\(\int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1137]

Optimal result

Integrand size = 27, antiderivative size = 173 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3 \left (4 a^2-b^2\right ) x}{2 b^5}-\frac {3 a \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \sqrt {a^2-b^2} d}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2}+\frac {3 \cos (c+d x) \left (4 a^2-b^2+2 a b \sin (c+d x)\right )}{2 b^4 d (a+b \sin (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2942, 2814, 2739, 632, 210} \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {3 a \left (4 a^2-3 b^2\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \sqrt {a^2-b^2}}+\frac {3 x \left (4 a^2-b^2\right )}{2 b^5}+\frac {3 \cos (c+d x) \left (4 a^2+2 a b \sin (c+d x)-b^2\right )}{2 b^4 d (a+b \sin (c+d x))}+\frac {\cos ^3(c+d x) (2 a+b \sin (c+d x))}{2 b^2 d (a+b \sin (c+d x))^2} \]

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 2.42 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.58 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {-\frac {48 a \left (4 a^2-3 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {96 a^4 c+24 a^2 b^2 c-12 b^4 c+96 a^4 d x+24 a^2 b^2 d x-12 b^4 d x+96 a^3 b \cos (c+d x)+12 b^2 \left (-4 a^2+b^2\right ) (c+d x) \cos (2 (c+d x))-8 a b^3 \cos (3 (c+d x))+192 a^3 b c \sin (c+d x)-48 a b^3 c \sin (c+d x)+192 a^3 b d x \sin (c+d x)-48 a b^3 d x \sin (c+d x)+72 a^2 b^2 \sin (2 (c+d x))-10 b^4 \sin (2 (c+d x))+b^4 \sin (4 (c+d x))}{(a+b \sin (c+d x))^2}}{16 b^5 d} \]

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

((-48*a*(4*a^2 - 3*b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (96*a^4*c + 24*a^2
*b^2*c - 12*b^4*c + 96*a^4*d*x + 24*a^2*b^2*d*x - 12*b^4*d*x + 96*a^3*b*Cos[c + d*x] + 12*b^2*(-4*a^2 + b^2)*(
c + d*x)*Cos[2*(c + d*x)] - 8*a*b^3*Cos[3*(c + d*x)] + 192*a^3*b*c*Sin[c + d*x] - 48*a*b^3*c*Sin[c + d*x] + 19
2*a^3*b*d*x*Sin[c + d*x] - 48*a*b^3*d*x*Sin[c + d*x] + 72*a^2*b^2*Sin[2*(c + d*x)] - 10*b^4*Sin[2*(c + d*x)] +
b^4*Sin[4*(c + d*x)])/(a + b*Sin[c + d*x])^2)/(16*b^5*d)

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.66

method	result	size
derivativedivides	\(\frac {-\frac {4 \left (\frac {-\frac {5 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {b \left (6 a^{4}+11 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {b^{2} \left (19 a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {a b \left (6 a^{2}-b^{2}\right )}{4}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 a \left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 \sqrt {a^{2}-b^{2}}}\right )}{b^{5}}+\frac {\frac {4 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{4}+\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b}{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{4}+\frac {3 a b}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+3 \left (4 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\)	\(288\)
default	\(\frac {-\frac {4 \left (\frac {-\frac {5 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}-\frac {b \left (6 a^{4}+11 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}-\frac {b^{2} \left (19 a^{2}-4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {a b \left (6 a^{2}-b^{2}\right )}{4}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {3 a \left (4 a^{2}-3 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 \sqrt {a^{2}-b^{2}}}\right )}{b^{5}}+\frac {\frac {4 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{4}+\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b}{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{4}+\frac {3 a b}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+3 \left (4 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5}}}{d}\)	\(288\)
risch	\(\frac {6 x \,a^{2}}{b^{5}}-\frac {3 x}{2 b^{3}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {3 a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}+\frac {3 a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {i \left (-8 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+3 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+20 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-5 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+3 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-7 a^{2} b^{2}+2 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,b^{5}}-\frac {6 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{5}}+\frac {9 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,b^{3}}+\frac {6 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, d \,b^{5}}-\frac {9 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,b^{3}}\)	\(541\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-4/b^5*((-5/4*a^2*b^2*tan(1/2*d*x+1/2*c)^3-1/4*b*(6*a^4+11*a^2*b^2-2*b^4)/a*tan(1/2*d*x+1/2*c)^2-1/4*b^2*
(19*a^2-4*b^2)*tan(1/2*d*x+1/2*c)-1/4*a*b*(6*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+3/4
*a*(4*a^2-3*b^2)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))+4/b^5*((1/4*tan(1/2
*d*x+1/2*c)^3*b^2+3/2*tan(1/2*d*x+1/2*c)^2*a*b-1/4*tan(1/2*d*x+1/2*c)*b^2+3/2*a*b)/(1+tan(1/2*d*x+1/2*c)^2)^2+
3/4*(4*a^2-b^2)*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. 377 vs. \(2 (162) = 324\).

Time = 0.35 (sec) , antiderivative size = 837, normalized size of antiderivative = 4.84 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\left [\frac {6 \, {\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 8 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 6 \, {\left (4 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + b^{6}\right )} d x - 3 \, {\left (4 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4} - {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, 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 ) - 6 \, {\left (4 \, a^{5} b - 3 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) - 2 \, {\left ({\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x + 3 \, {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{3} b^{6} - a b^{8}\right )} d \sin \left (d x + c\right ) - {\left (a^{4} b^{5} - b^{9}\right )} d\right )}}, \frac {3 \, {\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + b^{6}\right )} d x - 3 \, {\left (4 \, a^{5} + a^{3} b^{2} - 3 \, a b^{4} - {\left (4 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, 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 ) - 3 \, {\left (4 \, a^{5} b - 3 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x + 3 \, {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{3} b^{6} - a b^{8}\right )} d \sin \left (d x + c\right ) - {\left (a^{4} b^{5} - b^{9}\right )} d\right )}}\right ] \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^4*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (162) = 324\).

Time = 0.37 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.48 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (4 \, a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {6 \, {\left (4 \, 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 )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {2 \, {\left (6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 15 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 2 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 54 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 45 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 4 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 29 \, 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} + 42 \, 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 ) + 12 \, a^{4} - a^{2} b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, 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} a b^{4}}}{2 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*(3*(4*a^2 - b^2)*(d*x + c)/b^5 - 6*(4*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)))/(sqrt(a^2 - b^2)*b^5) + 2*(6*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 12*a^4
*tan(1/2*d*x + 1/2*c)^6 + 15*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 - 2*b^4*tan(1/2*d*x + 1/2*c)^6 + 54*a^3*b*tan(1/2*
d*x + 1/2*c)^5 + 36*a^4*tan(1/2*d*x + 1/2*c)^4 + 45*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 - 4*b^4*tan(1/2*d*x + 1/2*c
)^4 + 90*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 12*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 36*a^4*tan(1/2*d*x + 1/2*c)^2 + 29*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 + 42*a^3*b*tan(1/2*d*x + 1/2*c) - 4*a*b^3*tan(1/2
*d*x + 1/2*c) + 12*a^4 - a^2*b^2)/((a*tan(1/2*d*x + 1/2*c)^4 + 2*b*tan(1/2*d*x + 1/2*c)^3 + 2*a*tan(1/2*d*x +
1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a*b^4))/d

Mupad [B] (verification not implemented)

Time = 14.67 (sec) , antiderivative size = 1743, normalized size of antiderivative = 10.08 \[ \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)^4*sin(c + d*x))/(a + b*sin(c + d*x))^3,x)

[Out]

((54*a^2*tan(c/2 + (d*x)/2)^5)/b^3 - (a*b^2 - 12*a^3)/b^4 + (6*a^2*tan(c/2 + (d*x)/2)^7)/b^3 + (2*tan(c/2 + (d
*x)/2)*(21*a^2 - 2*b^2))/b^3 + (6*tan(c/2 + (d*x)/2)^3*(15*a^2 - 2*b^2))/b^3 + (tan(c/2 + (d*x)/2)^6*(12*a^4 -
2*b^4 + 15*a^2*b^2))/(a*b^4) + (tan(c/2 + (d*x)/2)^2*(36*a^4 - 2*b^4 + 29*a^2*b^2))/(a*b^4) + (tan(c/2 + (d*x
)/2)^4*(3*a^2 + 4*b^2)*(12*a^2 - b^2))/(a*b^4))/(d*(tan(c/2 + (d*x)/2)^2*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^
6*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(6*a^2 + 8*b^2) + a^2*tan(c/2 + (d*x)/2)^8 + a^2 + 12*a*b*tan(c/2 + (
d*x)/2)^3 + 12*a*b*tan(c/2 + (d*x)/2)^5 + 4*a*b*tan(c/2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2))) + (atan((864
*a^3*tan(c/2 + (d*x)/2))/(216*a*b^2 - 864*a^3) - (216*a*tan(c/2 + (d*x)/2))/(216*a - (864*a^3)/b^2))*(a^2*6i -
(b^2*3i)/2)*2i)/(b^5*d) + (a*atan(((a*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((8*(9*a^2*b^8 - 72*a^4*b^6 +
144*a^6*b^4))/b^11 + (8*tan(c/2 + (d*x)/2)*(18*a*b^10 - 234*a^3*b^8 + 576*a^5*b^6 - 288*a^7*b^4))/b^12 - (3*a*
(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((8*(6*a*b^12 - 12*a^3*b^10))/b^11 + (8*tan(c/2 + (d*x)/2)*(36*a^2*b^
12 - 48*a^4*b^10))/b^12 - (3*a*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*(32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(1
2*a*b^16 - 8*a^3*b^14))/b^12))/(2*(b^7 - a^2*b^5))))/(2*(b^7 - a^2*b^5)))*3i)/(2*(b^7 - a^2*b^5)) + (a*(-(a +
b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((8*(9*a^2*b^8 - 72*a^4*b^6 + 144*a^6*b^4))/b^11 + (8*tan(c/2 + (d*x)/2)*(18
*a*b^10 - 234*a^3*b^8 + 576*a^5*b^6 - 288*a^7*b^4))/b^12 + (3*a*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((8*(
6*a*b^12 - 12*a^3*b^10))/b^11 + (8*tan(c/2 + (d*x)/2)*(36*a^2*b^12 - 48*a^4*b^10))/b^12 + (3*a*(-(a + b)*(a -
b))^(1/2)*(4*a^2 - 3*b^2)*(32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(12*a*b^16 - 8*a^3*b^14))/b^12))/(2*(b^7 - a^2*b
^5))))/(2*(b^7 - a^2*b^5)))*3i)/(2*(b^7 - a^2*b^5)))/((16*(432*a^7 + 81*a^3*b^4 - 432*a^5*b^2))/b^11 + (16*tan
(c/2 + (d*x)/2)*(1728*a^8 - 81*a^2*b^6 + 756*a^4*b^4 - 2160*a^6*b^2))/b^12 + (3*a*(-(a + b)*(a - b))^(1/2)*(4*
a^2 - 3*b^2)*((8*(9*a^2*b^8 - 72*a^4*b^6 + 144*a^6*b^4))/b^11 + (8*tan(c/2 + (d*x)/2)*(18*a*b^10 - 234*a^3*b^8
+ 576*a^5*b^6 - 288*a^7*b^4))/b^12 - (3*a*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((8*(6*a*b^12 - 12*a^3*b^1
0))/b^11 + (8*tan(c/2 + (d*x)/2)*(36*a^2*b^12 - 48*a^4*b^10))/b^12 - (3*a*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*
b^2)*(32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(12*a*b^16 - 8*a^3*b^14))/b^12))/(2*(b^7 - a^2*b^5))))/(2*(b^7 - a^2*
b^5))))/(2*(b^7 - a^2*b^5)) - (3*a*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((8*(9*a^2*b^8 - 72*a^4*b^6 + 144*
a^6*b^4))/b^11 + (8*tan(c/2 + (d*x)/2)*(18*a*b^10 - 234*a^3*b^8 + 576*a^5*b^6 - 288*a^7*b^4))/b^12 + (3*a*(-(a
+ b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*((8*(6*a*b^12 - 12*a^3*b^10))/b^11 + (8*tan(c/2 + (d*x)/2)*(36*a^2*b^12 -
48*a^4*b^10))/b^12 + (3*a*(-(a + b)*(a - b))^(1/2)*(4*a^2 - 3*b^2)*(32*a^2*b^3 + (8*tan(c/2 + (d*x)/2)*(12*a*
b^16 - 8*a^3*b^14))/b^12))/(2*(b^7 - a^2*b^5))))/(2*(b^7 - a^2*b^5))))/(2*(b^7 - a^2*b^5))))*(-(a + b)*(a - b)
)^(1/2)*(4*a^2 - 3*b^2)*3i)/(d*(b^7 - a^2*b^5))