\(\int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx\) [433]

Optimal result

Integrand size = 23, antiderivative size = 187 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx=\frac {3 \left (2 c^2-2 c d+d^2\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d) \left (c^2-d^2\right )^{5/2} f}-\frac {\cos (e+f x)}{(c+d) f (c+d \sin (e+f x))^3}-\frac {(2 c-3 d) \cos (e+f x)}{2 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {(c-4 d) (2 c-d) \cos (e+f x)}{2 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2833, 12, 2739, 632, 210} \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx=\frac {a \left (2 c^2-2 c d+d^2\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f (c+d) \left (c^2-d^2\right )^{5/2}}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sin (e+f x))}-\frac {a (2 c-3 d) \cos (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sin (e+f x))^2}-\frac {a \cos (e+f x)}{3 f (c+d) (c+d \sin (e+f x))^3} \]

[In]

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

[Out]

(a*(2*c^2 - 2*c*d + d^2)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(5/2)*f) - (a*
Cos[e + f*x])/(3*(c + d)*f*(c + d*Sin[e + f*x])^3) - (a*(2*c - 3*d)*Cos[e + f*x])/(6*(c - d)*(c + d)^2*f*(c +
d*Sin[e + f*x])^2) - (a*(c - 4*d)*(2*c - d)*Cos[e + f*x])/(6*(c - d)^2*(c + d)^3*f*(c + d*Sin[e + f*x]))

Rule 12

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

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 2833

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.70 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.28 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx=\frac {(1+\sin (e+f x)) \left (\frac {24 \left (2 c^2-2 c d+d^2\right ) \arctan \left (\frac {\sec \left (\frac {f x}{2}\right ) (\cos (e)-i \sin (e)) \left (d \cos \left (e+\frac {f x}{2}\right )+c \sin \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (\cos (e)-i \sin (e))}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {2 c \left (4 c^4-18 c^3 d+14 c^2 d^2-27 c d^3+12 d^4\right ) \cot (e)-d \csc (e) \left (3 d \left (4 c^3-16 c^2 d+6 c d^2+d^3\right ) \cos (e+2 f x)-3 d^2 \left (2 c^2-2 c d+d^2\right ) \cos (3 e+2 f x)-24 c^4 \sin (f x)+78 c^3 d \sin (f x)-24 c^2 d^2 \sin (f x)+12 c d^3 \sin (f x)-12 d^4 \sin (f x)+30 c^3 d \sin (2 e+f x)-30 c^2 d^2 \sin (2 e+f x)+15 c d^3 \sin (2 e+f x)+2 c^2 d^2 \sin (2 e+3 f x)-9 c d^3 \sin (2 e+3 f x)+4 d^4 \sin (2 e+3 f x)\right )}{d (c+d \sin (e+f x))^3}\right )}{8 (c-d)^2 (c+d)^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \]

[In]

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

[Out]

((1 + Sin[e + f*x])*((24*(2*c^2 - 2*c*d + d^2)*ArcTan[(Sec[(f*x)/2]*(Cos[e] - I*Sin[e])*(d*Cos[e + (f*x)/2] +
c*Sin[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*(Cos[e] - I*Sin[e]))/(Sqrt[c^2 - d^2]*Sqrt[(Co
s[e] - I*Sin[e])^2]) + (2*c*(4*c^4 - 18*c^3*d + 14*c^2*d^2 - 27*c*d^3 + 12*d^4)*Cot[e] - d*Csc[e]*(3*d*(4*c^3
- 16*c^2*d + 6*c*d^2 + d^3)*Cos[e + 2*f*x] - 3*d^2*(2*c^2 - 2*c*d + d^2)*Cos[3*e + 2*f*x] - 24*c^4*Sin[f*x] +
78*c^3*d*Sin[f*x] - 24*c^2*d^2*Sin[f*x] + 12*c*d^3*Sin[f*x] - 12*d^4*Sin[f*x] + 30*c^3*d*Sin[2*e + f*x] - 30*c
^2*d^2*Sin[2*e + f*x] + 15*c*d^3*Sin[2*e + f*x] + 2*c^2*d^2*Sin[2*e + 3*f*x] - 9*c*d^3*Sin[2*e + 3*f*x] + 4*d^
4*Sin[2*e + 3*f*x]))/(d*(c + d*Sin[e + f*x])^3)))/(8*(c - d)^2*(c + d)^3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2
])^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(640\) vs. \(2(181)=362\).

Time = 1.59 (sec) , antiderivative size = 641, normalized size of antiderivative = 3.43

method	result	size
derivativedivides	\(\frac {2 a \left (\frac {-\frac {d \left (4 c^{4}-5 c^{3} d -4 c^{2} d^{2}+2 d^{3} c +2 d^{4}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}-\frac {\left (2 c^{6}-4 c^{5} d +10 c^{4} d^{2}-17 c^{3} d^{3}-6 c^{2} d^{4}+6 c \,d^{5}+4 d^{6}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right ) c^{2}}-\frac {d \left (18 c^{6}-36 c^{5} d +6 c^{4} d^{2}-15 c^{3} d^{3}+2 c^{2} d^{4}+6 c \,d^{5}+4 d^{6}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}-\frac {\left (2 c^{6}-4 c^{5} d +6 c^{4} d^{2}-14 c^{3} d^{3}+3 c \,d^{5}+2 d^{6}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right ) c^{2}}-\frac {d \left (8 c^{4}-19 c^{3} d +4 d^{3} c +2 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}-\frac {6 c^{4}-12 c^{3} d -2 c^{2} d^{2}+3 d^{3} c +2 d^{4}}{6 \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{3}}+\frac {\left (2 c^{2}-2 c d +d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\)	\(641\)
default	\(\frac {2 a \left (\frac {-\frac {d \left (4 c^{4}-5 c^{3} d -4 c^{2} d^{2}+2 d^{3} c +2 d^{4}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}-\frac {\left (2 c^{6}-4 c^{5} d +10 c^{4} d^{2}-17 c^{3} d^{3}-6 c^{2} d^{4}+6 c \,d^{5}+4 d^{6}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right ) c^{2}}-\frac {d \left (18 c^{6}-36 c^{5} d +6 c^{4} d^{2}-15 c^{3} d^{3}+2 c^{2} d^{4}+6 c \,d^{5}+4 d^{6}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}-\frac {\left (2 c^{6}-4 c^{5} d +6 c^{4} d^{2}-14 c^{3} d^{3}+3 c \,d^{5}+2 d^{6}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right ) c^{2}}-\frac {d \left (8 c^{4}-19 c^{3} d +4 d^{3} c +2 d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}-\frac {6 c^{4}-12 c^{3} d -2 c^{2} d^{2}+3 d^{3} c +2 d^{4}}{6 \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right )}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{3}}+\frac {\left (2 c^{2}-2 c d +d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{5}+c^{4} d -2 c^{3} d^{2}-2 c^{2} d^{3}+c \,d^{4}+d^{5}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\)	\(641\)
risch	\(\frac {i a \left (-12 i c \,d^{4} {\mathrm e}^{2 i \left (f x +e \right )}+9 i c \,d^{4}+24 i c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+6 c^{2} d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-6 c \,d^{4} {\mathrm e}^{5 i \left (f x +e \right )}+3 d^{5} {\mathrm e}^{5 i \left (f x +e \right )}+12 i d^{5} {\mathrm e}^{2 i \left (f x +e \right )}-4 i d^{5}+30 i c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-2 i c^{2} d^{3}+24 i c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+8 c^{5} {\mathrm e}^{3 i \left (f x +e \right )}-36 c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}+28 c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-54 c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+24 c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}+15 i c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}-78 i c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-30 i c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}-12 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}+48 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}-18 c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}-3 d^{5} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 \left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )^{3} \left (c +d \right )^{3} \left (c -d \right )^{2} f d}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c^{2}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right )^{2} f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c d}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right )^{2} f}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) d^{2}}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right )^{2} f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c^{2}}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right )^{2} f}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c d}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right )^{2} f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) d^{2}}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} \left (c -d \right )^{2} f}\)	\(889\)

[In]

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

[Out]

2/f*a*((-1/2*d*(4*c^4-5*c^3*d-4*c^2*d^2+2*c*d^3+2*d^4)/c/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x
+1/2*e)^5-1/2*(2*c^6-4*c^5*d+10*c^4*d^2-17*c^3*d^3-6*c^2*d^4+6*c*d^5+4*d^6)/(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d
^4+d^5)/c^2*tan(1/2*f*x+1/2*e)^4-1/3/c^3*d*(18*c^6-36*c^5*d+6*c^4*d^2-15*c^3*d^3+2*c^2*d^4+6*c*d^5+4*d^6)/(c^5
+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)^3-(2*c^6-4*c^5*d+6*c^4*d^2-14*c^3*d^3+3*c*d^5+2*d^6)/
(c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/c^2*tan(1/2*f*x+1/2*e)^2-1/2*d*(8*c^4-19*c^3*d+4*c*d^3+2*d^4)/c/(c^5
+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)*tan(1/2*f*x+1/2*e)-1/6*(6*c^4-12*c^3*d-2*c^2*d^2+3*c*d^3+2*d^4)/(c^5+c^4
*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5))/(tan(1/2*f*x+1/2*e)^2*c+2*d*tan(1/2*f*x+1/2*e)+c)^3+1/2*(2*c^2-2*c*d+d^2)/(
c^5+c^4*d-2*c^3*d^2-2*c^2*d^3+c*d^4+d^5)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/
2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (181) = 362\).

Time = 0.35 (sec) , antiderivative size = 1344, normalized size of antiderivative = 7.19 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/12*(2*(2*a*c^4*d^2 - 9*a*c^3*d^3 + 2*a*c^2*d^4 + 9*a*c*d^5 - 4*a*d^6)*cos(f*x + e)^3 - 6*(2*a*c^5*d - 7*a*
c^4*d^2 + 8*a*c^2*d^4 - 2*a*c*d^5 - a*d^6)*cos(f*x + e)*sin(f*x + e) - 3*(2*a*c^5 - 2*a*c^4*d + 7*a*c^3*d^2 -
6*a*c^2*d^3 + 3*a*c*d^4 - 3*(2*a*c^3*d^2 - 2*a*c^2*d^3 + a*c*d^4)*cos(f*x + e)^2 + (6*a*c^4*d - 6*a*c^3*d^2 +
5*a*c^2*d^3 - 2*a*c*d^4 + a*d^5 - (2*a*c^2*d^3 - 2*a*c*d^4 + a*d^5)*cos(f*x + e)^2)*sin(f*x + e))*sqrt(-c^2 +
d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*c
os(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) - 12*(a*c^6 - 2*a*c^5*d
- a*c^4*d^2 + a*c^3*d^3 + a*c^2*d^4 + a*c*d^5 - a*d^6)*cos(f*x + e))/(3*(c^8*d^2 + c^7*d^3 - 3*c^6*d^4 - 3*c^5
*d^5 + 3*c^4*d^6 + 3*c^3*d^7 - c^2*d^8 - c*d^9)*f*cos(f*x + e)^2 - (c^10 + c^9*d - 6*c^6*d^4 - 6*c^5*d^5 + 8*c
^4*d^6 + 8*c^3*d^7 - 3*c^2*d^8 - 3*c*d^9)*f + ((c^7*d^3 + c^6*d^4 - 3*c^5*d^5 - 3*c^4*d^6 + 3*c^3*d^7 + 3*c^2*
d^8 - c*d^9 - d^10)*f*cos(f*x + e)^2 - (3*c^9*d + 3*c^8*d^2 - 8*c^7*d^3 - 8*c^6*d^4 + 6*c^5*d^5 + 6*c^4*d^6 -
c*d^9 - d^10)*f)*sin(f*x + e)), -1/6*((2*a*c^4*d^2 - 9*a*c^3*d^3 + 2*a*c^2*d^4 + 9*a*c*d^5 - 4*a*d^6)*cos(f*x
+ e)^3 - 3*(2*a*c^5*d - 7*a*c^4*d^2 + 8*a*c^2*d^4 - 2*a*c*d^5 - a*d^6)*cos(f*x + e)*sin(f*x + e) - 3*(2*a*c^5
- 2*a*c^4*d + 7*a*c^3*d^2 - 6*a*c^2*d^3 + 3*a*c*d^4 - 3*(2*a*c^3*d^2 - 2*a*c^2*d^3 + a*c*d^4)*cos(f*x + e)^2 +
(6*a*c^4*d - 6*a*c^3*d^2 + 5*a*c^2*d^3 - 2*a*c*d^4 + a*d^5 - (2*a*c^2*d^3 - 2*a*c*d^4 + a*d^5)*cos(f*x + e)^2
)*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 6*(a*c^6 - 2*a*
c^5*d - a*c^4*d^2 + a*c^3*d^3 + a*c^2*d^4 + a*c*d^5 - a*d^6)*cos(f*x + e))/(3*(c^8*d^2 + c^7*d^3 - 3*c^6*d^4 -
3*c^5*d^5 + 3*c^4*d^6 + 3*c^3*d^7 - c^2*d^8 - c*d^9)*f*cos(f*x + e)^2 - (c^10 + c^9*d - 6*c^6*d^4 - 6*c^5*d^5
+ 8*c^4*d^6 + 8*c^3*d^7 - 3*c^2*d^8 - 3*c*d^9)*f + ((c^7*d^3 + c^6*d^4 - 3*c^5*d^5 - 3*c^4*d^6 + 3*c^3*d^7 +
3*c^2*d^8 - c*d^9 - d^10)*f*cos(f*x + e)^2 - (3*c^9*d + 3*c^8*d^2 - 8*c^7*d^3 - 8*c^6*d^4 + 6*c^5*d^5 + 6*c^4*
d^6 - c*d^9 - d^10)*f)*sin(f*x + e))]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^4,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*d^2-4*c^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 775 vs. \(2 (181) = 362\).

Time = 0.39 (sec) , antiderivative size = 775, normalized size of antiderivative = 4.14 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*(3*(2*a*c^2 - 2*a*c*d + a*d^2)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) +
d)/sqrt(c^2 - d^2)))/((c^5 + c^4*d - 2*c^3*d^2 - 2*c^2*d^3 + c*d^4 + d^5)*sqrt(c^2 - d^2)) - (12*a*c^6*d*tan(
1/2*f*x + 1/2*e)^5 - 15*a*c^5*d^2*tan(1/2*f*x + 1/2*e)^5 - 12*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^5 + 6*a*c^3*d^4*t
an(1/2*f*x + 1/2*e)^5 + 6*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^5 + 6*a*c^7*tan(1/2*f*x + 1/2*e)^4 - 12*a*c^6*d*tan(1
/2*f*x + 1/2*e)^4 + 30*a*c^5*d^2*tan(1/2*f*x + 1/2*e)^4 - 51*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^4 - 18*a*c^3*d^4*t
an(1/2*f*x + 1/2*e)^4 + 18*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^4 + 12*a*c*d^6*tan(1/2*f*x + 1/2*e)^4 + 36*a*c^6*d*t
an(1/2*f*x + 1/2*e)^3 - 72*a*c^5*d^2*tan(1/2*f*x + 1/2*e)^3 + 12*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^3 - 30*a*c^3*d
^4*tan(1/2*f*x + 1/2*e)^3 + 4*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^3 + 12*a*c*d^6*tan(1/2*f*x + 1/2*e)^3 + 8*a*d^7*t
an(1/2*f*x + 1/2*e)^3 + 12*a*c^7*tan(1/2*f*x + 1/2*e)^2 - 24*a*c^6*d*tan(1/2*f*x + 1/2*e)^2 + 36*a*c^5*d^2*tan
(1/2*f*x + 1/2*e)^2 - 84*a*c^4*d^3*tan(1/2*f*x + 1/2*e)^2 + 18*a*c^2*d^5*tan(1/2*f*x + 1/2*e)^2 + 12*a*c*d^6*t
an(1/2*f*x + 1/2*e)^2 + 24*a*c^6*d*tan(1/2*f*x + 1/2*e) - 57*a*c^5*d^2*tan(1/2*f*x + 1/2*e) + 12*a*c^3*d^4*tan
(1/2*f*x + 1/2*e) + 6*a*c^2*d^5*tan(1/2*f*x + 1/2*e) + 6*a*c^7 - 12*a*c^6*d - 2*a*c^5*d^2 + 3*a*c^4*d^3 + 2*a*
c^3*d^4)/((c^8 + c^7*d - 2*c^6*d^2 - 2*c^5*d^3 + c^4*d^4 + c^3*d^5)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*
x + 1/2*e) + c)^3))/f

Mupad [B] (verification not implemented)

Time = 10.56 (sec) , antiderivative size = 877, normalized size of antiderivative = 4.69 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx=\frac {a\,\mathrm {atan}\left (\frac {\left (\frac {a\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c^2-2\,c\,d+d^2\right )}{{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{5/2}}+\frac {a\,\left (2\,c^2-2\,c\,d+d^2\right )\,\left (2\,c^5\,d+2\,c^4\,d^2-4\,c^3\,d^3-4\,c^2\,d^4+2\,c\,d^5+2\,d^6\right )}{2\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{5/2}\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}\right )\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}{2\,a\,c^2-2\,a\,c\,d+a\,d^2}\right )\,\left (2\,c^2-2\,c\,d+d^2\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{5/2}}-\frac {\frac {6\,a\,c^4-12\,a\,c^3\,d-2\,a\,c^2\,d^2+3\,a\,c\,d^3+2\,a\,d^4}{3\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,c^6-4\,c^5\,d+10\,c^4\,d^2-17\,c^3\,d^3-6\,c^2\,d^4+6\,c\,d^5+4\,d^6\right )}{c^2\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {2\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,c^6-4\,c^5\,d+6\,c^4\,d^2-14\,c^3\,d^3+3\,c\,d^5+2\,d^6\right )}{c^2\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {a\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,c^4-19\,c^3\,d+4\,c\,d^3+2\,d^4\right )}{c\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {a\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (4\,c^4-5\,c^3\,d-4\,c^2\,d^2+2\,c\,d^3+2\,d^4\right )}{c\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {2\,a\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,c^2+2\,d^2\right )\,\left (6\,c^4-12\,c^3\,d-2\,c^2\,d^2+3\,c\,d^3+2\,d^4\right )}{3\,c^3\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}}{f\,\left (c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (12\,c^2\,d+8\,d^3\right )+c^3+6\,c^2\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+6\,c^2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )} \]

[In]

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

[Out]

(a*atan((((a*c*tan(e/2 + (f*x)/2)*(2*c^2 - 2*c*d + d^2))/((c + d)^(7/2)*(c - d)^(5/2)) + (a*(2*c^2 - 2*c*d + d
^2)*(2*c*d^5 + 2*c^5*d + 2*d^6 - 4*c^2*d^4 - 4*c^3*d^3 + 2*c^4*d^2))/(2*(c + d)^(7/2)*(c - d)^(5/2)*(c*d^4 + c
^4*d + c^5 + d^5 - 2*c^2*d^3 - 2*c^3*d^2)))*(c*d^4 + c^4*d + c^5 + d^5 - 2*c^2*d^3 - 2*c^3*d^2))/(2*a*c^2 + a*
d^2 - 2*a*c*d))*(2*c^2 - 2*c*d + d^2))/(f*(c + d)^(7/2)*(c - d)^(5/2)) - ((6*a*c^4 + 2*a*d^4 - 2*a*c^2*d^2 + 3
*a*c*d^3 - 12*a*c^3*d)/(3*(c*d^4 + c^4*d + c^5 + d^5 - 2*c^2*d^3 - 2*c^3*d^2)) + (a*tan(e/2 + (f*x)/2)^4*(6*c*
d^5 - 4*c^5*d + 2*c^6 + 4*d^6 - 6*c^2*d^4 - 17*c^3*d^3 + 10*c^4*d^2))/(c^2*(c*d^4 + c^4*d + c^5 + d^5 - 2*c^2*
d^3 - 2*c^3*d^2)) + (2*a*tan(e/2 + (f*x)/2)^2*(3*c*d^5 - 4*c^5*d + 2*c^6 + 2*d^6 - 14*c^3*d^3 + 6*c^4*d^2))/(c
^2*(c*d^4 + c^4*d + c^5 + d^5 - 2*c^2*d^3 - 2*c^3*d^2)) + (a*d*tan(e/2 + (f*x)/2)*(4*c*d^3 - 19*c^3*d + 8*c^4
+ 2*d^4))/(c*(c*d^4 + c^4*d + c^5 + d^5 - 2*c^2*d^3 - 2*c^3*d^2)) + (a*d*tan(e/2 + (f*x)/2)^5*(2*c*d^3 - 5*c^3
*d + 4*c^4 + 2*d^4 - 4*c^2*d^2))/(c*(c*d^4 + c^4*d + c^5 + d^5 - 2*c^2*d^3 - 2*c^3*d^2)) + (2*a*d*tan(e/2 + (f
*x)/2)^3*(3*c^2 + 2*d^2)*(3*c*d^3 - 12*c^3*d + 6*c^4 + 2*d^4 - 2*c^2*d^2))/(3*c^3*(c*d^4 + c^4*d + c^5 + d^5 -
2*c^2*d^3 - 2*c^3*d^2)))/(f*(c^3*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^2*(12*c*d^2 + 3*c^3) + tan(e/2 + (
f*x)/2)^4*(12*c*d^2 + 3*c^3) + tan(e/2 + (f*x)/2)^3*(12*c^2*d + 8*d^3) + c^3 + 6*c^2*d*tan(e/2 + (f*x)/2) + 6*
c^2*d*tan(e/2 + (f*x)/2)^5))