3.698 \(\int \frac {(c+d \sin (e+f x))^4}{a+b \sin (e+f x)} \, dx\)
Optimal. Leaf size=235 \[ \frac {2 (b c-a d)^4 \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 f \sqrt {a^2-b^2}}+\frac {d^2 \left (-3 a^2 d^2+12 a b c d-\left (b^2 \left (17 c^2+2 d^2\right )\right )\right ) \cos (e+f x)}{3 b^3 f}+\frac {d x \left (-2 a^3 d^3+8 a^2 b c d^2-a b^2 d \left (12 c^2+d^2\right )+4 b^3 c \left (2 c^2+d^2\right )\right )}{2 b^4}-\frac {d^3 (8 b c-3 a d) \sin (e+f x) \cos (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f} \]
[Out]
1/2*d*(8*a^2*b*c*d^2-2*a^3*d^3+4*b^3*c*(2*c^2+d^2)-a*b^2*d*(12*c^2+d^2))*x/b^4+1/3*d^2*(12*a*b*c*d-3*a^2*d^2-b
^2*(17*c^2+2*d^2))*cos(f*x+e)/b^3/f-1/6*d^3*(-3*a*d+8*b*c)*cos(f*x+e)*sin(f*x+e)/b^2/f-1/3*d^2*cos(f*x+e)*(c+d
*sin(f*x+e))^2/b/f+2*(-a*d+b*c)^4*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/b^4/f/(a^2-b^2)^(1/2)
________________________________________________________________________________________
Rubi [A] time = 0.65, antiderivative size = 235, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used =
{2793, 3033, 3023, 2735, 2660, 618, 204} \[ \frac {d^2 \left (-3 a^2 d^2+12 a b c d+b^2 \left (-\left (17 c^2+2 d^2\right )\right )\right ) \cos (e+f x)}{3 b^3 f}+\frac {d x \left (8 a^2 b c d^2-2 a^3 d^3-a b^2 d \left (12 c^2+d^2\right )+4 b^3 c \left (2 c^2+d^2\right )\right )}{2 b^4}+\frac {2 (b c-a d)^4 \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^4 f \sqrt {a^2-b^2}}-\frac {d^3 (8 b c-3 a d) \sin (e+f x) \cos (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x]),x]
[Out]
(d*(8*a^2*b*c*d^2 - 2*a^3*d^3 + 4*b^3*c*(2*c^2 + d^2) - a*b^2*d*(12*c^2 + d^2))*x)/(2*b^4) + (2*(b*c - a*d)^4*
ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*f) + (d^2*(12*a*b*c*d - 3*a^2*d^2 - b^2
*(17*c^2 + 2*d^2))*Cos[e + f*x])/(3*b^3*f) - (d^3*(8*b*c - 3*a*d)*Cos[e + f*x]*Sin[e + f*x])/(6*b^2*f) - (d^2*
Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(3*b*f)
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 2660
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]
Rule 2735
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]
Rule 2793
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d
*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a*d
*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n -
2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 2] && ( !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 3033
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]
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^4}{a+b \sin (e+f x)} \, dx &=-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\int \frac {(c+d \sin (e+f x)) \left (3 b c^3+2 a d^3+d \left (9 b c^2-a c d+2 b d^2\right ) \sin (e+f x)+d^2 (8 b c-3 a d) \sin ^2(e+f x)\right )}{a+b \sin (e+f x)} \, dx}{3 b}\\ &=-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\int \frac {3 \left (2 b^2 c^4+4 a b c d^3-a^2 d^4\right )-b d \left (a d \left (2 c^2-d^2\right )-12 b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)-2 d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \sin ^2(e+f x)}{a+b \sin (e+f x)} \, dx}{6 b^2}\\ &=\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\int \frac {3 b \left (2 b^2 c^4+4 a b c d^3-a^2 d^4\right )+3 d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) \sin (e+f x)}{a+b \sin (e+f x)} \, dx}{6 b^3}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {(b c-a d)^4 \int \frac {1}{a+b \sin (e+f x)} \, dx}{b^4}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}+\frac {\left (2 (b c-a d)^4\right ) \operatorname {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^4 f}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}-\frac {\left (4 (b c-a d)^4\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} (e+f x)\right )\right )}{b^4 f}\\ &=\frac {d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) x}{2 b^4}+\frac {2 (b c-a d)^4 \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2} f}+\frac {d^2 \left (12 a b c d-3 a^2 d^2-b^2 \left (17 c^2+2 d^2\right )\right ) \cos (e+f x)}{3 b^3 f}-\frac {d^3 (8 b c-3 a d) \cos (e+f x) \sin (e+f x)}{6 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))^2}{3 b f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.58, size = 203, normalized size = 0.86 \[ \frac {-3 b d^2 \left (4 a^2 d^2-16 a b c d+3 b^2 \left (8 c^2+d^2\right )\right ) \cos (e+f x)+\frac {24 (b c-a d)^4 \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-6 d (e+f x) \left (2 a^3 d^3-8 a^2 b c d^2+a b^2 d \left (12 c^2+d^2\right )-4 b^3 c \left (2 c^2+d^2\right )\right )-3 b^2 d^3 (4 b c-a d) \sin (2 (e+f x))+b^3 d^4 \cos (3 (e+f x))}{12 b^4 f} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*Sin[e + f*x])^4/(a + b*Sin[e + f*x]),x]
[Out]
(-6*d*(-8*a^2*b*c*d^2 + 2*a^3*d^3 - 4*b^3*c*(2*c^2 + d^2) + a*b^2*d*(12*c^2 + d^2))*(e + f*x) + (24*(b*c - a*d
)^4*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 3*b*d^2*(-16*a*b*c*d + 4*a^2*d^2 + 3*b
^2*(8*c^2 + d^2))*Cos[e + f*x] + b^3*d^4*Cos[3*(e + f*x)] - 3*b^2*d^3*(4*b*c - a*d)*Sin[2*(e + f*x)])/(12*b^4*
f)
________________________________________________________________________________________
fricas [A] time = 0.51, size = 795, normalized size = 3.38 \[ \left [\frac {2 \, {\left (a^{2} b^{3} - b^{5}\right )} d^{4} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{3} d - 12 \, {\left (a^{3} b^{2} - a b^{4}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{4} b - a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} f x - 3 \, {\left (4 \, {\left (a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) - 6 \, {\left (6 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{2} d^{2} - 4 \, {\left (a^{3} b^{2} - a b^{4}\right )} c d^{3} + {\left (a^{4} b - b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\right )} f}, \frac {2 \, {\left (a^{2} b^{3} - b^{5}\right )} d^{4} \cos \left (f x + e\right )^{3} + 3 \, {\left (8 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{3} d - 12 \, {\left (a^{3} b^{2} - a b^{4}\right )} c^{2} d^{2} + 4 \, {\left (2 \, a^{4} b - a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} f x - 3 \, {\left (4 \, {\left (a^{2} b^{3} - b^{5}\right )} c d^{3} - {\left (a^{3} b^{2} - a b^{4}\right )} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) - 6 \, {\left (6 \, {\left (a^{2} b^{3} - b^{5}\right )} c^{2} d^{2} - 4 \, {\left (a^{3} b^{2} - a b^{4}\right )} c d^{3} + {\left (a^{4} b - b^{5}\right )} d^{4}\right )} \cos \left (f x + e\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\right )} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e)),x, algorithm="fricas")
[Out]
[1/6*(2*(a^2*b^3 - b^5)*d^4*cos(f*x + e)^3 + 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*b^2 - a*b^4)*c^2*d^2 + 4*(2*
a^4*b - a^2*b^3 - b^5)*c*d^3 - (2*a^5 - a^3*b^2 - a*b^4)*d^4)*f*x - 3*(4*(a^2*b^3 - b^5)*c*d^3 - (a^3*b^2 - a*
b^4)*d^4)*cos(f*x + e)*sin(f*x + e) - 3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4
)*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)) - 6*(6*(a^
2*b^3 - b^5)*c^2*d^2 - 4*(a^3*b^2 - a*b^4)*c*d^3 + (a^4*b - b^5)*d^4)*cos(f*x + e))/((a^2*b^4 - b^6)*f), 1/6*(
2*(a^2*b^3 - b^5)*d^4*cos(f*x + e)^3 + 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*b^2 - a*b^4)*c^2*d^2 + 4*(2*a^4*b
- a^2*b^3 - b^5)*c*d^3 - (2*a^5 - a^3*b^2 - a*b^4)*d^4)*f*x - 3*(4*(a^2*b^3 - b^5)*c*d^3 - (a^3*b^2 - a*b^4)*d
^4)*cos(f*x + e)*sin(f*x + e) - 6*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt
(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) - 6*(6*(a^2*b^3 - b^5)*c^2*d^2 - 4*(a
^3*b^2 - a*b^4)*c*d^3 + (a^4*b - b^5)*d^4)*cos(f*x + e))/((a^2*b^4 - b^6)*f)]
________________________________________________________________________________________
giac [B] time = 0.23, size = 465, normalized size = 1.98 \[ \frac {\frac {3 \, {\left (8 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 8 \, a^{2} b c d^{3} + 4 \, b^{3} c d^{3} - 2 \, a^{3} d^{4} - a b^{2} d^{4}\right )} {\left (f x + e\right )}}{b^{4}} + \frac {12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{4}} + \frac {2 \, {\left (12 \, b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 36 \, b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 24 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 6 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 72 \, b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 48 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, b^{2} c^{2} d^{2} + 24 \, a b c d^{3} - 6 \, a^{2} d^{4} - 4 \, b^{2} d^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{3} b^{3}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e)),x, algorithm="giac")
[Out]
1/6*(3*(8*b^3*c^3*d - 12*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + 4*b^3*c*d^3 - 2*a^3*d^4 - a*b^2*d^4)*(f*x + e)/b^4 +
12*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*s
gn(a) + arctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^4) + 2*(12*b^2*c*d^3*tan(1/2*
f*x + 1/2*e)^5 - 3*a*b*d^4*tan(1/2*f*x + 1/2*e)^5 - 36*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^4 + 24*a*b*c*d^3*tan(1
/2*f*x + 1/2*e)^4 - 6*a^2*d^4*tan(1/2*f*x + 1/2*e)^4 - 72*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 + 48*a*b*c*d^3*ta
n(1/2*f*x + 1/2*e)^2 - 12*a^2*d^4*tan(1/2*f*x + 1/2*e)^2 - 12*b^2*d^4*tan(1/2*f*x + 1/2*e)^2 - 12*b^2*c*d^3*ta
n(1/2*f*x + 1/2*e) + 3*a*b*d^4*tan(1/2*f*x + 1/2*e) - 36*b^2*c^2*d^2 + 24*a*b*c*d^3 - 6*a^2*d^4 - 4*b^2*d^4)/(
(tan(1/2*f*x + 1/2*e)^2 + 1)^3*b^3))/f
________________________________________________________________________________________
maple [B] time = 0.24, size = 948, normalized size = 4.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e)),x)
[Out]
8/f*d^3/b^3*arctan(tan(1/2*f*x+1/2*e))*a^2*c-12/f*d^2/b^2*arctan(tan(1/2*f*x+1/2*e))*a*c^2+2/f/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))*a^4*d^4-1/f*d^4/b^2/(1+tan(1/2*f*x+1/2*e)^2)^3*t
an(1/2*f*x+1/2*e)^5*a-4/3/f*d^4/b/(1+tan(1/2*f*x+1/2*e)^2)^3+2/f/(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))*c^4-1/f*d^4/b^2*arctan(tan(1/2*f*x+1/2*e))*a+4/f*d^3/b*arctan(tan(1/2*f*x+1/2*e))*
c-4/f*d^4/b/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2-2/f*d^4/b^3/(1+tan(1/2*f*x+1/2*e)^2)^3*a^2-12/f*d^
2/b/(1+tan(1/2*f*x+1/2*e)^2)^3*c^2-2/f*d^4/b^4*arctan(tan(1/2*f*x+1/2*e))*a^3+8/f*d/b*arctan(tan(1/2*f*x+1/2*e
))*c^3+8/f*d^3/b^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*a*c-8/f/b/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*t
an(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2))*a*c^3*d+16/f*d^3/b^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*a
*c-8/f/b^3/(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))*a^3*c*d^3+12/f/b^2/(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))*a^2*c^2*d^2+4/f*d^3/b/(1+tan(1/2*f*x+1/2*e)^
2)^3*tan(1/2*f*x+1/2*e)^5*c-4/f*d^3/b/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)*c+8/f*d^3/b^2/(1+tan(1/2*f
*x+1/2*e)^2)^3*a*c-2/f*d^4/b^3/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*a^2-12/f*d^2/b/(1+tan(1/2*f*x+1
/2*e)^2)^3*tan(1/2*f*x+1/2*e)^4*c^2-4/f*d^4/b^3/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*a^2-24/f*d^2/b
/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2*c^2+1/f*d^4/b^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)
*a
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+b*sin(f*x+e)),x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?
________________________________________________________________________________________
mupad [B] time = 16.49, size = 8720, normalized size = 37.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c + d*sin(e + f*x))^4/(a + b*sin(e + f*x)),x)
[Out]
- ((2*(3*a^2*d^4 + 2*b^2*d^4 + 18*b^2*c^2*d^2 - 12*a*b*c*d^3))/(3*b^3) + (tan(e/2 + (f*x)/2)^5*(a*d^4 - 4*b*c*
d^3))/b^2 + (4*tan(e/2 + (f*x)/2)^2*(a^2*d^4 + b^2*d^4 + 6*b^2*c^2*d^2 - 4*a*b*c*d^3))/b^3 + (2*tan(e/2 + (f*x
)/2)^4*(a^2*d^4 + 6*b^2*c^2*d^2 - 4*a*b*c*d^3))/b^3 - (tan(e/2 + (f*x)/2)*(a*d^4 - 4*b*c*d^3))/b^2)/(f*(3*tan(
e/2 + (f*x)/2)^2 + 3*tan(e/2 + (f*x)/2)^4 + tan(e/2 + (f*x)/2)^6 + 1)) - (atan(((((8*(a^4*b^7*d^8 + 4*a^6*b^5*
d^8 + 4*a^8*b^3*d^8 - 8*a^3*b^8*c*d^7 - 32*a^5*b^6*c*d^7 - 32*a^7*b^4*c*d^7 + 16*a^2*b^9*c^2*d^6 + 64*a^2*b^9*
c^4*d^4 + 64*a^2*b^9*c^6*d^2 - 112*a^3*b^8*c^3*d^5 - 192*a^3*b^8*c^5*d^3 + 88*a^4*b^7*c^2*d^6 + 272*a^4*b^7*c^
4*d^4 - 224*a^5*b^6*c^3*d^5 + 112*a^6*b^5*c^2*d^6))/b^8 + (((8*(4*a^2*b^10*c^4 + 2*a^2*b^10*d^4 + 2*a^4*b^8*d^
4 - 8*a^3*b^9*c*d^3 + 24*a^2*b^10*c^2*d^2 - 8*a*b^11*c*d^3 - 16*a*b^11*c^3*d))/b^8 + (8*tan(e/2 + (f*x)/2)*(8*
a*b^12*c^4 + 8*a^5*b^8*d^4 - 32*a^2*b^11*c^3*d - 32*a^4*b^9*c*d^3 + 48*a^3*b^10*c^2*d^2))/b^9 + ((32*a^2*b^3 +
(8*tan(e/2 + (f*x)/2)*(12*a*b^13 - 8*a^3*b^11))/b^9)*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d
*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4)*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c
*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4 + (8*tan(e/2 + (f*x)/2)*(2*a^3*b^9*d^8 - 4*a*b^11*c^8 + 7*a^5*b^7*d
^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8 + 32*a*b^11*c^2*d^6 + 128*a*b^11*c^4*d^4 + 128*a*b^11*c^6*d^2 - 16*a^2*b^10
*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4*b^8*c*d^7 - 32*a^6*b^6*c*d^7 + 64*a^8*b^4*c*d^7 - 224*a^2*b^10*c^3*d^5 - 3
84*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c^2*d^6 + 480*a^3*b^9*c^4*d^4 - 176*a^3*b^9*c^6*d^2 - 336*a^4*b^8*c^3*d^5 +
416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c^2*d^6 - 552*a^5*b^7*c^4*d^4 + 448*a^6*b^6*c^3*d^5 - 224*a^7*b^5*c^2*d^6))/
b^9)*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i)*1i)/b^
4 + (((8*(a^4*b^7*d^8 + 4*a^6*b^5*d^8 + 4*a^8*b^3*d^8 - 8*a^3*b^8*c*d^7 - 32*a^5*b^6*c*d^7 - 32*a^7*b^4*c*d^7
+ 16*a^2*b^9*c^2*d^6 + 64*a^2*b^9*c^4*d^4 + 64*a^2*b^9*c^6*d^2 - 112*a^3*b^8*c^3*d^5 - 192*a^3*b^8*c^5*d^3 + 8
8*a^4*b^7*c^2*d^6 + 272*a^4*b^7*c^4*d^4 - 224*a^5*b^6*c^3*d^5 + 112*a^6*b^5*c^2*d^6))/b^8 - (((8*(4*a^2*b^10*c
^4 + 2*a^2*b^10*d^4 + 2*a^4*b^8*d^4 - 8*a^3*b^9*c*d^3 + 24*a^2*b^10*c^2*d^2 - 8*a*b^11*c*d^3 - 16*a*b^11*c^3*d
))/b^8 + (8*tan(e/2 + (f*x)/2)*(8*a*b^12*c^4 + 8*a^5*b^8*d^4 - 32*a^2*b^11*c^3*d - 32*a^4*b^9*c*d^3 + 48*a^3*b
^10*c^2*d^2))/b^9 - ((32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^13 - 8*a^3*b^11))/b^9)*(a^3*d^4*1i + (b^2*d*(
a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4)*(a^3*d^4*1i + (b^2*d*(a*d^3
+ 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4 + (8*tan(e/2 + (f*x)/2)*(2*a^3*b^
9*d^8 - 4*a*b^11*c^8 + 7*a^5*b^7*d^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8 + 32*a*b^11*c^2*d^6 + 128*a*b^11*c^4*d^4
+ 128*a*b^11*c^6*d^2 - 16*a^2*b^10*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4*b^8*c*d^7 - 32*a^6*b^6*c*d^7 + 64*a^8*b^
4*c*d^7 - 224*a^2*b^10*c^3*d^5 - 384*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c^2*d^6 + 480*a^3*b^9*c^4*d^4 - 176*a^3*b^
9*c^6*d^2 - 336*a^4*b^8*c^3*d^5 + 416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c^2*d^6 - 552*a^5*b^7*c^4*d^4 + 448*a^6*b^
6*c^3*d^5 - 224*a^7*b^5*c^2*d^6))/b^9)*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c
^3)*1i)/2 - a^2*b*c*d^3*4i)*1i)/b^4)/((16*(2*a^10*d^12 + a^8*b^2*d^12 + 8*a*b^9*c^9*d^3 - 12*a^7*b^3*c*d^11 +
16*a^2*b^8*c^6*d^6 - 2*a^2*b^8*c^8*d^4 - 88*a^2*b^8*c^10*d^2 - 72*a^3*b^7*c^5*d^7 - 128*a^3*b^7*c^7*d^5 + 208*
a^3*b^7*c^9*d^3 + 129*a^4*b^6*c^4*d^8 + 416*a^4*b^6*c^6*d^6 - 276*a^4*b^6*c^8*d^4 - 116*a^5*b^5*c^3*d^9 - 640*
a^5*b^5*c^5*d^7 + 224*a^5*b^5*c^7*d^5 + 54*a^6*b^4*c^2*d^10 + 584*a^6*b^4*c^4*d^8 - 112*a^6*b^4*c^6*d^6 - 336*
a^7*b^3*c^3*d^9 + 32*a^7*b^3*c^5*d^7 + 120*a^8*b^2*c^2*d^10 - 4*a^8*b^2*c^4*d^8 + 16*a*b^9*c^11*d - 24*a^9*b*c
*d^11))/b^8 + (((8*(a^4*b^7*d^8 + 4*a^6*b^5*d^8 + 4*a^8*b^3*d^8 - 8*a^3*b^8*c*d^7 - 32*a^5*b^6*c*d^7 - 32*a^7*
b^4*c*d^7 + 16*a^2*b^9*c^2*d^6 + 64*a^2*b^9*c^4*d^4 + 64*a^2*b^9*c^6*d^2 - 112*a^3*b^8*c^3*d^5 - 192*a^3*b^8*c
^5*d^3 + 88*a^4*b^7*c^2*d^6 + 272*a^4*b^7*c^4*d^4 - 224*a^5*b^6*c^3*d^5 + 112*a^6*b^5*c^2*d^6))/b^8 + (((8*(4*
a^2*b^10*c^4 + 2*a^2*b^10*d^4 + 2*a^4*b^8*d^4 - 8*a^3*b^9*c*d^3 + 24*a^2*b^10*c^2*d^2 - 8*a*b^11*c*d^3 - 16*a*
b^11*c^3*d))/b^8 + (8*tan(e/2 + (f*x)/2)*(8*a*b^12*c^4 + 8*a^5*b^8*d^4 - 32*a^2*b^11*c^3*d - 32*a^4*b^9*c*d^3
+ 48*a^3*b^10*c^2*d^2))/b^9 + ((32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^13 - 8*a^3*b^11))/b^9)*(a^3*d^4*1i
+ (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4)*(a^3*d^4*1i + (b^
2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4 + (8*tan(e/2 + (f*x)/2)
*(2*a^3*b^9*d^8 - 4*a*b^11*c^8 + 7*a^5*b^7*d^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8 + 32*a*b^11*c^2*d^6 + 128*a*b^1
1*c^4*d^4 + 128*a*b^11*c^6*d^2 - 16*a^2*b^10*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4*b^8*c*d^7 - 32*a^6*b^6*c*d^7 +
64*a^8*b^4*c*d^7 - 224*a^2*b^10*c^3*d^5 - 384*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c^2*d^6 + 480*a^3*b^9*c^4*d^4 -
176*a^3*b^9*c^6*d^2 - 336*a^4*b^8*c^3*d^5 + 416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c^2*d^6 - 552*a^5*b^7*c^4*d^4 +
448*a^6*b^6*c^3*d^5 - 224*a^7*b^5*c^2*d^6))/b^9)*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c
*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4 - (((8*(a^4*b^7*d^8 + 4*a^6*b^5*d^8 + 4*a^8*b^3*d^8 - 8*a^3*b^8*c*d
^7 - 32*a^5*b^6*c*d^7 - 32*a^7*b^4*c*d^7 + 16*a^2*b^9*c^2*d^6 + 64*a^2*b^9*c^4*d^4 + 64*a^2*b^9*c^6*d^2 - 112*
a^3*b^8*c^3*d^5 - 192*a^3*b^8*c^5*d^3 + 88*a^4*b^7*c^2*d^6 + 272*a^4*b^7*c^4*d^4 - 224*a^5*b^6*c^3*d^5 + 112*a
^6*b^5*c^2*d^6))/b^8 - (((8*(4*a^2*b^10*c^4 + 2*a^2*b^10*d^4 + 2*a^4*b^8*d^4 - 8*a^3*b^9*c*d^3 + 24*a^2*b^10*c
^2*d^2 - 8*a*b^11*c*d^3 - 16*a*b^11*c^3*d))/b^8 + (8*tan(e/2 + (f*x)/2)*(8*a*b^12*c^4 + 8*a^5*b^8*d^4 - 32*a^2
*b^11*c^3*d - 32*a^4*b^9*c*d^3 + 48*a^3*b^10*c^2*d^2))/b^9 - ((32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^13 -
8*a^3*b^11))/b^9)*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*
d^3*4i))/b^4)*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4
i))/b^4 + (8*tan(e/2 + (f*x)/2)*(2*a^3*b^9*d^8 - 4*a*b^11*c^8 + 7*a^5*b^7*d^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8
+ 32*a*b^11*c^2*d^6 + 128*a*b^11*c^4*d^4 + 128*a*b^11*c^6*d^2 - 16*a^2*b^10*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4
*b^8*c*d^7 - 32*a^6*b^6*c*d^7 + 64*a^8*b^4*c*d^7 - 224*a^2*b^10*c^3*d^5 - 384*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c
^2*d^6 + 480*a^3*b^9*c^4*d^4 - 176*a^3*b^9*c^6*d^2 - 336*a^4*b^8*c^3*d^5 + 416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c
^2*d^6 - 552*a^5*b^7*c^4*d^4 + 448*a^6*b^6*c^3*d^5 - 224*a^7*b^5*c^2*d^6))/b^9)*(a^3*d^4*1i + (b^2*d*(a*d^3 +
12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^2*b*c*d^3*4i))/b^4 + (16*tan(e/2 + (f*x)/2)*(8*a^11*d^1
2 + 2*a^7*b^4*d^12 + 8*a^9*b^2*d^12 + 32*a*b^10*c^6*d^6 + 128*a*b^10*c^8*d^4 + 128*a*b^10*c^10*d^2 - 24*a^6*b^
5*c*d^11 - 96*a^8*b^3*c*d^11 - 144*a^2*b^9*c^5*d^7 - 736*a^2*b^9*c^7*d^5 - 896*a^2*b^9*c^9*d^3 + 258*a^3*b^8*c
^4*d^8 + 1840*a^3*b^8*c^6*d^6 + 2848*a^3*b^8*c^8*d^4 - 232*a^4*b^7*c^3*d^9 - 2624*a^4*b^7*c^5*d^7 - 5440*a^4*b
^7*c^7*d^5 + 108*a^5*b^6*c^2*d^10 + 2344*a^5*b^6*c^4*d^8 + 6944*a^5*b^6*c^6*d^6 - 1344*a^6*b^5*c^3*d^9 - 6208*
a^6*b^5*c^5*d^7 + 480*a^7*b^4*c^2*d^10 + 3944*a^7*b^4*c^4*d^8 - 1760*a^8*b^3*c^3*d^9 + 528*a^9*b^2*c^2*d^10 -
96*a^10*b*c*d^11))/b^9))*(a^3*d^4*1i + (b^2*d*(a*d^3 + 12*a*c^2*d)*1i)/2 - (b^3*d*(4*c*d^2 + 8*c^3)*1i)/2 - a^
2*b*c*d^3*4i)*2i)/(b^4*f) - (atan((((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(a^4*b^7*d^8 + 4*a^6*b^5*d^8 +
4*a^8*b^3*d^8 - 8*a^3*b^8*c*d^7 - 32*a^5*b^6*c*d^7 - 32*a^7*b^4*c*d^7 + 16*a^2*b^9*c^2*d^6 + 64*a^2*b^9*c^4*d^
4 + 64*a^2*b^9*c^6*d^2 - 112*a^3*b^8*c^3*d^5 - 192*a^3*b^8*c^5*d^3 + 88*a^4*b^7*c^2*d^6 + 272*a^4*b^7*c^4*d^4
- 224*a^5*b^6*c^3*d^5 + 112*a^6*b^5*c^2*d^6))/b^8 + (8*tan(e/2 + (f*x)/2)*(2*a^3*b^9*d^8 - 4*a*b^11*c^8 + 7*a^
5*b^7*d^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8 + 32*a*b^11*c^2*d^6 + 128*a*b^11*c^4*d^4 + 128*a*b^11*c^6*d^2 - 16*a
^2*b^10*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4*b^8*c*d^7 - 32*a^6*b^6*c*d^7 + 64*a^8*b^4*c*d^7 - 224*a^2*b^10*c^3*
d^5 - 384*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c^2*d^6 + 480*a^3*b^9*c^4*d^4 - 176*a^3*b^9*c^6*d^2 - 336*a^4*b^8*c^3
*d^5 + 416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c^2*d^6 - 552*a^5*b^7*c^4*d^4 + 448*a^6*b^6*c^3*d^5 - 224*a^7*b^5*c^2
*d^6))/b^9 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(4*a^2*b^10*c^4 + 2*a^2*b^10*d^4 + 2*a^4*b^8*d^4 - 8*
a^3*b^9*c*d^3 + 24*a^2*b^10*c^2*d^2 - 8*a*b^11*c*d^3 - 16*a*b^11*c^3*d))/b^8 + (8*tan(e/2 + (f*x)/2)*(8*a*b^12
*c^4 + 8*a^5*b^8*d^4 - 32*a^2*b^11*c^3*d - 32*a^4*b^9*c*d^3 + 48*a^3*b^10*c^2*d^2))/b^9 + ((-(a + b)*(a - b))^
(1/2)*(a*d - b*c)^4*(32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^13 - 8*a^3*b^11))/b^9))/(b^6 - a^2*b^4)))/(b^6
- a^2*b^4))*1i)/(b^6 - a^2*b^4) + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(a^4*b^7*d^8 + 4*a^6*b^5*d^8 +
4*a^8*b^3*d^8 - 8*a^3*b^8*c*d^7 - 32*a^5*b^6*c*d^7 - 32*a^7*b^4*c*d^7 + 16*a^2*b^9*c^2*d^6 + 64*a^2*b^9*c^4*d^
4 + 64*a^2*b^9*c^6*d^2 - 112*a^3*b^8*c^3*d^5 - 192*a^3*b^8*c^5*d^3 + 88*a^4*b^7*c^2*d^6 + 272*a^4*b^7*c^4*d^4
- 224*a^5*b^6*c^3*d^5 + 112*a^6*b^5*c^2*d^6))/b^8 + (8*tan(e/2 + (f*x)/2)*(2*a^3*b^9*d^8 - 4*a*b^11*c^8 + 7*a^
5*b^7*d^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8 + 32*a*b^11*c^2*d^6 + 128*a*b^11*c^4*d^4 + 128*a*b^11*c^6*d^2 - 16*a
^2*b^10*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4*b^8*c*d^7 - 32*a^6*b^6*c*d^7 + 64*a^8*b^4*c*d^7 - 224*a^2*b^10*c^3*
d^5 - 384*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c^2*d^6 + 480*a^3*b^9*c^4*d^4 - 176*a^3*b^9*c^6*d^2 - 336*a^4*b^8*c^3
*d^5 + 416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c^2*d^6 - 552*a^5*b^7*c^4*d^4 + 448*a^6*b^6*c^3*d^5 - 224*a^7*b^5*c^2
*d^6))/b^9 - ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(4*a^2*b^10*c^4 + 2*a^2*b^10*d^4 + 2*a^4*b^8*d^4 - 8*
a^3*b^9*c*d^3 + 24*a^2*b^10*c^2*d^2 - 8*a*b^11*c*d^3 - 16*a*b^11*c^3*d))/b^8 + (8*tan(e/2 + (f*x)/2)*(8*a*b^12
*c^4 + 8*a^5*b^8*d^4 - 32*a^2*b^11*c^3*d - 32*a^4*b^9*c*d^3 + 48*a^3*b^10*c^2*d^2))/b^9 - ((-(a + b)*(a - b))^
(1/2)*(a*d - b*c)^4*(32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^13 - 8*a^3*b^11))/b^9))/(b^6 - a^2*b^4)))/(b^6
- a^2*b^4))*1i)/(b^6 - a^2*b^4))/((16*(2*a^10*d^12 + a^8*b^2*d^12 + 8*a*b^9*c^9*d^3 - 12*a^7*b^3*c*d^11 + 16*
a^2*b^8*c^6*d^6 - 2*a^2*b^8*c^8*d^4 - 88*a^2*b^8*c^10*d^2 - 72*a^3*b^7*c^5*d^7 - 128*a^3*b^7*c^7*d^5 + 208*a^3
*b^7*c^9*d^3 + 129*a^4*b^6*c^4*d^8 + 416*a^4*b^6*c^6*d^6 - 276*a^4*b^6*c^8*d^4 - 116*a^5*b^5*c^3*d^9 - 640*a^5
*b^5*c^5*d^7 + 224*a^5*b^5*c^7*d^5 + 54*a^6*b^4*c^2*d^10 + 584*a^6*b^4*c^4*d^8 - 112*a^6*b^4*c^6*d^6 - 336*a^7
*b^3*c^3*d^9 + 32*a^7*b^3*c^5*d^7 + 120*a^8*b^2*c^2*d^10 - 4*a^8*b^2*c^4*d^8 + 16*a*b^9*c^11*d - 24*a^9*b*c*d^
11))/b^8 + (16*tan(e/2 + (f*x)/2)*(8*a^11*d^12 + 2*a^7*b^4*d^12 + 8*a^9*b^2*d^12 + 32*a*b^10*c^6*d^6 + 128*a*b
^10*c^8*d^4 + 128*a*b^10*c^10*d^2 - 24*a^6*b^5*c*d^11 - 96*a^8*b^3*c*d^11 - 144*a^2*b^9*c^5*d^7 - 736*a^2*b^9*
c^7*d^5 - 896*a^2*b^9*c^9*d^3 + 258*a^3*b^8*c^4*d^8 + 1840*a^3*b^8*c^6*d^6 + 2848*a^3*b^8*c^8*d^4 - 232*a^4*b^
7*c^3*d^9 - 2624*a^4*b^7*c^5*d^7 - 5440*a^4*b^7*c^7*d^5 + 108*a^5*b^6*c^2*d^10 + 2344*a^5*b^6*c^4*d^8 + 6944*a
^5*b^6*c^6*d^6 - 1344*a^6*b^5*c^3*d^9 - 6208*a^6*b^5*c^5*d^7 + 480*a^7*b^4*c^2*d^10 + 3944*a^7*b^4*c^4*d^8 - 1
760*a^8*b^3*c^3*d^9 + 528*a^9*b^2*c^2*d^10 - 96*a^10*b*c*d^11))/b^9 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*
((8*(a^4*b^7*d^8 + 4*a^6*b^5*d^8 + 4*a^8*b^3*d^8 - 8*a^3*b^8*c*d^7 - 32*a^5*b^6*c*d^7 - 32*a^7*b^4*c*d^7 + 16*
a^2*b^9*c^2*d^6 + 64*a^2*b^9*c^4*d^4 + 64*a^2*b^9*c^6*d^2 - 112*a^3*b^8*c^3*d^5 - 192*a^3*b^8*c^5*d^3 + 88*a^4
*b^7*c^2*d^6 + 272*a^4*b^7*c^4*d^4 - 224*a^5*b^6*c^3*d^5 + 112*a^6*b^5*c^2*d^6))/b^8 + (8*tan(e/2 + (f*x)/2)*(
2*a^3*b^9*d^8 - 4*a*b^11*c^8 + 7*a^5*b^7*d^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8 + 32*a*b^11*c^2*d^6 + 128*a*b^11*
c^4*d^4 + 128*a*b^11*c^6*d^2 - 16*a^2*b^10*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4*b^8*c*d^7 - 32*a^6*b^6*c*d^7 + 6
4*a^8*b^4*c*d^7 - 224*a^2*b^10*c^3*d^5 - 384*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c^2*d^6 + 480*a^3*b^9*c^4*d^4 - 17
6*a^3*b^9*c^6*d^2 - 336*a^4*b^8*c^3*d^5 + 416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c^2*d^6 - 552*a^5*b^7*c^4*d^4 + 44
8*a^6*b^6*c^3*d^5 - 224*a^7*b^5*c^2*d^6))/b^9 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(4*a^2*b^10*c^4 +
2*a^2*b^10*d^4 + 2*a^4*b^8*d^4 - 8*a^3*b^9*c*d^3 + 24*a^2*b^10*c^2*d^2 - 8*a*b^11*c*d^3 - 16*a*b^11*c^3*d))/b^
8 + (8*tan(e/2 + (f*x)/2)*(8*a*b^12*c^4 + 8*a^5*b^8*d^4 - 32*a^2*b^11*c^3*d - 32*a^4*b^9*c*d^3 + 48*a^3*b^10*c
^2*d^2))/b^9 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*(32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^13 - 8*a^3*
b^11))/b^9))/(b^6 - a^2*b^4)))/(b^6 - a^2*b^4)))/(b^6 - a^2*b^4) - ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8
*(a^4*b^7*d^8 + 4*a^6*b^5*d^8 + 4*a^8*b^3*d^8 - 8*a^3*b^8*c*d^7 - 32*a^5*b^6*c*d^7 - 32*a^7*b^4*c*d^7 + 16*a^2
*b^9*c^2*d^6 + 64*a^2*b^9*c^4*d^4 + 64*a^2*b^9*c^6*d^2 - 112*a^3*b^8*c^3*d^5 - 192*a^3*b^8*c^5*d^3 + 88*a^4*b^
7*c^2*d^6 + 272*a^4*b^7*c^4*d^4 - 224*a^5*b^6*c^3*d^5 + 112*a^6*b^5*c^2*d^6))/b^8 + (8*tan(e/2 + (f*x)/2)*(2*a
^3*b^9*d^8 - 4*a*b^11*c^8 + 7*a^5*b^7*d^8 + 4*a^7*b^5*d^8 - 8*a^9*b^3*d^8 + 32*a*b^11*c^2*d^6 + 128*a*b^11*c^4
*d^4 + 128*a*b^11*c^6*d^2 - 16*a^2*b^10*c*d^7 + 32*a^2*b^10*c^7*d - 56*a^4*b^8*c*d^7 - 32*a^6*b^6*c*d^7 + 64*a
^8*b^4*c*d^7 - 224*a^2*b^10*c^3*d^5 - 384*a^2*b^10*c^5*d^3 + 160*a^3*b^9*c^2*d^6 + 480*a^3*b^9*c^4*d^4 - 176*a
^3*b^9*c^6*d^2 - 336*a^4*b^8*c^3*d^5 + 416*a^4*b^8*c^5*d^3 + 136*a^5*b^7*c^2*d^6 - 552*a^5*b^7*c^4*d^4 + 448*a
^6*b^6*c^3*d^5 - 224*a^7*b^5*c^2*d^6))/b^9 - ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(4*a^2*b^10*c^4 + 2*a
^2*b^10*d^4 + 2*a^4*b^8*d^4 - 8*a^3*b^9*c*d^3 + 24*a^2*b^10*c^2*d^2 - 8*a*b^11*c*d^3 - 16*a*b^11*c^3*d))/b^8 +
(8*tan(e/2 + (f*x)/2)*(8*a*b^12*c^4 + 8*a^5*b^8*d^4 - 32*a^2*b^11*c^3*d - 32*a^4*b^9*c*d^3 + 48*a^3*b^10*c^2*
d^2))/b^9 - ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*(32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^13 - 8*a^3*b^1
1))/b^9))/(b^6 - a^2*b^4)))/(b^6 - a^2*b^4)))/(b^6 - a^2*b^4)))*(-(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*2i)/(f*
(b^6 - a^2*b^4))
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))**4/(a+b*sin(f*x+e)),x)
[Out]
Timed out
________________________________________________________________________________________