\(\int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1087]

Optimal result

Integrand size = 27, antiderivative size = 167 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {x}{b^3}+\frac {a \left (2 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^3 \left (a^2-b^2\right )^{3/2} d}-\frac {a \cos ^3(c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac {\cos (c+d x) \left (2 \left (a^2-b^2\right )+a b \sin (c+d x)\right )}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2943, 2942, 2814, 2739, 632, 210} \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {a \left (2 a^2-3 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^3 d \left (a^2-b^2\right )^{3/2}}-\frac {a \cos ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {\cos (c+d x) \left (2 \left (a^2-b^2\right )+a b \sin (c+d x)\right )}{2 b^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {x}{b^3} \]

[In]

Int[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

-(x/b^3) + (a*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*(a^2 - b^2)^(3/2)*d) - (a
*Cos[c + d*x]^3)/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (Cos[c + d*x]*(2*(a^2 - b^2) + a*b*Sin[c + d*x]))/
(2*b^2*(a^2 - b^2)*d*(a + b*Sin[c + d*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 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 2942

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

Rule 2943

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

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

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {-8 (c+d x)+\frac {2 a \left (8 a^4-20 a^2 b^2+15 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {a b \left (4 a^2-3 b^2\right ) \cos (c+d x)}{(a-b) (a+b) (a+b \sin (c+d x))^2}-\frac {3 b \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}}{b^3}-\frac {\frac {6 a b \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {\cos (c+d x) \left (a \left (2 a^2+b^2\right )+b \left (a^2+2 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}}{(a-b)^2 (a+b)^2}}{8 d} \]

[In]

Integrate[(Cos[c + d*x]^2*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

((-8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b
^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Cos[c + d*x])/((a - b)*(a + b)*(a + b*Sin[c + d*x])^2) - (3*b*(4*a^4 - 7*a^2*
b^2 + 2*b^4)*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/b^3 - ((6*a*b*ArcTan[(b + a*Tan[(c + d*
x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Sin[c + d*x]))/(a
+ b*Sin[c + d*x])^2)/((a - b)^2*(a + b)^2))/(8*d)

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.51

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a^{2}-b^{2}\right )}-\frac {b^{2} \left (7 a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a b \left (2 a^{2}-b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {a \left (2 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}}{b^{3}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}}{d}\)	\(252\)
default	\(\frac {\frac {\frac {2 \left (-\frac {a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (a^{2}-b^{2}\right )}-\frac {b^{2} \left (7 a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a b \left (2 a^{2}-b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {a \left (2 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}}{b^{3}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}}{d}\)	\(252\)
risch	\(-\frac {x}{b^{3}}+\frac {i \left (-4 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+8 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-5 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{2} b^{2}+2 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right ) d \,b^{3}}-\frac {i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}+\frac {i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}-\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}\)	\(519\)

[In]

int(cos(d*x+c)^2*sin(d*x+c)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^3*((-1/2*a^2*b^2/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3-1/2*b*(2*a^4+3*a^2*b^2-2*b^4)/a/(a^2-b^2)*tan(1/2*d*x
+1/2*c)^2-1/2*b^2*(7*a^2-4*b^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)-1/2*a*b*(2*a^2-b^2)/(a^2-b^2))/(tan(1/2*d*x+1/2*c
)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*a*(2*a^2-3*b^2)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/
(a^2-b^2)^(1/2)))-2/b^3*arctan(tan(1/2*d*x+1/2*c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (157) = 314\).

Time = 0.34 (sec) , antiderivative size = 793, normalized size of antiderivative = 4.75 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\left [-\frac {4 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} - 4 \, {\left (a^{6} - a^{4} b^{2} - a^{2} b^{4} + b^{6}\right )} d x - {\left (2 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4} - {\left (2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - 2 \, {\left (4 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d x + {\left (3 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} d \sin \left (d x + c\right ) - {\left (a^{6} b^{3} - a^{4} b^{5} - a^{2} b^{7} + b^{9}\right )} d\right )}}, -\frac {2 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{6} - a^{4} b^{2} - a^{2} b^{4} + b^{6}\right )} d x - {\left (2 \, a^{5} - a^{3} b^{2} - 3 \, a b^{4} - {\left (2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right ) - {\left (4 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} d x + {\left (3 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{5} b^{4} - 2 \, a^{3} b^{6} + a b^{8}\right )} d \sin \left (d x + c\right ) - {\left (a^{6} b^{3} - a^{4} b^{5} - a^{2} b^{7} + b^{9}\right )} d\right )}}\right ] \]

[In]

integrate(cos(d*x+c)^2*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**2*sin(d*x+c)/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^2*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.53 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {{\left (2 \, a^{3} - 3 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )} \sqrt {a^{2} - b^{2}}} - \frac {a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{4} - a^{2} b^{2}}{{\left (a^{3} b^{2} - a b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2}} - \frac {d x + c}{b^{3}}}{d} \]

[In]

integrate(cos(d*x+c)^2*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

((2*a^3 - 3*a*b^2)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b
^2)))/((a^2*b^3 - b^5)*sqrt(a^2 - b^2)) - (a^3*b*tan(1/2*d*x + 1/2*c)^3 + 2*a^4*tan(1/2*d*x + 1/2*c)^2 + 3*a^2
*b^2*tan(1/2*d*x + 1/2*c)^2 - 2*b^4*tan(1/2*d*x + 1/2*c)^2 + 7*a^3*b*tan(1/2*d*x + 1/2*c) - 4*a*b^3*tan(1/2*d*
x + 1/2*c) + 2*a^4 - a^2*b^2)/((a^3*b^2 - a*b^4)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2)
- (d*x + c)/b^3)/d

Mupad [B] (verification not implemented)

Time = 14.35 (sec) , antiderivative size = 2709, normalized size of antiderivative = 16.22 \[ \int \frac {\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)^2*sin(c + d*x))/(a + b*sin(c + d*x))^3,x)

[Out]

(2*atan((64*a*b^5*tan(c/2 + (d*x)/2))/((176*a^3*b^12)/(b^9 - 2*a^2*b^7 + a^4*b^5) - (160*a^5*b^10)/(b^9 - 2*a^
2*b^7 + a^4*b^5) + (48*a^7*b^8)/(b^9 - 2*a^2*b^7 + a^4*b^5) - (64*a*b^14)/(b^9 - 2*a^2*b^7 + a^4*b^5)) - (48*a
^3*b^3*tan(c/2 + (d*x)/2))/((176*a^3*b^12)/(b^9 - 2*a^2*b^7 + a^4*b^5) - (160*a^5*b^10)/(b^9 - 2*a^2*b^7 + a^4
*b^5) + (48*a^7*b^8)/(b^9 - 2*a^2*b^7 + a^4*b^5) - (64*a*b^14)/(b^9 - 2*a^2*b^7 + a^4*b^5))))/(b^3*d) - ((tan(
c/2 + (d*x)/2)*(7*a^2 - 4*b^2))/(b*(a^2 - b^2)) - (a*b^2 - 2*a^3)/(b^2*(a^2 - b^2)) + (a^2*tan(c/2 + (d*x)/2)^
3)/(b*(a^2 - b^2)) + (tan(c/2 + (d*x)/2)^2*(a^2 + 2*b^2)*(2*a^2 - b^2))/(a*b^2*(a^2 - b^2)))/(d*(tan(c/2 + (d*
x)/2)^2*(2*a^2 + 4*b^2) + a^2*tan(c/2 + (d*x)/2)^4 + a^2 + 4*a*b*tan(c/2 + (d*x)/2)^3 + 4*a*b*tan(c/2 + (d*x)/
2))) - (a*atan(((a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^6 - 8*a^4*b^4 + 4*a^6*b^2))/(b^9
- 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(8*a*b^8 - 29*a^3*b^6 + 28*a^5*b^4 - 8*a^7*b^2))/(b^10 - 2*a^2*
b^8 + a^4*b^6) - (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a*b^10 - 6*a^3*b^8 + 2*a^5*b^6))/(b^9
- 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(12*a^2*b^10 - 20*a^4*b^8 + 8*a^6*b^6))/(b^10 - 2*a^2*b^8 + a^4
*b^6) - (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^12 - 8*a^4*b^10 + 4*a^6*b^8))/(b^9 - 2*a^
2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(12*a*b^14 - 32*a^3*b^12 + 28*a^5*b^10 - 8*a^7*b^8))/(b^10 - 2*a^2*b^
8 + a^4*b^6)))/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))))/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))*1i)/
(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)) + (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^6 -
8*a^4*b^4 + 4*a^6*b^2))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(8*a*b^8 - 29*a^3*b^6 + 28*a^5*b^
4 - 8*a^7*b^2))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a*b^10 -
6*a^3*b^8 + 2*a^5*b^6))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(12*a^2*b^10 - 20*a^4*b^8 + 8*a^6
*b^6))/(b^10 - 2*a^2*b^8 + a^4*b^6) + (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^12 - 8*a^4*
b^10 + 4*a^6*b^8))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(12*a*b^14 - 32*a^3*b^12 + 28*a^5*b^10
- 8*a^7*b^8))/(b^10 - 2*a^2*b^8 + a^4*b^6)))/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))))/(2*(b^9 - 3*a^2*b^7
+ 3*a^4*b^5 - a^6*b^3)))*1i)/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))/((16*(2*a^5 - 3*a^3*b^2))/(b^9 - 2*
a^2*b^7 + a^4*b^5) + (16*tan(c/2 + (d*x)/2)*(8*a^6 + 12*a^2*b^4 - 20*a^4*b^2))/(b^10 - 2*a^2*b^8 + a^4*b^6) -
(a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^6 - 8*a^4*b^4 + 4*a^6*b^2))/(b^9 - 2*a^2*b^7 + a^
4*b^5) + (8*tan(c/2 + (d*x)/2)*(8*a*b^8 - 29*a^3*b^6 + 28*a^5*b^4 - 8*a^7*b^2))/(b^10 - 2*a^2*b^8 + a^4*b^6) -
(a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a*b^10 - 6*a^3*b^8 + 2*a^5*b^6))/(b^9 - 2*a^2*b^7 + a^
4*b^5) + (8*tan(c/2 + (d*x)/2)*(12*a^2*b^10 - 20*a^4*b^8 + 8*a^6*b^6))/(b^10 - 2*a^2*b^8 + a^4*b^6) - (a*(2*a^
2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^12 - 8*a^4*b^10 + 4*a^6*b^8))/(b^9 - 2*a^2*b^7 + a^4*b^5)
+ (8*tan(c/2 + (d*x)/2)*(12*a*b^14 - 32*a^3*b^12 + 28*a^5*b^10 - 8*a^7*b^8))/(b^10 - 2*a^2*b^8 + a^4*b^6)))/(
2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))))/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))))/(2*(b^9 - 3*a^2*b^7
+ 3*a^4*b^5 - a^6*b^3)) + (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^6 - 8*a^4*b^4 + 4*a^6*
b^2))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(8*a*b^8 - 29*a^3*b^6 + 28*a^5*b^4 - 8*a^7*b^2))/(b^
10 - 2*a^2*b^8 + a^4*b^6) + (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a*b^10 - 6*a^3*b^8 + 2*a^5*
b^6))/(b^9 - 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(12*a^2*b^10 - 20*a^4*b^8 + 8*a^6*b^6))/(b^10 - 2*a^
2*b^8 + a^4*b^6) + (a*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(4*a^2*b^12 - 8*a^4*b^10 + 4*a^6*b^8))/
(b^9 - 2*a^2*b^7 + a^4*b^5) + (8*tan(c/2 + (d*x)/2)*(12*a*b^14 - 32*a^3*b^12 + 28*a^5*b^10 - 8*a^7*b^8))/(b^10
- 2*a^2*b^8 + a^4*b^6)))/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))))/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*
b^3))))/(2*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))))*(2*a^2 - 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*1i)/(d*(b^9
- 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))