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

Optimal. Leaf size=325 \[ -\frac{(a c-b d) \left (a^2 \left (-\left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{7/2}}-\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )+5 a b c d \left (c^2-7 d^2\right )+b^2 \left (-5 c^2 d^2+2 c^4+18 d^4\right )\right ) (b c-a d) \cos (e+f x)}{6 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))}+\frac{\left (5 a c d+2 b c^2-7 b d^2\right ) (b c-a d)^2 \cos (e+f x)}{6 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3} \]

[Out]

-(((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]))

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Rubi [A]  time = 0.724919, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {2792, 3021, 2754, 12, 2660, 618, 204} \[ -\frac{(a c-b d) \left (a^2 \left (-\left (2 c^2+3 d^2\right )\right )+10 a b c d-b^2 \left (3 c^2+2 d^2\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{7/2}}-\frac{\left (a^2 d^2 \left (11 c^2+4 d^2\right )+5 a b c d \left (c^2-7 d^2\right )+b^2 \left (-5 c^2 d^2+2 c^4+18 d^4\right )\right ) (b c-a d) \cos (e+f x)}{6 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))}+\frac{\left (5 a c d+2 b c^2-7 b d^2\right ) (b c-a d)^2 \cos (e+f x)}{6 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}+\frac{(b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((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]))

Rule 2792

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

Rule 3021

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]

Rule 2754

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]

Rule 12

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

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{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \tan ^{-1}\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]  time = 5.15598, size = 345, normalized size = 1.06 \[ \frac{\frac{6 \left (-3 a^2 b d \left (4 c^2+d^2\right )+a^3 \left (2 c^3+3 c d^2\right )+3 a b^2 c \left (c^2+4 d^2\right )-b^3 d \left (3 c^2+2 d^2\right )\right ) \tan ^{-1}\left (\frac{c \tan \left (\frac{1}{2} (e+f x)\right )+d}{\sqrt{c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+\frac{\left (3 a^2 b c d^2 \left (2 c^2+13 d^2\right )-a^3 d^3 \left (11 c^2+4 d^2\right )+3 a b^2 d \left (-10 c^2 d^2+c^4-6 d^4\right )+b^3 \left (-5 c^3 d^2+2 c^5+18 c d^4\right )\right ) \cos (e+f x)}{d^2 \left (d^2-c^2\right )^3 (c+d \sin (e+f x))}+\frac{2 (b c-a d)^3 \cos (e+f x)}{d^2 \left (d^2-c^2\right ) (c+d \sin (e+f x))^3}+\frac{\left (5 a c d+4 b c^2-9 b d^2\right ) (b c-a d)^2 \cos (e+f x)}{d^2 \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2}}{6 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*(-3*a^2*b*d*(4*c^2 + d^2) - b^3*d*(3*c^2 + 2*d^2) + 3*a*b^2*c*(c^2 + 4*d^2) + a^3*(2*c^3 + 3*c*d^2))*ArcTa
n[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(c^2 - d^2)^(7/2) + (2*(b*c - a*d)^3*Cos[e + f*x])/(d^2*(-c^2 + d
^2)*(c + d*Sin[e + f*x])^3) + ((b*c - a*d)^2*(4*b*c^2 + 5*a*c*d - 9*b*d^2)*Cos[e + f*x])/(d^2*(c^2 - d^2)^2*(c
+ d*Sin[e + f*x])^2) + ((-(a^3*d^3*(11*c^2 + 4*d^2)) + 3*a^2*b*c*d^2*(2*c^2 + 13*d^2) + 3*a*b^2*d*(c^4 - 10*c
^2*d^2 - 6*d^4) + b^3*(2*c^5 - 5*c^3*d^2 + 18*c*d^4))*Cos[e + f*x])/(d^2*(-c^2 + d^2)^3*(c + d*Sin[e + f*x])))
/(6*f)

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Maple [B]  time = 0.125, size = 6128, normalized size = 18.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.10513, size = 4540, normalized size = 13.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-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))]

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

[Out]

Timed out

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Giac [B]  time = 1.48679, size = 2264, normalized size = 6.97 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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