3.5.32 \(\int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx\) [432]

Optimal. Leaf size=134 \[ \frac {a (2 c-d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d) \left (c^2-d^2\right )^{3/2} f}-\frac {a \cos (e+f x)}{2 (c+d) f (c+d \sin (e+f x))^2}-\frac {a (c-2 d) \cos (e+f x)}{2 (c-d) (c+d)^2 f (c+d \sin (e+f x))} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2833, 12, 2739, 632, 210} \begin {gather*} \frac {a (2 c-d) \text {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 )^{3/2}}-\frac {a (c-2 d) \cos (e+f x)}{2 f (c-d) (c+d)^2 (c+d \sin (e+f x))}-\frac {a \cos (e+f x)}{2 f (c+d) (c+d \sin (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*c - d)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((c + d)*(c^2 - d^2)^(3/2)*f) - (a*Cos[e + f*x]
)/(2*(c + d)*f*(c + d*Sin[e + f*x])^2) - (a*(c - 2*d)*Cos[e + f*x])/(2*(c - d)*(c + d)^2*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*} \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx &=-\frac {a \cos (e+f x)}{2 (c+d) f (c+d \sin (e+f x))^2}-\frac {\int \frac {-2 a (c-d)-a (c-d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{2 \left (c^2-d^2\right )}\\ &=-\frac {a \cos (e+f x)}{2 (c+d) f (c+d \sin (e+f x))^2}-\frac {a (c-2 d) \cos (e+f x)}{2 (c-d) (c+d)^2 f (c+d \sin (e+f x))}+\frac {\int \frac {a (c-d) (2 c-d)}{c+d \sin (e+f x)} \, dx}{2 \left (c^2-d^2\right )^2}\\ &=-\frac {a \cos (e+f x)}{2 (c+d) f (c+d \sin (e+f x))^2}-\frac {a (c-2 d) \cos (e+f x)}{2 (c-d) (c+d)^2 f (c+d \sin (e+f x))}+\frac {(a (2 c-d)) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 (c-d) (c+d)^2}\\ &=-\frac {a \cos (e+f x)}{2 (c+d) f (c+d \sin (e+f x))^2}-\frac {a (c-2 d) \cos (e+f x)}{2 (c-d) (c+d)^2 f (c+d \sin (e+f x))}+\frac {(a (2 c-d)) \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) (c+d)^2 f}\\ &=-\frac {a \cos (e+f x)}{2 (c+d) f (c+d \sin (e+f x))^2}-\frac {a (c-2 d) \cos (e+f x)}{2 (c-d) (c+d)^2 f (c+d \sin (e+f x))}-\frac {(2 a (2 c-d)) \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) (c+d)^2 f}\\ &=\frac {a (2 c-d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c-d) (c+d)^2 \sqrt {c^2-d^2} f}-\frac {a \cos (e+f x)}{2 (c+d) f (c+d \sin (e+f x))^2}-\frac {a (c-2 d) \cos (e+f x)}{2 (c-d) (c+d)^2 f (c+d \sin (e+f x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.79, size = 242, normalized size = 1.81 \begin {gather*} \frac {a (1+\sin (e+f x)) \left (\frac {4 (2 c-d) \tan ^{-1}\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))}{(c-d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {2 (c+d) \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d \sin (e+f x))^2}+\frac {(-4 c+2 d) \cot (e)+2 (c-2 d) \csc (e) \sin (f x)}{(c-d) (c+d \sin (e+f x))}\right )}{4 (c+d)^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(1 + Sin[e + f*x])*((4*(2*c - d)*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]))/((c - d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e
] - I*Sin[e])^2]) + (2*(c + d)*Csc[e]*(c*Cos[e] + d*Sin[f*x]))/(d*(c + d*Sin[e + f*x])^2) + ((-4*c + 2*d)*Cot[
e] + 2*(c - 2*d)*Csc[e]*Sin[f*x])/((c - d)*(c + d*Sin[e + f*x]))))/(4*(c + d)^2*f*(Cos[(e + f*x)/2] + Sin[(e +
f*x)/2])^2)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(324\) vs. \(2(125)=250\).
time = 0.54, size = 325, normalized size = 2.43

method	result	size
derivativedivides	\(\frac {2 a \left (\frac {-\frac {d \left (3 c^{2}-2 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right ) c}-\frac {\left (2 c^{4}-2 c^{3} d +3 c^{2} d^{2}-4 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right ) c^{2}}-\frac {d \left (5 c^{2}-6 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right )}-\frac {2 c^{2}-2 c d -d^{2}}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 c -d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\)	\(325\)
default	\(\frac {2 a \left (\frac {-\frac {d \left (3 c^{2}-2 c d -2 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right ) c}-\frac {\left (2 c^{4}-2 c^{3} d +3 c^{2} d^{2}-4 d^{3} c -2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right ) c^{2}}-\frac {d \left (5 c^{2}-6 c d -2 d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right )}-\frac {2 c^{2}-2 c d -d^{2}}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 c -d \right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{3}+c^{2} d -c \,d^{2}-d^{3}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\)	\(325\)
risch	\(\frac {i a \left (2 i c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}-i d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+4 i c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}-6 i c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}-i d^{3} {\mathrm e}^{i \left (f x +e \right )}+2 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-4 d \,c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+d^{2} {\mathrm e}^{2 i \left (f x +e \right )} c -2 d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-c \,d^{2}+2 d^{3}\right )}{\left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )^{2} \left (c +d \right )^{2} \left (c -d \right ) f d}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right ) f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) d}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right ) f}+\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) c}{\sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right ) f}-\frac {a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right ) d}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{2} \left (c -d \right ) f}\)	\(522\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (130) = 260\).
time = 0.38, size = 826, normalized size = 6.16 \begin {gather*} \left [\frac {2 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} - a c d^{3} + 2 \, a d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (2 \, a c^{3} - a c^{2} d + 2 \, a c d^{2} - a d^{3} - {\left (2 \, a c d^{2} - a d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, a c^{2} d - a c d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) + 2 \, {\left (2 \, a c^{4} - 2 \, a c^{3} d - 3 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right )} \cos \left (f x + e\right )}{4 \, {\left ({\left (c^{5} d^{2} + c^{4} d^{3} - 2 \, c^{3} d^{4} - 2 \, c^{2} d^{5} + c d^{6} + d^{7}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{6} d + c^{5} d^{2} - 2 \, c^{4} d^{3} - 2 \, c^{3} d^{4} + c^{2} d^{5} + c d^{6}\right )} f \sin \left (f x + e\right ) - {\left (c^{7} + c^{6} d - c^{5} d^{2} - c^{4} d^{3} - c^{3} d^{4} - c^{2} d^{5} + c d^{6} + d^{7}\right )} f\right )}}, \frac {{\left (a c^{3} d - 2 \, a c^{2} d^{2} - a c d^{3} + 2 \, a d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (2 \, a c^{3} - a c^{2} d + 2 \, a c d^{2} - a d^{3} - {\left (2 \, a c d^{2} - a d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (2 \, a c^{2} d - a c d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) + {\left (2 \, a c^{4} - 2 \, a c^{3} d - 3 \, a c^{2} d^{2} + 2 \, a c d^{3} + a d^{4}\right )} \cos \left (f x + e\right )}{2 \, {\left ({\left (c^{5} d^{2} + c^{4} d^{3} - 2 \, c^{3} d^{4} - 2 \, c^{2} d^{5} + c d^{6} + d^{7}\right )} f \cos \left (f x + e\right )^{2} - 2 \, {\left (c^{6} d + c^{5} d^{2} - 2 \, c^{4} d^{3} - 2 \, c^{3} d^{4} + c^{2} d^{5} + c d^{6}\right )} f \sin \left (f x + e\right ) - {\left (c^{7} + c^{6} d - c^{5} d^{2} - c^{4} d^{3} - c^{3} d^{4} - c^{2} d^{5} + c d^{6} + d^{7}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(2*(a*c^3*d - 2*a*c^2*d^2 - a*c*d^3 + 2*a*d^4)*cos(f*x + e)*sin(f*x + e) + (2*a*c^3 - a*c^2*d + 2*a*c*d^2
- a*d^3 - (2*a*c*d^2 - a*d^3)*cos(f*x + e)^2 + 2*(2*a*c^2*d - a*c*d^2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2
*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*cos(f*x + e)*sin(f*x + e) + d*cos(f*x + e))
*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2)) + 2*(2*a*c^4 - 2*a*c^3*d - 3*a*c^2*d
^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e))/((c^5*d^2 + c^4*d^3 - 2*c^3*d^4 - 2*c^2*d^5 + c*d^6 + d^7)*f*cos(f*x + e
)^2 - 2*(c^6*d + c^5*d^2 - 2*c^4*d^3 - 2*c^3*d^4 + c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^7 + c^6*d - c^5*d^2 -
c^4*d^3 - c^3*d^4 - c^2*d^5 + c*d^6 + d^7)*f), 1/2*((a*c^3*d - 2*a*c^2*d^2 - a*c*d^3 + 2*a*d^4)*cos(f*x + e)*s
in(f*x + e) + (2*a*c^3 - a*c^2*d + 2*a*c*d^2 - a*d^3 - (2*a*c*d^2 - a*d^3)*cos(f*x + e)^2 + 2*(2*a*c^2*d - a*c
*d^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))) + (2*a*c^4 -
2*a*c^3*d - 3*a*c^2*d^2 + 2*a*c*d^3 + a*d^4)*cos(f*x + e))/((c^5*d^2 + c^4*d^3 - 2*c^3*d^4 - 2*c^2*d^5 + c*d^6
+ d^7)*f*cos(f*x + e)^2 - 2*(c^6*d + c^5*d^2 - 2*c^4*d^3 - 2*c^3*d^4 + c^2*d^5 + c*d^6)*f*sin(f*x + e) - (c^7
+ c^6*d - c^5*d^2 - c^4*d^3 - c^3*d^4 - c^2*d^5 + c*d^6 + d^7)*f)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (130) = 260\).
time = 0.53, size = 384, normalized size = 2.87 \begin {gather*} \frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )} {\left (2 \, a c - a d\right )}}{{\left (c^{3} + c^{2} d - c d^{2} - d^{3}\right )} \sqrt {c^{2} - d^{2}}} - \frac {3 \, a c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, a c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, a c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, a c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a c^{4} - 2 \, a c^{3} d - a c^{2} d^{2}}{{\left (c^{5} + c^{4} d - c^{3} d^{2} - c^{2} d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 8.97, size = 445, normalized size = 3.32 \begin {gather*} -\frac {\frac {-2\,a\,c^2+2\,a\,c\,d+a\,d^2}{-c^3-c^2\,d+c\,d^2+d^3}+\frac {a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (c^2+2\,d^2\right )\,\left (-2\,c^2+2\,c\,d+d^2\right )}{c^2\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}+\frac {a\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (-5\,c^2+6\,c\,d+2\,d^2\right )}{c\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}+\frac {a\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-3\,c^2+2\,c\,d+2\,d^2\right )}{c\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,c^2+4\,d^2\right )+c^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+c^2+4\,c\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+4\,c\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}-\frac {a\,\mathrm {atan}\left (\frac {\left (\frac {a\,\left (2\,c-d\right )\,\left (-2\,c^3\,d-2\,c^2\,d^2+2\,c\,d^3+2\,d^4\right )}{2\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{3/2}\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}+\frac {a\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c-d\right )}{{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{3/2}}\right )\,\left (-c^3-c^2\,d+c\,d^2+d^3\right )}{2\,a\,c-a\,d}\right )\,\left (2\,c-d\right )}{f\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((a*d^2 - 2*a*c^2 + 2*a*c*d)/(c*d^2 - c^2*d - c^3 + d^3) + (a*tan(e/2 + (f*x)/2)^2*(c^2 + 2*d^2)*(2*c*d - 2*
c^2 + d^2))/(c^2*(c*d^2 - c^2*d - c^3 + d^3)) + (a*d*tan(e/2 + (f*x)/2)*(6*c*d - 5*c^2 + 2*d^2))/(c*(c*d^2 - c
^2*d - c^3 + d^3)) + (a*d*tan(e/2 + (f*x)/2)^3*(2*c*d - 3*c^2 + 2*d^2))/(c*(c*d^2 - c^2*d - c^3 + d^3)))/(f*(t
an(e/2 + (f*x)/2)^2*(2*c^2 + 4*d^2) + c^2*tan(e/2 + (f*x)/2)^4 + c^2 + 4*c*d*tan(e/2 + (f*x)/2)^3 + 4*c*d*tan(
e/2 + (f*x)/2))) - (a*atan((((a*(2*c - d)*(2*c*d^3 - 2*c^3*d + 2*d^4 - 2*c^2*d^2))/(2*(c + d)^(5/2)*(c - d)^(3
/2)*(c*d^2 - c^2*d - c^3 + d^3)) + (a*c*tan(e/2 + (f*x)/2)*(2*c - d))/((c + d)^(5/2)*(c - d)^(3/2)))*(c*d^2 -
c^2*d - c^3 + d^3))/(2*a*c - a*d))*(2*c - d))/(f*(c + d)^(5/2)*(c - d)^(3/2))

________________________________________________________________________________________