3.8.14 \(\int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx\) [714]

Optimal. Leaf size=534 \[ -\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}+\frac {(b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2} f}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2} \]

[Out]

-1/2*d^3*(30*a*b*c*d-12*a^2*d^2-b^2*(20*c^2+d^2))*x/b^5+(-a*d+b*c)^3*(6*a^3*b*c*d-12*a*b^3*c*d+12*a^4*d^2+a^2*
b^2*(2*c^2-29*d^2)+b^4*(c^2+20*d^2))*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^5/(a^2-b^2)^(5/2)/f-1/
2*d*(30*a^4*b*c*d^3-12*a^5*d^4-a^3*b^2*d^2*(16*c^2-21*d^2)-b^5*c*d*(17*c^2-10*d^2)-a^2*b^3*c*d*(4*c^2+55*d^2)+
a*b^4*(6*c^4+43*c^2*d^2-6*d^4))*cos(f*x+e)/b^4/(a^2-b^2)^2/f+1/2*d^2*(7*a^3*b*c*d^2-6*a^4*d^3+b^4*d*(8*c^2-d^2
)+a^2*b^2*d*(c^2+10*d^2)-a*b^3*c*(3*c^2+16*d^2))*cos(f*x+e)*sin(f*x+e)/b^3/(a^2-b^2)^2/f+1/2*(-a*d+b*c)^2*(4*a
^2*d+3*a*b*c-7*b^2*d)*cos(f*x+e)*(c+d*sin(f*x+e))^2/b^2/(a^2-b^2)^2/f/(a+b*sin(f*x+e))+1/2*(-a*d+b*c)^2*cos(f*
x+e)*(c+d*sin(f*x+e))^3/b/(a^2-b^2)/f/(a+b*sin(f*x+e))^2

________________________________________________________________________________________

Rubi [A]
time = 1.44, antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2871, 3126, 3112, 3102, 2814, 2739, 632, 210} \begin {gather*} \frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b f \left (a^2-b^2\right ) (a+b \sin (e+f x))^2}+\frac {(b c-a d)^2 \left (4 a^2 d+3 a b c-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))}-\frac {d^3 x \left (-12 a^2 d^2+30 a b c d-\left (b^2 \left (20 c^2+d^2\right )\right )\right )}{2 b^5}+\frac {(b c-a d)^3 \left (12 a^4 d^2+6 a^3 b c d+a^2 b^2 \left (2 c^2-29 d^2\right )-12 a b^3 c d+b^4 \left (c^2+20 d^2\right )\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 f \left (a^2-b^2\right )^{5/2}}+\frac {d^2 \left (-6 a^4 d^3+7 a^3 b c d^2+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )+b^4 d \left (8 c^2-d^2\right )\right ) \sin (e+f x) \cos (e+f x)}{2 b^3 f \left (a^2-b^2\right )^2}-\frac {d \left (-12 a^5 d^4+30 a^4 b c d^3-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )-b^5 c d \left (17 c^2-10 d^2\right )\right ) \cos (e+f x)}{2 b^4 f \left (a^2-b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*Sin[e + f*x])^5/(a + b*Sin[e + f*x])^3,x]

[Out]

-1/2*(d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*x)/b^5 + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d +
12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])
/(b^5*(a^2 - b^2)^(5/2)*f) - (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2
- 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(2*b^4*(a^2 - b^
2)^2*f) + (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 +
16*d^2))*Cos[e + f*x]*Sin[e + f*x])/(2*b^3*(a^2 - b^2)^2*f) + ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*Cos
[e + f*x]*(c + d*Sin[e + f*x])^2)/(2*b^2*(a^2 - b^2)^2*f*(a + b*Sin[e + f*x])) + ((b*c - a*d)^2*Cos[e + f*x]*(
c + d*Sin[e + f*x])^3)/(2*b*(a^2 - b^2)*f*(a + b*Sin[e + f*x])^2)

Rule 210

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])

Rule 632

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 2739

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 2814

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 2871

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])

Rule 3102

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 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3126

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

Rubi steps

\begin {align*} \int \frac {(c+d \sin (e+f x))^5}{(a+b \sin (e+f x))^3} \, dx &=\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x))^2 \left (7 b^2 c^2 d+3 a^2 d^3-2 a b c \left (c^2+4 d^2\right )-\left (a^2 c d^2+2 a b d \left (2 c^2+d^2\right )-b^2 \left (c^3+6 c d^2\right )\right ) \sin (e+f x)+2 d \left (2 a b c d-2 a^2 d^2-b^2 \left (c^2-d^2\right )\right ) \sin ^2(e+f x)\right )}{(a+b \sin (e+f x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {(c+d \sin (e+f x)) \left (11 a^3 b c d^3-8 a^4 d^4+b^4 c^2 \left (c^2+20 d^2\right )-a b^3 c d \left (15 c^2+32 d^2\right )+a^2 b^2 \left (2 c^4+7 c^2 d^2+14 d^4\right )+d \left (4 a^4 c d^2-b^4 c \left (c^2-8 d^2\right )+a^2 b^2 c \left (4 c^2-3 d^2\right )-a^3 b d \left (4 c^2-d^2\right )-a b^3 d \left (5 c^2+4 d^2\right )\right ) \sin (e+f x)-2 d \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {-2 \left (15 a^4 b c d^4-6 a^5 d^5-10 a^3 b^2 d^3 \left (c^2-d^2\right )-b^5 c^3 \left (c^2+20 d^2\right )+a b^4 d \left (15 c^4+40 c^2 d^2-d^4\right )-2 a^2 b^3 c \left (c^4+5 c^2 d^2+15 d^4\right )\right )+2 b d \left (b^4 d^2 \left (20 c^2+d^2\right )-a b^3 c d \left (17 c^2+20 d^2\right )-a^3 b \left (4 c^3 d-5 c d^3\right )+a^4 \left (4 c^2 d^2-2 d^4\right )+a^2 b^2 \left (6 c^4+3 c^2 d^2+4 d^4\right )\right ) \sin (e+f x)+2 d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\int \frac {-2 b \left (15 a^4 b c d^4-6 a^5 d^5-10 a^3 b^2 d^3 \left (c^2-d^2\right )-b^5 c^3 \left (c^2+20 d^2\right )+a b^4 d \left (15 c^4+40 c^2 d^2-d^4\right )-2 a^2 b^3 c \left (c^4+5 c^2 d^2+15 d^4\right )\right )-2 \left (a^2-b^2\right )^2 d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=-\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left ((b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}+\frac {\left ((b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^5 \left (a^2-b^2\right )^2 f}\\ &=-\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}-\frac {\left (2 (b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right )\right ) \text {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} (e+f x)\right )\right )}{b^5 \left (a^2-b^2\right )^2 f}\\ &=-\frac {d^3 \left (30 a b c d-12 a^2 d^2-b^2 \left (20 c^2+d^2\right )\right ) x}{2 b^5}+\frac {(b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2} f}-\frac {d \left (30 a^4 b c d^3-12 a^5 d^4-a^3 b^2 d^2 \left (16 c^2-21 d^2\right )-b^5 c d \left (17 c^2-10 d^2\right )-a^2 b^3 c d \left (4 c^2+55 d^2\right )+a b^4 \left (6 c^4+43 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{2 b^4 \left (a^2-b^2\right )^2 f}+\frac {d^2 \left (7 a^3 b c d^2-6 a^4 d^3+b^4 d \left (8 c^2-d^2\right )+a^2 b^2 d \left (c^2+10 d^2\right )-a b^3 c \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sin (e+f x)}{2 b^3 \left (a^2-b^2\right )^2 f}+\frac {(b c-a d)^2 \left (3 a b c+4 a^2 d-7 b^2 d\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{2 b^2 \left (a^2-b^2\right )^2 f (a+b \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x) (c+d \sin (e+f x))^3}{2 b \left (a^2-b^2\right ) f (a+b \sin (e+f x))^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 4.06, size = 341, normalized size = 0.64 \begin {gather*} \frac {2 d^3 \left (-30 a b c d+12 a^2 d^2+b^2 \left (20 c^2+d^2\right )\right ) (e+f x)+\frac {4 (b c-a d)^3 \left (6 a^3 b c d-12 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-29 d^2\right )+b^4 \left (c^2+20 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+2 b d^4 (-5 b c+3 a d) (\cos (e+f x)-i \sin (e+f x))+2 b d^4 (-5 b c+3 a d) (\cos (e+f x)+i \sin (e+f x))-\frac {2 b (b c-a d)^5 \cos (e+f x)}{\left (-a^2+b^2\right ) (a+b \sin (e+f x))^2}+\frac {2 b (b c-a d)^4 \left (3 a b c+7 a^2 d-10 b^2 d\right ) \cos (e+f x)}{\left (a^2-b^2\right )^2 (a+b \sin (e+f x))}-b^2 d^5 \sin (2 (e+f x))}{4 b^5 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Sin[e + f*x])^5/(a + b*Sin[e + f*x])^3,x]

[Out]

(2*d^3*(-30*a*b*c*d + 12*a^2*d^2 + b^2*(20*c^2 + d^2))*(e + f*x) + (4*(b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*
d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2
]])/(a^2 - b^2)^(5/2) + 2*b*d^4*(-5*b*c + 3*a*d)*(Cos[e + f*x] - I*Sin[e + f*x]) + 2*b*d^4*(-5*b*c + 3*a*d)*(C
os[e + f*x] + I*Sin[e + f*x]) - (2*b*(b*c - a*d)^5*Cos[e + f*x])/((-a^2 + b^2)*(a + b*Sin[e + f*x])^2) + (2*b*
(b*c - a*d)^4*(3*a*b*c + 7*a^2*d - 10*b^2*d)*Cos[e + f*x])/((a^2 - b^2)^2*(a + b*Sin[e + f*x])) - b^2*d^5*Sin[
2*(e + f*x)])/(4*b^5*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1122\) vs. \(2(519)=1038\).
time = 1.16, size = 1123, normalized size = 2.10 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(2*d^3/b^5*((1/2*d^2*b^2*tan(1/2*f*x+1/2*e)^3+(3*a*b*d^2-5*b^2*c*d)*tan(1/2*f*x+1/2*e)^2-1/2*d^2*b^2*tan(1
/2*f*x+1/2*e)+3*a*b*d^2-5*b^2*c*d)/(1+tan(1/2*f*x+1/2*e)^2)^2+1/2*(12*a^2*d^2-30*a*b*c*d+20*b^2*c^2+b^2*d^2)*a
rctan(tan(1/2*f*x+1/2*e)))-2/b^5*((-1/2*b^2*(5*a^7*d^5-15*a^6*b*c*d^4+10*a^5*b^2*c^2*d^3-8*a^5*b^2*d^5+10*a^4*
b^3*c^3*d^2+30*a^4*b^3*c*d^4-15*a^3*b^4*c^4*d-40*a^3*b^4*c^2*d^3+5*a^2*b^5*c^5+20*a^2*b^5*c^3*d^2-2*b^7*c^5)/(
a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3-1/2*b*(6*a^9*d^5-20*a^8*b*c*d^4+20*a^7*b^2*c^2*d^3+3*a^7*b^2*d^5-5*a
^6*b^3*c*d^4-10*a^5*b^4*c^4*d-10*a^5*b^4*c^2*d^3-18*a^5*b^4*d^5+4*a^4*b^5*c^5+30*a^4*b^5*c^3*d^2+70*a^4*b^5*c*
d^4-25*a^3*b^6*c^4*d-100*a^3*b^6*c^2*d^3+7*a^2*b^7*c^5+60*a^2*b^7*c^3*d^2-10*a*b^8*c^4*d-2*b^9*c^5)/(a^4-2*a^2
*b^2+b^4)/a^2*tan(1/2*f*x+1/2*e)^2-1/2*b^2*(19*a^7*d^5-65*a^6*b*c*d^4+70*a^5*b^2*c^2*d^3-28*a^5*b^2*d^5-10*a^4
*b^3*c^3*d^2+110*a^4*b^3*c*d^4-25*a^3*b^4*c^4*d-160*a^3*b^4*c^2*d^3+11*a^2*b^5*c^5+100*a^2*b^5*c^3*d^2-20*a*b^
6*c^4*d-2*b^7*c^5)/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)-1/2*b*(6*a^7*d^5-20*a^6*b*c*d^4+20*a^5*b^2*c^2*d^3
-9*a^5*b^2*d^5+35*a^4*b^3*c*d^4-10*a^3*b^4*c^4*d-50*a^3*b^4*c^2*d^3+4*a^2*b^5*c^5+30*a^2*b^5*c^3*d^2-5*a*b^6*c
^4*d-b^7*c^5)/(a^4-2*a^2*b^2+b^4))/(a*tan(1/2*f*x+1/2*e)^2+2*b*tan(1/2*f*x+1/2*e)+a)^2+1/2*(12*a^7*d^5-30*a^6*
b*c*d^4+20*a^5*b^2*c^2*d^3-29*a^5*b^2*d^5+75*a^4*b^3*c*d^4-50*a^3*b^4*c^2*d^3+20*a^3*b^4*d^5-2*a^2*b^5*c^5-10*
a^2*b^5*c^3*d^2-60*a^2*b^5*c*d^4+15*a*b^6*c^4*d+60*a*b^6*c^2*d^3-b^7*c^5-20*b^7*c^3*d^2)/(a^4-2*a^2*b^2+b^4)/(
a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,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 de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1556 vs. \(2 (529) = 1058\).
time = 0.53, size = 3203, normalized size = 6.00 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/4*(2*(20*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c*
d^4 + (12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 - 9*a^2*b^8 - b^10)*d^5)*f*x*cos(f*x + e)^2 - 4*(5*(a^6*b^4 - 3*a^
4*b^6 + 3*a^2*b^8 - b^10)*c*d^4 - 2*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*d^5)*cos(f*x + e)^3 - 2*(20*(a^8
*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10)*c^2*d^3 - 30*(a^9*b - 2*a^7*b^3 + 2*a^3*b^7 - a*b^9)*c*d^4 + (12*a^10 - 2
3*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8 - b^10)*d^5)*f*x - ((2*a^4*b^5 + 3*a^2*b^7 + b^9)*c^5 - 15*(a^
3*b^6 + a*b^8)*c^4*d + 10*(a^4*b^5 + 3*a^2*b^7 + 2*b^9)*c^3*d^2 - 10*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6 + 6*a*b^
8)*c^2*d^3 + 15*(2*a^8*b - 3*a^6*b^3 - a^4*b^5 + 4*a^2*b^7)*c*d^4 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*
b^6)*d^5 + (15*a*b^8*c^4*d - (2*a^2*b^7 + b^9)*c^5 - 10*(a^2*b^7 + 2*b^9)*c^3*d^2 + 10*(2*a^5*b^4 - 5*a^3*b^6
+ 6*a*b^8)*c^2*d^3 - 15*(2*a^6*b^3 - 5*a^4*b^5 + 4*a^2*b^7)*c*d^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*d^5
)*cos(f*x + e)^2 - 2*(15*a^2*b^7*c^4*d - (2*a^3*b^6 + a*b^8)*c^5 - 10*(a^3*b^6 + 2*a*b^8)*c^3*d^2 + 10*(2*a^6*
b^3 - 5*a^4*b^5 + 6*a^2*b^7)*c^2*d^3 - 15*(2*a^7*b^2 - 5*a^5*b^4 + 4*a^3*b^6)*c*d^4 + (12*a^8*b - 29*a^6*b^3 +
20*a^4*b^5)*d^5)*sin(f*x + e))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2
- b^2 - 2*(a*cos(f*x + e)*sin(f*x + e) + b*cos(f*x + e))*sqrt(-a^2 + b^2))/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*
x + e) - a^2 - b^2)) - 2*((4*a^4*b^6 - 5*a^2*b^8 + b^10)*c^5 - 5*(2*a^5*b^5 - a^3*b^7 - a*b^9)*c^4*d + 30*(a^4
*b^6 - a^2*b^8)*c^3*d^2 + 10*(2*a^7*b^3 - 7*a^5*b^5 + 5*a^3*b^7)*c^2*d^3 - 5*(6*a^8*b^2 - 15*a^6*b^4 + 7*a^4*b
^6 + 4*a^2*b^8 - 2*b^10)*c*d^4 + (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6*a^3*b^7 - 4*a*b^9)*d^5)*cos(f*x + e)
- 2*((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*d^5*cos(f*x + e)^3 + 2*(20*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*
b^9)*c^2*d^3 - 30*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*c*d^4 + (12*a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*
a^3*b^7 - a*b^9)*d^5)*f*x + (3*(a^3*b^7 - a*b^9)*c^5 - 5*(a^4*b^6 + a^2*b^8 - 2*b^10)*c^4*d - 10*(a^5*b^5 - 5*
a^3*b^7 + 4*a*b^9)*c^3*d^2 + 30*(a^6*b^4 - 3*a^4*b^6 + 2*a^2*b^8)*c^2*d^3 - 5*(9*a^7*b^3 - 25*a^5*b^5 + 20*a^3
*b^7 - 4*a*b^9)*c*d^4 + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8 + b^10)*d^5)*cos(f*x + e))*sin(f*x
+ e))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*f*cos(f*x + e)^2 - 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^
12)*f*sin(f*x + e) - (a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13)*f), 1/2*((20*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 -
b^10)*c^2*d^3 - 30*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c*d^4 + (12*a^8*b^2 - 35*a^6*b^4 + 33*a^4*b^6 -
9*a^2*b^8 - b^10)*d^5)*f*x*cos(f*x + e)^2 - 2*(5*(a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*c*d^4 - 2*(a^7*b^3 -
3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*d^5)*cos(f*x + e)^3 - (20*(a^8*b^2 - 2*a^6*b^4 + 2*a^2*b^8 - b^10)*c^2*d^3 - 3
0*(a^9*b - 2*a^7*b^3 + 2*a^3*b^7 - a*b^9)*c*d^4 + (12*a^10 - 23*a^8*b^2 - 2*a^6*b^4 + 24*a^4*b^6 - 10*a^2*b^8
- b^10)*d^5)*f*x + ((2*a^4*b^5 + 3*a^2*b^7 + b^9)*c^5 - 15*(a^3*b^6 + a*b^8)*c^4*d + 10*(a^4*b^5 + 3*a^2*b^7 +
2*b^9)*c^3*d^2 - 10*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6 + 6*a*b^8)*c^2*d^3 + 15*(2*a^8*b - 3*a^6*b^3 - a^4*b^5 +
4*a^2*b^7)*c*d^4 - (12*a^9 - 17*a^7*b^2 - 9*a^5*b^4 + 20*a^3*b^6)*d^5 + (15*a*b^8*c^4*d - (2*a^2*b^7 + b^9)*c
^5 - 10*(a^2*b^7 + 2*b^9)*c^3*d^2 + 10*(2*a^5*b^4 - 5*a^3*b^6 + 6*a*b^8)*c^2*d^3 - 15*(2*a^6*b^3 - 5*a^4*b^5 +
4*a^2*b^7)*c*d^4 + (12*a^7*b^2 - 29*a^5*b^4 + 20*a^3*b^6)*d^5)*cos(f*x + e)^2 - 2*(15*a^2*b^7*c^4*d - (2*a^3*
b^6 + a*b^8)*c^5 - 10*(a^3*b^6 + 2*a*b^8)*c^3*d^2 + 10*(2*a^6*b^3 - 5*a^4*b^5 + 6*a^2*b^7)*c^2*d^3 - 15*(2*a^7
*b^2 - 5*a^5*b^4 + 4*a^3*b^6)*c*d^4 + (12*a^8*b - 29*a^6*b^3 + 20*a^4*b^5)*d^5)*sin(f*x + e))*sqrt(a^2 - b^2)*
arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - ((4*a^4*b^6 - 5*a^2*b^8 + b^10)*c^5 - 5*(2*a^5*
b^5 - a^3*b^7 - a*b^9)*c^4*d + 30*(a^4*b^6 - a^2*b^8)*c^3*d^2 + 10*(2*a^7*b^3 - 7*a^5*b^5 + 5*a^3*b^7)*c^2*d^3
- 5*(6*a^8*b^2 - 15*a^6*b^4 + 7*a^4*b^6 + 4*a^2*b^8 - 2*b^10)*c*d^4 + (12*a^9*b - 29*a^7*b^3 + 15*a^5*b^5 + 6
*a^3*b^7 - 4*a*b^9)*d^5)*cos(f*x + e) - ((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*d^5*cos(f*x + e)^3 + 2*(20*(
a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9)*c^2*d^3 - 30*(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8)*c*d^4 + (12*
a^9*b - 35*a^7*b^3 + 33*a^5*b^5 - 9*a^3*b^7 - a*b^9)*d^5)*f*x + (3*(a^3*b^7 - a*b^9)*c^5 - 5*(a^4*b^6 + a^2*b^
8 - 2*b^10)*c^4*d - 10*(a^5*b^5 - 5*a^3*b^7 + 4*a*b^9)*c^3*d^2 + 30*(a^6*b^4 - 3*a^4*b^6 + 2*a^2*b^8)*c^2*d^3
- 5*(9*a^7*b^3 - 25*a^5*b^5 + 20*a^3*b^7 - 4*a*b^9)*c*d^4 + (18*a^8*b^2 - 51*a^6*b^4 + 46*a^4*b^6 - 14*a^2*b^8
+ b^10)*d^5)*cos(f*x + e))*sin(f*x + e))/((a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*f*cos(f*x + e)^2 - 2*(a^7
*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*f*sin(f*x + e) - (a^8*b^5 - 2*a^6*b^7 + 2*a^2*b^11 - b^13)*f)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))**5/(a+b*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3162 vs. \(2 (529) = 1058\).
time = 0.60, size = 3162, normalized size = 5.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*sin(f*x+e))^5/(a+b*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/2*(2*(2*a^2*b^5*c^5 + b^7*c^5 - 15*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 + 20*b^7*c^3*d^2 - 20*a^5*b^2*c^2*d^3 +
50*a^3*b^4*c^2*d^3 - 60*a*b^6*c^2*d^3 + 30*a^6*b*c*d^4 - 75*a^4*b^3*c*d^4 + 60*a^2*b^5*c*d^4 - 12*a^7*d^5 + 29
*a^5*b^2*d^5 - 20*a^3*b^4*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*f*x + 1/2*e) + b)/
sqrt(a^2 - b^2)))/((a^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(a^2 - b^2)) + 2*(5*a^3*b^6*c^5*tan(1/2*f*x + 1/2*e)^7 - 2*
a*b^8*c^5*tan(1/2*f*x + 1/2*e)^7 - 15*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e)^7 + 10*a^5*b^4*c^3*d^2*tan(1/2*f*x +
1/2*e)^7 + 20*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*e)^7 + 10*a^6*b^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^7 - 40*a^4*b^5*
c^2*d^3*tan(1/2*f*x + 1/2*e)^7 - 15*a^7*b^2*c*d^4*tan(1/2*f*x + 1/2*e)^7 + 30*a^5*b^4*c*d^4*tan(1/2*f*x + 1/2*
e)^7 + 6*a^8*b*d^5*tan(1/2*f*x + 1/2*e)^7 - 10*a^6*b^3*d^5*tan(1/2*f*x + 1/2*e)^7 + a^4*b^5*d^5*tan(1/2*f*x +
1/2*e)^7 + 4*a^4*b^5*c^5*tan(1/2*f*x + 1/2*e)^6 + 7*a^2*b^7*c^5*tan(1/2*f*x + 1/2*e)^6 - 2*b^9*c^5*tan(1/2*f*x
+ 1/2*e)^6 - 10*a^5*b^4*c^4*d*tan(1/2*f*x + 1/2*e)^6 - 25*a^3*b^6*c^4*d*tan(1/2*f*x + 1/2*e)^6 - 10*a*b^8*c^4
*d*tan(1/2*f*x + 1/2*e)^6 + 30*a^4*b^5*c^3*d^2*tan(1/2*f*x + 1/2*e)^6 + 60*a^2*b^7*c^3*d^2*tan(1/2*f*x + 1/2*e
)^6 + 20*a^7*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^6 - 10*a^5*b^4*c^2*d^3*tan(1/2*f*x + 1/2*e)^6 - 100*a^3*b^6*c^2*
d^3*tan(1/2*f*x + 1/2*e)^6 - 30*a^8*b*c*d^4*tan(1/2*f*x + 1/2*e)^6 + 15*a^6*b^3*c*d^4*tan(1/2*f*x + 1/2*e)^6 +
60*a^4*b^5*c*d^4*tan(1/2*f*x + 1/2*e)^6 + 12*a^9*d^5*tan(1/2*f*x + 1/2*e)^6 - 5*a^7*b^2*d^5*tan(1/2*f*x + 1/2
*e)^6 - 20*a^5*b^4*d^5*tan(1/2*f*x + 1/2*e)^6 + 4*a^3*b^6*d^5*tan(1/2*f*x + 1/2*e)^6 + 21*a^3*b^6*c^5*tan(1/2*
f*x + 1/2*e)^5 - 6*a*b^8*c^5*tan(1/2*f*x + 1/2*e)^5 - 55*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e)^5 - 20*a^2*b^7*c^4
*d*tan(1/2*f*x + 1/2*e)^5 + 10*a^5*b^4*c^3*d^2*tan(1/2*f*x + 1/2*e)^5 + 140*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*
e)^5 + 90*a^6*b^3*c^2*d^3*tan(1/2*f*x + 1/2*e)^5 - 240*a^4*b^5*c^2*d^3*tan(1/2*f*x + 1/2*e)^5 - 135*a^7*b^2*c*
d^4*tan(1/2*f*x + 1/2*e)^5 + 250*a^5*b^4*c*d^4*tan(1/2*f*x + 1/2*e)^5 - 40*a^3*b^6*c*d^4*tan(1/2*f*x + 1/2*e)^
5 + 54*a^8*b*d^5*tan(1/2*f*x + 1/2*e)^5 - 90*a^6*b^3*d^5*tan(1/2*f*x + 1/2*e)^5 + 17*a^4*b^5*d^5*tan(1/2*f*x +
1/2*e)^5 + 4*a^2*b^7*d^5*tan(1/2*f*x + 1/2*e)^5 + 12*a^4*b^5*c^5*tan(1/2*f*x + 1/2*e)^4 + 13*a^2*b^7*c^5*tan(
1/2*f*x + 1/2*e)^4 - 4*b^9*c^5*tan(1/2*f*x + 1/2*e)^4 - 30*a^5*b^4*c^4*d*tan(1/2*f*x + 1/2*e)^4 - 55*a^3*b^6*c
^4*d*tan(1/2*f*x + 1/2*e)^4 - 20*a*b^8*c^4*d*tan(1/2*f*x + 1/2*e)^4 + 90*a^4*b^5*c^3*d^2*tan(1/2*f*x + 1/2*e)^
4 + 120*a^2*b^7*c^3*d^2*tan(1/2*f*x + 1/2*e)^4 + 60*a^7*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^4 - 70*a^5*b^4*c^2*d^
3*tan(1/2*f*x + 1/2*e)^4 - 200*a^3*b^6*c^2*d^3*tan(1/2*f*x + 1/2*e)^4 - 90*a^8*b*c*d^4*tan(1/2*f*x + 1/2*e)^4
+ 45*a^6*b^3*c*d^4*tan(1/2*f*x + 1/2*e)^4 + 190*a^4*b^5*c*d^4*tan(1/2*f*x + 1/2*e)^4 - 40*a^2*b^7*c*d^4*tan(1/
2*f*x + 1/2*e)^4 + 36*a^9*d^5*tan(1/2*f*x + 1/2*e)^4 - 15*a^7*b^2*d^5*tan(1/2*f*x + 1/2*e)^4 - 66*a^5*b^4*d^5*
tan(1/2*f*x + 1/2*e)^4 + 24*a^3*b^6*d^5*tan(1/2*f*x + 1/2*e)^4 + 27*a^3*b^6*c^5*tan(1/2*f*x + 1/2*e)^3 - 6*a*b
^8*c^5*tan(1/2*f*x + 1/2*e)^3 - 65*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e)^3 - 40*a^2*b^7*c^4*d*tan(1/2*f*x + 1/2*e
)^3 - 10*a^5*b^4*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 220*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 150*a^6*b^3*c^2
*d^3*tan(1/2*f*x + 1/2*e)^3 - 360*a^4*b^5*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - 225*a^7*b^2*c*d^4*tan(1/2*f*x + 1/2
*e)^3 + 410*a^5*b^4*c*d^4*tan(1/2*f*x + 1/2*e)^3 - 80*a^3*b^6*c*d^4*tan(1/2*f*x + 1/2*e)^3 + 90*a^8*b*d^5*tan(
1/2*f*x + 1/2*e)^3 - 162*a^6*b^3*d^5*tan(1/2*f*x + 1/2*e)^3 + 55*a^4*b^5*d^5*tan(1/2*f*x + 1/2*e)^3 - 4*a^2*b^
7*d^5*tan(1/2*f*x + 1/2*e)^3 + 12*a^4*b^5*c^5*tan(1/2*f*x + 1/2*e)^2 + 5*a^2*b^7*c^5*tan(1/2*f*x + 1/2*e)^2 -
2*b^9*c^5*tan(1/2*f*x + 1/2*e)^2 - 30*a^5*b^4*c^4*d*tan(1/2*f*x + 1/2*e)^2 - 35*a^3*b^6*c^4*d*tan(1/2*f*x + 1/
2*e)^2 - 10*a*b^8*c^4*d*tan(1/2*f*x + 1/2*e)^2 + 90*a^4*b^5*c^3*d^2*tan(1/2*f*x + 1/2*e)^2 + 60*a^2*b^7*c^3*d^
2*tan(1/2*f*x + 1/2*e)^2 + 60*a^7*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 110*a^5*b^4*c^2*d^3*tan(1/2*f*x + 1/2*e
)^2 - 100*a^3*b^6*c^2*d^3*tan(1/2*f*x + 1/2*e)^2 - 90*a^8*b*c*d^4*tan(1/2*f*x + 1/2*e)^2 + 85*a^6*b^3*c*d^4*ta
n(1/2*f*x + 1/2*e)^2 + 120*a^4*b^5*c*d^4*tan(1/2*f*x + 1/2*e)^2 - 40*a^2*b^7*c*d^4*tan(1/2*f*x + 1/2*e)^2 + 36
*a^9*d^5*tan(1/2*f*x + 1/2*e)^2 - 31*a^7*b^2*d^5*tan(1/2*f*x + 1/2*e)^2 - 40*a^5*b^4*d^5*tan(1/2*f*x + 1/2*e)^
2 + 20*a^3*b^6*d^5*tan(1/2*f*x + 1/2*e)^2 + 11*a^3*b^6*c^5*tan(1/2*f*x + 1/2*e) - 2*a*b^8*c^5*tan(1/2*f*x + 1/
2*e) - 25*a^4*b^5*c^4*d*tan(1/2*f*x + 1/2*e) - 20*a^2*b^7*c^4*d*tan(1/2*f*x + 1/2*e) - 10*a^5*b^4*c^3*d^2*tan(
1/2*f*x + 1/2*e) + 100*a^3*b^6*c^3*d^2*tan(1/2*f*x + 1/2*e) + 70*a^6*b^3*c^2*d^3*tan(1/2*f*x + 1/2*e) - 160*a^
4*b^5*c^2*d^3*tan(1/2*f*x + 1/2*e) - 105*a^7*b^2*c*d^4*tan(1/2*f*x + 1/2*e) + 190*a^5*b^4*c*d^4*tan(1/2*f*x +
1/2*e) - 40*a^3*b^6*c*d^4*tan(1/2*f*x + 1/2*e) + 42*a^8*b*d^5*tan(1/2*f*x + 1/2*e) - 74*a^6*b^3*d^5*tan(1/2*f*
x + 1/2*e) + 23*a^4*b^5*d^5*tan(1/2*f*x + 1/2*e...

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Mupad [B]
time = 25.76, size = 2500, normalized size = 4.68 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*sin(e + f*x))^5/(a + b*sin(e + f*x))^3,x)

[Out]

- ((b^7*c^5 - 12*a^7*d^5 - 4*a^2*b^5*c^5 - 6*a^3*b^4*d^5 + 21*a^5*b^2*d^5 + 10*a^2*b^5*c*d^4 + 10*a^3*b^4*c^4*
d - 55*a^4*b^3*c*d^4 - 30*a^2*b^5*c^3*d^2 + 50*a^3*b^4*c^2*d^3 - 20*a^5*b^2*c^2*d^3 + 5*a*b^6*c^4*d + 30*a^6*b
*c*d^4)/(b^4*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^6*(12*a^9*d^5 - 2*b^9*c^5 + 7*a^2*b^7*c^5 + 4*a^4*
b^5*c^5 + 4*a^3*b^6*d^5 - 20*a^5*b^4*d^5 - 5*a^7*b^2*d^5 - 25*a^3*b^6*c^4*d + 60*a^4*b^5*c*d^4 - 10*a^5*b^4*c^
4*d + 15*a^6*b^3*c*d^4 + 60*a^2*b^7*c^3*d^2 - 100*a^3*b^6*c^2*d^3 + 30*a^4*b^5*c^3*d^2 - 10*a^5*b^4*c^2*d^3 +
20*a^7*b^2*c^2*d^3 - 10*a*b^8*c^4*d - 30*a^8*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)
^2*(2*b^9*c^5 - 36*a^9*d^5 - 5*a^2*b^7*c^5 - 12*a^4*b^5*c^5 - 20*a^3*b^6*d^5 + 40*a^5*b^4*d^5 + 31*a^7*b^2*d^5
+ 40*a^2*b^7*c*d^4 + 35*a^3*b^6*c^4*d - 120*a^4*b^5*c*d^4 + 30*a^5*b^4*c^4*d - 85*a^6*b^3*c*d^4 - 60*a^2*b^7*
c^3*d^2 + 100*a^3*b^6*c^2*d^3 - 90*a^4*b^5*c^3*d^2 + 110*a^5*b^4*c^2*d^3 - 60*a^7*b^2*c^2*d^3 + 10*a*b^8*c^4*d
+ 90*a^8*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^5*(54*a^7*d^5 - 6*b^7*c^5 + 4*a*b^
6*d^5 + 21*a^2*b^5*c^5 + 17*a^3*b^4*d^5 - 90*a^5*b^2*d^5 - 40*a^2*b^5*c*d^4 - 55*a^3*b^4*c^4*d + 250*a^4*b^3*c
*d^4 + 140*a^2*b^5*c^3*d^2 - 240*a^3*b^4*c^2*d^3 + 10*a^4*b^3*c^3*d^2 + 90*a^5*b^2*c^2*d^3 - 20*a*b^6*c^4*d -
135*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)^3*(6*b^7*c^5 - 90*a^7*d^5 + 4*a*b^6*d^
5 - 27*a^2*b^5*c^5 - 55*a^3*b^4*d^5 + 162*a^5*b^2*d^5 + 80*a^2*b^5*c*d^4 + 65*a^3*b^4*c^4*d - 410*a^4*b^3*c*d^
4 - 220*a^2*b^5*c^3*d^2 + 360*a^3*b^4*c^2*d^3 + 10*a^4*b^3*c^3*d^2 - 150*a^5*b^2*c^2*d^3 + 40*a*b^6*c^4*d + 22
5*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) - (tan(e/2 + (f*x)/2)^7*(6*a^7*d^5 - 2*b^7*c^5 + 5*a^2*b^5*c^5
+ a^3*b^4*d^5 - 10*a^5*b^2*d^5 - 15*a^3*b^4*c^4*d + 30*a^4*b^3*c*d^4 + 20*a^2*b^5*c^3*d^2 - 40*a^3*b^4*c^2*d^
3 + 10*a^4*b^3*c^3*d^2 + 10*a^5*b^2*c^2*d^3 - 15*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f
*x)/2)*(2*b^7*c^5 - 42*a^7*d^5 - 11*a^2*b^5*c^5 - 23*a^3*b^4*d^5 + 74*a^5*b^2*d^5 + 40*a^2*b^5*c*d^4 + 25*a^3*
b^4*c^4*d - 190*a^4*b^3*c*d^4 - 100*a^2*b^5*c^3*d^2 + 160*a^3*b^4*c^2*d^3 + 10*a^4*b^3*c^3*d^2 - 70*a^5*b^2*c^
2*d^3 + 20*a*b^6*c^4*d + 105*a^6*b*c*d^4))/(a*b^3*(a^4 + b^4 - 2*a^2*b^2)) + (tan(e/2 + (f*x)/2)^4*(3*a^2 + 4*
b^2)*(b^7*c^5 - 12*a^7*d^5 - 4*a^2*b^5*c^5 - 6*a^3*b^4*d^5 + 21*a^5*b^2*d^5 + 10*a^2*b^5*c*d^4 + 10*a^3*b^4*c^
4*d - 55*a^4*b^3*c*d^4 - 30*a^2*b^5*c^3*d^2 + 50*a^3*b^4*c^2*d^3 - 20*a^5*b^2*c^2*d^3 + 5*a*b^6*c^4*d + 30*a^6
*b*c*d^4))/(a^2*b^4*(a^4 + b^4 - 2*a^2*b^2)))/(f*(tan(e/2 + (f*x)/2)^2*(4*a^2 + 4*b^2) + tan(e/2 + (f*x)/2)^6*
(4*a^2 + 4*b^2) + tan(e/2 + (f*x)/2)^4*(6*a^2 + 8*b^2) + a^2*tan(e/2 + (f*x)/2)^8 + a^2 + 12*a*b*tan(e/2 + (f*
x)/2)^3 + 12*a*b*tan(e/2 + (f*x)/2)^5 + 4*a*b*tan(e/2 + (f*x)/2)^7 + 4*a*b*tan(e/2 + (f*x)/2))) - (atan(((((4*
(2*a^2*b^16*d^10 + 40*a^4*b^14*d^10 + 108*a^6*b^12*d^10 - 872*a^8*b^10*d^10 + 1538*a^10*b^8*d^10 - 1104*a^12*b
^6*d^10 + 288*a^14*b^4*d^10 - 120*a^3*b^15*c*d^9 - 960*a^5*b^13*c*d^9 + 5040*a^7*b^11*c*d^9 - 8160*a^9*b^9*c*d
^9 + 5640*a^11*b^7*c*d^9 - 1440*a^13*b^5*c*d^9 + 80*a^2*b^16*c^2*d^8 + 800*a^2*b^16*c^4*d^6 - 2400*a^3*b^15*c^
3*d^7 + 2440*a^4*b^14*c^2*d^8 - 3200*a^4*b^14*c^4*d^6 + 9600*a^5*b^13*c^3*d^7 - 10560*a^6*b^12*c^2*d^8 + 4800*
a^6*b^12*c^4*d^6 - 14400*a^7*b^11*c^3*d^7 + 16240*a^8*b^10*c^2*d^8 - 3200*a^8*b^10*c^4*d^6 + 9600*a^9*b^9*c^3*
d^7 - 10960*a^10*b^8*c^2*d^8 + 800*a^10*b^8*c^4*d^6 - 2400*a^11*b^7*c^3*d^7 + 2760*a^12*b^6*c^2*d^8))/(b^19 -
4*a^2*b^17 + 6*a^4*b^15 - 4*a^6*b^13 + a^8*b^11) - (8*tan(e/2 + (f*x)/2)*(a*b^18*c^10 - 2*a*b^18*d^10 + 4*a^3*
b^16*c^10 + 4*a^5*b^14*c^10 - 39*a^3*b^16*d^10 - 88*a^5*b^14*d^10 + 1326*a^7*b^12*d^10 - 3134*a^9*b^10*d^10 +
3194*a^11*b^8*d^10 - 1536*a^13*b^6*d^10 + 288*a^15*b^4*d^10 - 80*a*b^18*c^2*d^8 - 800*a*b^18*c^4*d^6 + 400*a*b
^18*c^6*d^4 + 40*a*b^18*c^8*d^2 + 120*a^2*b^17*c*d^9 - 30*a^2*b^17*c^9*d + 900*a^4*b^15*c*d^9 - 60*a^4*b^15*c^
9*d - 7920*a^6*b^13*c*d^9 + 17160*a^8*b^11*c*d^9 - 16710*a^10*b^9*c*d^9 + 7800*a^12*b^7*c*d^9 - 1440*a^14*b^5*
c*d^9 + 2400*a^2*b^17*c^3*d^7 - 2400*a^2*b^17*c^5*d^5 - 720*a^2*b^17*c^7*d^3 - 2400*a^3*b^16*c^2*d^8 + 9600*a^
3*b^16*c^4*d^6 + 2320*a^3*b^16*c^6*d^4 + 325*a^3*b^16*c^8*d^2 - 18800*a^4*b^15*c^3*d^7 - 1040*a^4*b^15*c^5*d^5
- 440*a^4*b^15*c^7*d^3 + 17780*a^5*b^14*c^2*d^8 - 13600*a^5*b^14*c^4*d^6 - 1310*a^5*b^14*c^6*d^4 + 40*a^5*b^1
4*c^8*d^2 + 34960*a^6*b^13*c^3*d^7 + 2428*a^6*b^13*c^5*d^5 + 160*a^6*b^13*c^7*d^3 - 36000*a^7*b^12*c^2*d^8 + 9
330*a^7*b^12*c^4*d^6 + 360*a^7*b^12*c^6*d^4 - 30200*a^8*b^11*c^3*d^7 - 1208*a^8*b^11*c^5*d^5 - 80*a^8*b^11*c^7
*d^3 + 33445*a^9*b^10*c^2*d^8 - 3440*a^9*b^10*c^4*d^6 + 120*a^9*b^10*c^6*d^4 + 12960*a^10*b^9*c^3*d^7 - 48*a^1
0*b^9*c^5*d^5 - 15100*a^11*b^8*c^2*d^8 + 800*a^11*b^8*c^4*d^6 - 2400*a^12*b^7*c^3*d^7 + 2760*a^13*b^6*c^2*d^8)
)/(b^20 - 4*a^2*b^18 + 6*a^4*b^16 - 4*a^6*b^14 + a^8*b^12) + ((a^2*d^5*6i + (b^2*d^3*(20*c^2 + d^2)*1i)/2 - a*
b*c*d^4*15i)*((8*tan(e/2 + (f*x)/2)*(4*a*b^21*c...

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