*c*d^5 - 3*((2*a^3 + 3*a*b^2)*c^4*d^2 - 3*(4*a^2

*b + b^3)*c^3*d^3 + 3*(a^3 + 4*a*b^2)*c^2*d^4 - (3*a^2*b + 2*b^3)*c*d^5)*cos(f*x + e)^2 + (3*(2*a^3 + 3*a*b^2)

*c^5*d - 9*(4*a^2*b + b^3)*c^4*d^2 + (11*a^3 + 39*a*b^2)*c^3*d^3 - 3*(7*a^2*b + 3*b^3)*c^2*d^4 + 3*(a^3 + 4*a*

b^2)*c*d^5 - (3*a^2*b + 2*b^3)*d^6 - ((2*a^3 + 3*a*b^2)*c^3*d^3 - 3*(4*a^2*b + b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2

)*c*d^5 - (3*a^2*b + 2*b^3)*d^6)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e

)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2

*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 12*(3*a^2*b*c^5*d^2 + 2*a^3*c^4*d^3 + 2*b^3*c^3*d^4 + 3*a

*b^2*c^2*d^5 + (3*a^2*b + b^3)*c^7 - 3*(a^3 + 2*a*b^2)*c^6*d - 3*(2*a^2*b + b^3)*c*d^6 + (a^3 + 3*a*b^2)*d^7)*

cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6

*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos

(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e)), -1/6*((2*b^3

*c^7 + 3*a*b^2*c^6*d + (6*a^2*b - 7*b^3)*c^5*d^2 - 11*(a^3 + 3*a*b^2)*c^4*d^3 + (33*a^2*b + 23*b^3)*c^3*d^4 +

(7*a^3 + 12*a*b^2)*c^2*d^5 - 3*(13*a^2*b + 6*b^3)*c*d^6 + 2*(2*a^3 + 9*a*b^2)*d^7)*cos(f*x + e)^3 - 3*(3*a*b^2

*c^7 + 3*a^2*b*d^7 + (6*a^2*b + b^3)*c^6*d - 3*(3*a^3 + 10*a*b^2)*c^5*d^2 + (21*a^2*b + 8*b^3)*c^4*d^3 + (8*a^

3 + 21*a*b^2)*c^3*d^4 - 3*(10*a^2*b + 3*b^3)*c^2*d^5 + (a^3 + 6*a*b^2)*c*d^6)*cos(f*x + e)*sin(f*x + e) - 3*((

2*a^3 + 3*a*b^2)*c^6 - 3*(4*a^2*b + b^3)*c^5*d + 3*(3*a^3 + 7*a*b^2)*c^4*d^2 - (39*a^2*b + 11*b^3)*c^3*d^3 + 9

*(a^3 + 4*a*b^2)*c^2*d^4 - 3*(3*a^2*b + 2*b^3)*c*d^5 - 3*((2*a^3 + 3*a*b^2)*c^4*d^2 - 3*(4*a^2*b + b^3)*c^3*d^

3 + 3*(a^3 + 4*a*b^2)*c^2*d^4 - (3*a^2*b + 2*b^3)*c*d^5)*cos(f*x + e)^2 + (3*(2*a^3 + 3*a*b^2)*c^5*d - 9*(4*a^

2*b + b^3)*c^4*d^2 + (11*a^3 + 39*a*b^2)*c^3*d^3 - 3*(7*a^2*b + 3*b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d^5 - (3*

a^2*b + 2*b^3)*d^6 - ((2*a^3 + 3*a*b^2)*c^3*d^3 - 3*(4*a^2*b + b^3)*c^2*d^4 + 3*(a^3 + 4*a*b^2)*c*d^5 - (3*a^2

*b + 2*b^3)*d^6)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*c

os(f*x + e))) - 6*(3*a^2*b*c^5*d^2 + 2*a^3*c^4*d^3 + 2*b^3*c^3*d^4 + 3*a*b^2*c^2*d^5 + (3*a^2*b + b^3)*c^7 - 3

*(a^3 + 2*a*b^2)*c^6*d - 3*(2*a^2*b + b^3)*c*d^6 + (a^3 + 3*a*b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4

+ 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3

*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 +

14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e))]

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Timed out} \]

integrate((a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**4,x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Exception raised: ValueError} \]

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

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*d^2-4*c^2>0)', see `assume?`

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1613 vs. \(2 (314) = 628\).

Time = 0.36 (sec) , antiderivative size = 1613, normalized size of antiderivative = 5.14 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]

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

1/3*(3*(2*a^3*c^3 + 3*a*b^2*c^3 - 12*a^2*b*c^2*d - 3*b^3*c^2*d + 3*a^3*c*d^2 + 12*a*b^2*c*d^2 - 3*a^2*b*d^3 -

2*b^3*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((

c^6 - 3*c^4*d^2 + 3*c^2*d^4 - d^6)*sqrt(c^2 - d^2)) + (9*a*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 36*a^2*b*c^7*d*tan

(1/2*f*x + 1/2*e)^5 - 9*b^3*c^7*d*tan(1/2*f*x + 1/2*e)^5 + 27*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 + 36*a*b^2*c^

6*d^2*tan(1/2*f*x + 1/2*e)^5 - 9*a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 6*b^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 -

18*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^5 + 6*a^3*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 18*a^2*b*c^8*tan(1/2*f*x + 1/2

*e)^4 + 18*a^3*c^7*d*tan(1/2*f*x + 1/2*e)^4 + 45*a*b^2*c^7*d*tan(1/2*f*x + 1/2*e)^4 - 126*a^2*b*c^6*d^2*tan(1/

2*f*x + 1/2*e)^4 - 45*b^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^4 + 81*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 + 180*a*b^2*c

^5*d^3*tan(1/2*f*x + 1/2*e)^4 - 99*a^2*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^4 - 30*b^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^

4 - 36*a^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^4 + 18*a^2*b*c^2*d^6*tan(1/2*f*x + 1/2*e)^4 + 12*a^3*c*d^7*tan(1/2*f*x

+ 1/2*e)^4 - 108*a^2*b*c^7*d*tan(1/2*f*x + 1/2*e)^3 - 24*b^3*c^7*d*tan(1/2*f*x + 1/2*e)^3 + 108*a^3*c^6*d^2*t

an(1/2*f*x + 1/2*e)^3 + 234*a*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 - 252*a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 -

82*b^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 + 42*a^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 192*a*b^2*c^4*d^4*tan(1/2*f*x

+ 1/2*e)^3 - 102*a^2*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 44*b^3*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 8*a^3*c^2*d^6*

tan(1/2*f*x + 1/2*e)^3 + 24*a*b^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 12*a^2*b*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 8*a

^3*d^8*tan(1/2*f*x + 1/2*e)^3 - 36*a^2*b*c^8*tan(1/2*f*x + 1/2*e)^2 - 12*b^3*c^8*tan(1/2*f*x + 1/2*e)^2 + 36*a

^3*c^7*d*tan(1/2*f*x + 1/2*e)^2 + 72*a*b^2*c^7*d*tan(1/2*f*x + 1/2*e)^2 - 180*a^2*b*c^6*d^2*tan(1/2*f*x + 1/2*

e)^2 - 36*b^3*c^6*d^2*tan(1/2*f*x + 1/2*e)^2 + 120*a^3*c^5*d^3*tan(1/2*f*x + 1/2*e)^2 + 306*a*b^2*c^5*d^3*tan(

1/2*f*x + 1/2*e)^2 - 252*a^2*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 - 102*b^3*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 - 18*a^

3*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 72*a*b^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 18*a^2*b*c^2*d^6*tan(1/2*f*x + 1/

2*e)^2 + 12*a^3*c*d^7*tan(1/2*f*x + 1/2*e)^2 - 9*a*b^2*c^8*tan(1/2*f*x + 1/2*e) - 72*a^2*b*c^7*d*tan(1/2*f*x +

1/2*e) - 15*b^3*c^7*d*tan(1/2*f*x + 1/2*e) + 81*a^3*c^6*d^2*tan(1/2*f*x + 1/2*e) + 198*a*b^2*c^6*d^2*tan(1/2*

f*x + 1/2*e) - 171*a^2*b*c^5*d^3*tan(1/2*f*x + 1/2*e) - 60*b^3*c^5*d^3*tan(1/2*f*x + 1/2*e) - 12*a^3*c^4*d^4*t

an(1/2*f*x + 1/2*e) + 36*a*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e) + 18*a^2*b*c^3*d^5*tan(1/2*f*x + 1/2*e) + 6*a^3*c^

2*d^6*tan(1/2*f*x + 1/2*e) - 18*a^2*b*c^8 - 4*b^3*c^8 + 18*a^3*c^7*d + 39*a*b^2*c^7*d - 30*a^2*b*c^6*d^2 - 11*

b^3*c^6*d^2 - 5*a^3*c^5*d^3 + 6*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 + 2*a^3*c^3*d^5)/((c^9 - 3*c^7*d^2 + 3*c^5*d^4

- c^3*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^3))/f

Mupad [B] (veriﬁcation not implemented)

Time = 12.65 (sec) , antiderivative size = 1423, normalized size of antiderivative = 4.53 \[ \int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]

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

((2*a^3*d^5 - 4*b^3*c^5 - 18*a^2*b*c^5 + 18*a^3*c^4*d - 5*a^3*c^2*d^3 - 11*b^3*c^3*d^2 + 6*a*b^2*c^2*d^3 - 30*

a^2*b*c^3*d^2 + 39*a*b^2*c^4*d + 3*a^2*b*c*d^4)/(3*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) - (tan(e/2 + (f*x)/2)^

5*(3*b^3*c^5*d - 3*a*b^2*c^6 - 2*a^3*d^6 + 6*a^3*c^2*d^4 - 9*a^3*c^4*d^2 + 2*b^3*c^3*d^3 - 12*a*b^2*c^4*d^2 +

3*a^2*b*c^3*d^3 + 12*a^2*b*c^5*d))/(c*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (2*tan(e/2 + (f*x)/2)^2*(2*a^3*d^

7 - 2*b^3*c^7 - 6*a^2*b*c^7 + 6*a^3*c^6*d - 3*a^3*c^2*d^5 + 20*a^3*c^4*d^3 - 17*b^3*c^3*d^4 - 6*b^3*c^5*d^2 +

12*a*b^2*c^2*d^5 + 51*a*b^2*c^4*d^3 - 42*a^2*b*c^3*d^4 - 30*a^2*b*c^5*d^2 + 12*a*b^2*c^6*d + 3*a^2*b*c*d^6))/(

c^2*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) - (tan(e/2 + (f*x)/2)*(3*a*b^2*c^6 - 2*a^3*d^6 + 5*b^3*c^5*d + 4*a^3*

c^2*d^4 - 27*a^3*c^4*d^2 + 20*b^3*c^3*d^3 - 12*a*b^2*c^2*d^4 - 66*a*b^2*c^4*d^2 + 57*a^2*b*c^3*d^3 - 6*a^2*b*c

*d^5 + 24*a^2*b*c^5*d))/(c*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (tan(e/2 + (f*x)/2)^4*(4*a^3*d^7 - 6*a^2*b*c

^7 + 6*a^3*c^6*d - 12*a^3*c^2*d^5 + 27*a^3*c^4*d^3 - 10*b^3*c^3*d^4 - 15*b^3*c^5*d^2 + 60*a*b^2*c^4*d^3 - 33*a

^2*b*c^3*d^4 - 42*a^2*b*c^5*d^2 + 15*a*b^2*c^6*d + 6*a^2*b*c*d^6))/(c^2*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) +

(2*d*tan(e/2 + (f*x)/2)^3*(3*c^2 + 2*d^2)*(2*a^3*d^5 - 4*b^3*c^5 - 18*a^2*b*c^5 + 18*a^3*c^4*d - 5*a^3*c^2*d^

3 - 11*b^3*c^3*d^2 + 6*a*b^2*c^2*d^3 - 30*a^2*b*c^3*d^2 + 39*a*b^2*c^4*d + 3*a^2*b*c*d^4))/(3*c^3*(c^6 - d^6 +

3*c^2*d^4 - 3*c^4*d^2)))/(f*(c^3*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^2*(12*c*d^2 + 3*c^3) + tan(e/2 + (

f*x)/2)^4*(12*c*d^2 + 3*c^3) + tan(e/2 + (f*x)/2)^3*(12*c^2*d + 8*d^3) + c^3 + 6*c^2*d*tan(e/2 + (f*x)/2) + 6*

c^2*d*tan(e/2 + (f*x)/2)^5)) + (atan((((c*tan(e/2 + (f*x)/2)*(a*c - b*d)*(2*a^2*c^2 + 3*a^2*d^2 + 3*b^2*c^2 +

2*b^2*d^2 - 10*a*b*c*d))/((c + d)^(7/2)*(c - d)^(7/2)) + ((a*c - b*d)*(2*c^6*d - 2*d^7 + 6*c^2*d^5 - 6*c^4*d^3

)*(2*a^2*c^2 + 3*a^2*d^2 + 3*b^2*c^2 + 2*b^2*d^2 - 10*a*b*c*d))/(2*(c + d)^(7/2)*(c - d)^(7/2)*(c^6 - d^6 + 3*

c^2*d^4 - 3*c^4*d^2)))*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2))/(2*a^3*c^3 - 2*b^3*d^3 + 3*a*b^2*c^3 - 3*a^2*b*d^3

+ 3*a^3*c*d^2 - 3*b^3*c^2*d + 12*a*b^2*c*d^2 - 12*a^2*b*c^2*d))*(a*c - b*d)*(2*a^2*c^2 + 3*a^2*d^2 + 3*b^2*c^

2 + 2*b^2*d^2 - 10*a*b*c*d))/(f*(c + d)^(7/2)*(c - d)^(7/2))

\(\int \frac {(3+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx\) [693]

Optimal result