\(\int \frac {(c+d \sin (e+f x))^3}{3+b \sin (e+f x)} \, dx\) [699]

Optimal result

Integrand size = 25, antiderivative size = 146 \[ \int \frac {(c+d \sin (e+f x))^3}{3+b \sin (e+f x)} \, dx=-\frac {d \left (18 b c d-18 d^2-b^2 \left (6 c^2+d^2\right )\right ) x}{2 b^3}+\frac {2 (b c-3 d)^3 \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{b^3 \sqrt {9-b^2} f}-\frac {(5 b c-6 d) d^2 \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2872, 3102, 2814, 2739, 632, 210} \[ \int \frac {(c+d \sin (e+f x))^3}{3+b \sin (e+f x)} \, dx=\frac {2 (b c-a d)^3 \arctan \left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 f \sqrt {a^2-b^2}}-\frac {d x \left (-2 a^2 d^2+6 a b c d-\left (b^2 \left (6 c^2+d^2\right )\right )\right )}{2 b^3}-\frac {d^2 (5 b c-2 a d) \cos (e+f x)}{2 b^2 f}-\frac {d^2 \cos (e+f x) (c+d \sin (e+f x))}{2 b f} \]

[In]

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

[Out]

-1/2*(d*(6*a*b*c*d - 2*a^2*d^2 - b^2*(6*c^2 + d^2))*x)/b^3 + (2*(b*c - a*d)^3*ArcTan[(b + a*Tan[(e + f*x)/2])/
Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*f) - (d^2*(5*b*c - 2*a*d)*Cos[e + f*x])/(2*b^2*f) - (d^2*Cos[e + f*x]*(
c + d*Sin[e + f*x]))/(2*b*f)

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 2872

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

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

Mathematica [A] (verified)

Time = 2.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d \sin (e+f x))^3}{3+b \sin (e+f x)} \, dx=\frac {2 d \left (-18 b c d+18 d^2+b^2 \left (6 c^2+d^2\right )\right ) (e+f x)+\frac {8 (b c-3 d)^3 \arctan \left (\frac {b+3 \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {9-b^2}}\right )}{\sqrt {9-b^2}}-12 b (b c-d) d^2 \cos (e+f x)-b^2 d^3 \sin (2 (e+f x))}{4 b^3 f} \]

[In]

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

[Out]

(2*d*(-18*b*c*d + 18*d^2 + b^2*(6*c^2 + d^2))*(e + f*x) + (8*(b*c - 3*d)^3*ArcTan[(b + 3*Tan[(e + f*x)/2])/Sqr
t[9 - b^2]])/Sqrt[9 - b^2] - 12*b*(b*c - d)*d^2*Cos[e + f*x] - b^2*d^3*Sin[2*(e + f*x)])/(4*b^3*f)

Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.57

method	result	size
derivativedivides	\(\frac {\frac {2 \left (-d^{3} a^{3}+3 c \,d^{2} a^{2} b -3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{3} \sqrt {a^{2}-b^{2}}}+\frac {2 d \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b^{2} d^{2}}{2}+\left (a b \,d^{2}-3 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{2} d^{2}}{2}+a b \,d^{2}-3 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (2 d^{2} a^{2}-6 a b c d +6 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{3}}}{f}\)	\(229\)
default	\(\frac {\frac {2 \left (-d^{3} a^{3}+3 c \,d^{2} a^{2} b -3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{3} \sqrt {a^{2}-b^{2}}}+\frac {2 d \left (\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) b^{2} d^{2}}{2}+\left (a b \,d^{2}-3 b^{2} c d \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{2} d^{2}}{2}+a b \,d^{2}-3 b^{2} c d}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {\left (2 d^{2} a^{2}-6 a b c d +6 b^{2} c^{2}+d^{2} b^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )}{b^{3}}}{f}\)	\(229\)
risch	\(\frac {d^{3} x \,a^{2}}{b^{3}}-\frac {3 d^{2} x a c}{b^{2}}+\frac {3 d x \,c^{2}}{b}+\frac {d^{3} x}{2 b}+\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )} a}{2 b^{2} f}-\frac {3 d^{2} {\mathrm e}^{i \left (f x +e \right )} c}{2 b f}+\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )} a}{2 b^{2} f}-\frac {3 d^{2} {\mathrm e}^{-i \left (f x +e \right )} c}{2 b f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) d^{3} a^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) c \,d^{2} a^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a \,c^{2} d}{\sqrt {-a^{2}+b^{2}}\, f b}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) c^{3}}{\sqrt {-a^{2}+b^{2}}\, f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) d^{3} a^{3}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) c \,d^{2} a^{2}}{\sqrt {-a^{2}+b^{2}}\, f \,b^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) a \,c^{2} d}{\sqrt {-a^{2}+b^{2}}\, f b}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) c^{3}}{\sqrt {-a^{2}+b^{2}}\, f}-\frac {d^{3} \sin \left (2 f x +2 e \right )}{4 b f}\)	\(733\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 566, normalized size of antiderivative = 3.88 \[ \int \frac {(c+d \sin (e+f x))^3}{3+b \sin (e+f x)} \, dx=\left [-\frac {{\left (a^{2} b^{2} - b^{4}\right )} d^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} f x - {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (f x + e\right ) \sin \left (f x + e\right ) + b \cos \left (f x + e\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (3 \, {\left (a^{2} b^{2} - b^{4}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} f}, -\frac {{\left (a^{2} b^{2} - b^{4}\right )} d^{3} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} f x + 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (f x + e\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )}\right ) + 2 \, {\left (3 \, {\left (a^{2} b^{2} - b^{4}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} f}\right ] \]

[In]

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

[Out]

[-1/2*((a^2*b^2 - b^4)*d^3*cos(f*x + e)*sin(f*x + e) - (6*(a^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (2
*a^4 - a^2*b^2 - b^4)*d^3)*f*x - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*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*(3*(a^2*b^2 - b^4)*c*d^2 - (a^3*
b - a*b^3)*d^3)*cos(f*x + e))/((a^2*b^3 - b^5)*f), -1/2*((a^2*b^2 - b^4)*d^3*cos(f*x + e)*sin(f*x + e) - (6*(a
^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (2*a^4 - a^2*b^2 - b^4)*d^3)*f*x + 2*(b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a^2 - b^2)*arctan(-(a*sin(f*x + e) + b)/(sqrt(a^2 - b^2)*cos(f*x + e))) + 2*(3
*(a^2*b^2 - b^4)*c*d^2 - (a^3*b - a*b^3)*d^3)*cos(f*x + e))/((a^2*b^3 - b^5)*f)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((c+d*sin(f*x+e))^3/(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 de

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.67 \[ \int \frac {(c+d \sin (e+f x))^3}{3+b \sin (e+f x)} \, dx=\frac {\frac {{\left (6 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 2 \, a^{2} d^{3} + b^{2} d^{3}\right )} {\left (f x + e\right )}}{b^{3}} + \frac {4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{3}} + \frac {2 \, {\left (b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, b c d^{2} + 2 \, a d^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} b^{2}}}{2 \, f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 15.47 (sec) , antiderivative size = 5902, normalized size of antiderivative = 40.42 \[ \int \frac {(c+d \sin (e+f x))^3}{3+b \sin (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*(a*d^3 - 3*b*c*d^2))/b^2 + (d^3*tan(e/2 + (f*x)/2)^3)/b + (2*tan(e/2 + (f*x)/2)^2*(a*d^3 - 3*b*c*d^2))/b^2
- (d^3*tan(e/2 + (f*x)/2))/b)/(f*(2*tan(e/2 + (f*x)/2)^2 + tan(e/2 + (f*x)/2)^4 + 1)) + (atan((((a^2*d^3*1i +
(b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i)*((8*(a^2*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a^6*b^2*d^6 - 12*a^3*b^5*c*d^
5 - 24*a^5*b^3*c*d^5 + 12*a^2*b^6*c^2*d^4 + 36*a^2*b^6*c^4*d^2 - 72*a^3*b^5*c^3*d^3 + 60*a^4*b^4*c^2*d^4))/b^5
+ (8*tan(e/2 + (f*x)/2)*(2*a*b^8*d^6 - 4*a*b^8*c^6 + 7*a^3*b^6*d^6 + 4*a^5*b^4*d^6 - 8*a^7*b^2*d^6 + 24*a*b^8
*c^2*d^4 + 72*a*b^8*c^4*d^2 - 24*a^2*b^7*c*d^5 + 24*a^2*b^7*c^5*d - 36*a^4*b^5*c*d^5 + 48*a^6*b^3*c*d^5 - 144*
a^2*b^7*c^3*d^3 + 108*a^3*b^6*c^2*d^4 - 96*a^3*b^6*c^4*d^2 + 152*a^4*b^5*c^3*d^3 - 120*a^5*b^4*c^2*d^4))/b^6 +
((a^2*d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i)*((8*tan(e/2 + (f*x)/2)*(8*a*b^9*c^3 - 8*a^4*b^6*d^3
- 24*a^2*b^8*c^2*d + 24*a^3*b^7*c*d^2))/b^6 - (8*(2*a*b^8*d^3 - 4*a^2*b^7*c^3 + 2*a^3*b^6*d^3 - 12*a^2*b^7*c*
d^2 + 12*a*b^8*c^2*d))/b^5 + ((32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^10 - 8*a^3*b^8))/b^6)*(a^2*d^3*1i +
(b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i))/b^3))/b^3)*1i)/b^3 + ((a^2*d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a
*b*c*d^2*3i)*((8*(a^2*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a^6*b^2*d^6 - 12*a^3*b^5*c*d^5 - 24*a^5*b^3*c*d^5 + 12*a^2*b
^6*c^2*d^4 + 36*a^2*b^6*c^4*d^2 - 72*a^3*b^5*c^3*d^3 + 60*a^4*b^4*c^2*d^4))/b^5 + (8*tan(e/2 + (f*x)/2)*(2*a*b
^8*d^6 - 4*a*b^8*c^6 + 7*a^3*b^6*d^6 + 4*a^5*b^4*d^6 - 8*a^7*b^2*d^6 + 24*a*b^8*c^2*d^4 + 72*a*b^8*c^4*d^2 - 2
4*a^2*b^7*c*d^5 + 24*a^2*b^7*c^5*d - 36*a^4*b^5*c*d^5 + 48*a^6*b^3*c*d^5 - 144*a^2*b^7*c^3*d^3 + 108*a^3*b^6*c
^2*d^4 - 96*a^3*b^6*c^4*d^2 + 152*a^4*b^5*c^3*d^3 - 120*a^5*b^4*c^2*d^4))/b^6 + ((a^2*d^3*1i + (b^2*d*(6*c^2 +
d^2)*1i)/2 - a*b*c*d^2*3i)*((8*(2*a*b^8*d^3 - 4*a^2*b^7*c^3 + 2*a^3*b^6*d^3 - 12*a^2*b^7*c*d^2 + 12*a*b^8*c^2
*d))/b^5 - (8*tan(e/2 + (f*x)/2)*(8*a*b^9*c^3 - 8*a^4*b^6*d^3 - 24*a^2*b^8*c^2*d + 24*a^3*b^7*c*d^2))/b^6 + ((
32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^10 - 8*a^3*b^8))/b^6)*(a^2*d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a*
b*c*d^2*3i))/b^3))/b^3)*1i)/b^3)/((16*(2*a^7*d^9 + a^5*b^2*d^9 - 2*a*b^6*c^6*d^3 - 3*a^4*b^3*c*d^8 + 4*a^6*b*c
^3*d^6 - a^2*b^5*c^3*d^6 + 48*a^2*b^5*c^7*d^2 + 3*a^3*b^4*c^2*d^7 + 18*a^3*b^4*c^4*d^5 - 76*a^3*b^4*c^6*d^3 -
36*a^4*b^3*c^3*d^6 + 60*a^4*b^3*c^5*d^4 + 30*a^5*b^2*c^2*d^7 - 24*a^5*b^2*c^4*d^5 - 12*a*b^6*c^8*d - 12*a^6*b*
c*d^8))/b^5 - ((a^2*d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i)*((8*(a^2*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a
^6*b^2*d^6 - 12*a^3*b^5*c*d^5 - 24*a^5*b^3*c*d^5 + 12*a^2*b^6*c^2*d^4 + 36*a^2*b^6*c^4*d^2 - 72*a^3*b^5*c^3*d^
3 + 60*a^4*b^4*c^2*d^4))/b^5 + (8*tan(e/2 + (f*x)/2)*(2*a*b^8*d^6 - 4*a*b^8*c^6 + 7*a^3*b^6*d^6 + 4*a^5*b^4*d^
6 - 8*a^7*b^2*d^6 + 24*a*b^8*c^2*d^4 + 72*a*b^8*c^4*d^2 - 24*a^2*b^7*c*d^5 + 24*a^2*b^7*c^5*d - 36*a^4*b^5*c*d
^5 + 48*a^6*b^3*c*d^5 - 144*a^2*b^7*c^3*d^3 + 108*a^3*b^6*c^2*d^4 - 96*a^3*b^6*c^4*d^2 + 152*a^4*b^5*c^3*d^3 -
120*a^5*b^4*c^2*d^4))/b^6 + ((a^2*d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i)*((8*tan(e/2 + (f*x)/2)*
(8*a*b^9*c^3 - 8*a^4*b^6*d^3 - 24*a^2*b^8*c^2*d + 24*a^3*b^7*c*d^2))/b^6 - (8*(2*a*b^8*d^3 - 4*a^2*b^7*c^3 + 2
*a^3*b^6*d^3 - 12*a^2*b^7*c*d^2 + 12*a*b^8*c^2*d))/b^5 + ((32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^10 - 8*a
^3*b^8))/b^6)*(a^2*d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i))/b^3))/b^3))/b^3 + ((a^2*d^3*1i + (b^2*
d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i)*((8*(a^2*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a^6*b^2*d^6 - 12*a^3*b^5*c*d^5 - 24
*a^5*b^3*c*d^5 + 12*a^2*b^6*c^2*d^4 + 36*a^2*b^6*c^4*d^2 - 72*a^3*b^5*c^3*d^3 + 60*a^4*b^4*c^2*d^4))/b^5 + (8*
tan(e/2 + (f*x)/2)*(2*a*b^8*d^6 - 4*a*b^8*c^6 + 7*a^3*b^6*d^6 + 4*a^5*b^4*d^6 - 8*a^7*b^2*d^6 + 24*a*b^8*c^2*d
^4 + 72*a*b^8*c^4*d^2 - 24*a^2*b^7*c*d^5 + 24*a^2*b^7*c^5*d - 36*a^4*b^5*c*d^5 + 48*a^6*b^3*c*d^5 - 144*a^2*b^
7*c^3*d^3 + 108*a^3*b^6*c^2*d^4 - 96*a^3*b^6*c^4*d^2 + 152*a^4*b^5*c^3*d^3 - 120*a^5*b^4*c^2*d^4))/b^6 + ((a^2
*d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i)*((8*(2*a*b^8*d^3 - 4*a^2*b^7*c^3 + 2*a^3*b^6*d^3 - 12*a^2
*b^7*c*d^2 + 12*a*b^8*c^2*d))/b^5 - (8*tan(e/2 + (f*x)/2)*(8*a*b^9*c^3 - 8*a^4*b^6*d^3 - 24*a^2*b^8*c^2*d + 24
*a^3*b^7*c*d^2))/b^6 + ((32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^10 - 8*a^3*b^8))/b^6)*(a^2*d^3*1i + (b^2*d
*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i))/b^3))/b^3))/b^3 + (16*tan(e/2 + (f*x)/2)*(8*a^8*d^9 + 2*a^4*b^4*d^9 + 8*
a^6*b^2*d^9 - 2*a*b^7*c^3*d^6 - 24*a*b^7*c^5*d^4 - 72*a*b^7*c^7*d^2 - 6*a^3*b^5*c*d^8 - 48*a^5*b^3*c*d^8 + 6*a
^2*b^6*c^2*d^7 + 96*a^2*b^6*c^4*d^5 + 360*a^2*b^6*c^6*d^3 - 152*a^3*b^5*c^3*d^6 - 768*a^3*b^5*c^5*d^4 + 120*a^
4*b^4*c^2*d^7 + 912*a^4*b^4*c^4*d^5 - 656*a^5*b^3*c^3*d^6 + 288*a^6*b^2*c^2*d^7 - 72*a^7*b*c*d^8))/b^6))*(a^2*
d^3*1i + (b^2*d*(6*c^2 + d^2)*1i)/2 - a*b*c*d^2*3i)*2i)/(b^3*f) + (atan((((-(a + b)*(a - b))^(1/2)*(a*d - b*c)
^3*((8*(a^2*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a^6*b^2*d^6 - 12*a^3*b^5*c*d^5 - 24*a^5*b^3*c*d^5 + 12*a^2*b^6*c^2*d^4
+ 36*a^2*b^6*c^4*d^2 - 72*a^3*b^5*c^3*d^3 + 60*a^4*b^4*c^2*d^4))/b^5 + (8*tan(e/2 + (f*x)/2)*(2*a*b^8*d^6 - 4
*a*b^8*c^6 + 7*a^3*b^6*d^6 + 4*a^5*b^4*d^6 - 8*a^7*b^2*d^6 + 24*a*b^8*c^2*d^4 + 72*a*b^8*c^4*d^2 - 24*a^2*b^7*
c*d^5 + 24*a^2*b^7*c^5*d - 36*a^4*b^5*c*d^5 + 48*a^6*b^3*c*d^5 - 144*a^2*b^7*c^3*d^3 + 108*a^3*b^6*c^2*d^4 - 9
6*a^3*b^6*c^4*d^2 + 152*a^4*b^5*c^3*d^3 - 120*a^5*b^4*c^2*d^4))/b^6 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*
((8*tan(e/2 + (f*x)/2)*(8*a*b^9*c^3 - 8*a^4*b^6*d^3 - 24*a^2*b^8*c^2*d + 24*a^3*b^7*c*d^2))/b^6 - (8*(2*a*b^8*
d^3 - 4*a^2*b^7*c^3 + 2*a^3*b^6*d^3 - 12*a^2*b^7*c*d^2 + 12*a*b^8*c^2*d))/b^5 + ((-(a + b)*(a - b))^(1/2)*(a*d
- b*c)^3*(32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^10 - 8*a^3*b^8))/b^6))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3)
)*1i)/(b^5 - a^2*b^3) + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*(a^2*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a^6*b^2*d
^6 - 12*a^3*b^5*c*d^5 - 24*a^5*b^3*c*d^5 + 12*a^2*b^6*c^2*d^4 + 36*a^2*b^6*c^4*d^2 - 72*a^3*b^5*c^3*d^3 + 60*a
^4*b^4*c^2*d^4))/b^5 + (8*tan(e/2 + (f*x)/2)*(2*a*b^8*d^6 - 4*a*b^8*c^6 + 7*a^3*b^6*d^6 + 4*a^5*b^4*d^6 - 8*a^
7*b^2*d^6 + 24*a*b^8*c^2*d^4 + 72*a*b^8*c^4*d^2 - 24*a^2*b^7*c*d^5 + 24*a^2*b^7*c^5*d - 36*a^4*b^5*c*d^5 + 48*
a^6*b^3*c*d^5 - 144*a^2*b^7*c^3*d^3 + 108*a^3*b^6*c^2*d^4 - 96*a^3*b^6*c^4*d^2 + 152*a^4*b^5*c^3*d^3 - 120*a^5
*b^4*c^2*d^4))/b^6 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*(2*a*b^8*d^3 - 4*a^2*b^7*c^3 + 2*a^3*b^6*d^3
- 12*a^2*b^7*c*d^2 + 12*a*b^8*c^2*d))/b^5 - (8*tan(e/2 + (f*x)/2)*(8*a*b^9*c^3 - 8*a^4*b^6*d^3 - 24*a^2*b^8*c^
2*d + 24*a^3*b^7*c*d^2))/b^6 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*(32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12
*a*b^10 - 8*a^3*b^8))/b^6))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3))*1i)/(b^5 - a^2*b^3))/((16*(2*a^7*d^9 + a^5*b^2*
d^9 - 2*a*b^6*c^6*d^3 - 3*a^4*b^3*c*d^8 + 4*a^6*b*c^3*d^6 - a^2*b^5*c^3*d^6 + 48*a^2*b^5*c^7*d^2 + 3*a^3*b^4*c
^2*d^7 + 18*a^3*b^4*c^4*d^5 - 76*a^3*b^4*c^6*d^3 - 36*a^4*b^3*c^3*d^6 + 60*a^4*b^3*c^5*d^4 + 30*a^5*b^2*c^2*d^
7 - 24*a^5*b^2*c^4*d^5 - 12*a*b^6*c^8*d - 12*a^6*b*c*d^8))/b^5 + (16*tan(e/2 + (f*x)/2)*(8*a^8*d^9 + 2*a^4*b^4
*d^9 + 8*a^6*b^2*d^9 - 2*a*b^7*c^3*d^6 - 24*a*b^7*c^5*d^4 - 72*a*b^7*c^7*d^2 - 6*a^3*b^5*c*d^8 - 48*a^5*b^3*c*
d^8 + 6*a^2*b^6*c^2*d^7 + 96*a^2*b^6*c^4*d^5 + 360*a^2*b^6*c^6*d^3 - 152*a^3*b^5*c^3*d^6 - 768*a^3*b^5*c^5*d^4
+ 120*a^4*b^4*c^2*d^7 + 912*a^4*b^4*c^4*d^5 - 656*a^5*b^3*c^3*d^6 + 288*a^6*b^2*c^2*d^7 - 72*a^7*b*c*d^8))/b^
6 - ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*(a^2*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a^6*b^2*d^6 - 12*a^3*b^5*c*d^
5 - 24*a^5*b^3*c*d^5 + 12*a^2*b^6*c^2*d^4 + 36*a^2*b^6*c^4*d^2 - 72*a^3*b^5*c^3*d^3 + 60*a^4*b^4*c^2*d^4))/b^5
+ (8*tan(e/2 + (f*x)/2)*(2*a*b^8*d^6 - 4*a*b^8*c^6 + 7*a^3*b^6*d^6 + 4*a^5*b^4*d^6 - 8*a^7*b^2*d^6 + 24*a*b^8
*c^2*d^4 + 72*a*b^8*c^4*d^2 - 24*a^2*b^7*c*d^5 + 24*a^2*b^7*c^5*d - 36*a^4*b^5*c*d^5 + 48*a^6*b^3*c*d^5 - 144*
a^2*b^7*c^3*d^3 + 108*a^3*b^6*c^2*d^4 - 96*a^3*b^6*c^4*d^2 + 152*a^4*b^5*c^3*d^3 - 120*a^5*b^4*c^2*d^4))/b^6 +
((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*tan(e/2 + (f*x)/2)*(8*a*b^9*c^3 - 8*a^4*b^6*d^3 - 24*a^2*b^8*c^2*
d + 24*a^3*b^7*c*d^2))/b^6 - (8*(2*a*b^8*d^3 - 4*a^2*b^7*c^3 + 2*a^3*b^6*d^3 - 12*a^2*b^7*c*d^2 + 12*a*b^8*c^2
*d))/b^5 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*(32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^10 - 8*a^3*b^8)
)/b^6))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3) + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*(a^2
*b^6*d^6 + 4*a^4*b^4*d^6 + 4*a^6*b^2*d^6 - 12*a^3*b^5*c*d^5 - 24*a^5*b^3*c*d^5 + 12*a^2*b^6*c^2*d^4 + 36*a^2*b
^6*c^4*d^2 - 72*a^3*b^5*c^3*d^3 + 60*a^4*b^4*c^2*d^4))/b^5 + (8*tan(e/2 + (f*x)/2)*(2*a*b^8*d^6 - 4*a*b^8*c^6
+ 7*a^3*b^6*d^6 + 4*a^5*b^4*d^6 - 8*a^7*b^2*d^6 + 24*a*b^8*c^2*d^4 + 72*a*b^8*c^4*d^2 - 24*a^2*b^7*c*d^5 + 24*
a^2*b^7*c^5*d - 36*a^4*b^5*c*d^5 + 48*a^6*b^3*c*d^5 - 144*a^2*b^7*c^3*d^3 + 108*a^3*b^6*c^2*d^4 - 96*a^3*b^6*c
^4*d^2 + 152*a^4*b^5*c^3*d^3 - 120*a^5*b^4*c^2*d^4))/b^6 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*(2*a*b^
8*d^3 - 4*a^2*b^7*c^3 + 2*a^3*b^6*d^3 - 12*a^2*b^7*c*d^2 + 12*a*b^8*c^2*d))/b^5 - (8*tan(e/2 + (f*x)/2)*(8*a*b
^9*c^3 - 8*a^4*b^6*d^3 - 24*a^2*b^8*c^2*d + 24*a^3*b^7*c*d^2))/b^6 + ((-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*(
32*a^2*b^3 + (8*tan(e/2 + (f*x)/2)*(12*a*b^10 - 8*a^3*b^8))/b^6))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3)))/(b^5 - a
^2*b^3)))*(-(a + b)*(a - b))^(1/2)*(a*d - b*c)^3*2i)/(f*(b^5 - a^2*b^3))