\(\int \frac {\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [388]

Optimal result

Integrand size = 23, antiderivative size = 764 \[ \int \frac {\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {2 (-1)^{2/3} a^{2/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} b^{4/3} d}+\frac {2 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 a^{2/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{4/3} d}-\frac {4 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{2/3} d}+\frac {2 \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {2 \sqrt [3]{-1} a^{2/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} b^{4/3} d}-\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} b^{2/3} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} b^{2/3} d}-\frac {\cos (c+d x)}{b d} \]

[Out]

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

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3305, 3292, 2739, 632, 210, 3299, 212, 2718} \[ \int \frac {\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {2 (-1)^{2/3} a^{2/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 b^{4/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} d \sqrt {a^{2/3}-b^{2/3}}}-\frac {4 \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 b^{2/3} d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 a^{2/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 b^{4/3} d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{2/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}-\frac {2 \sqrt [3]{-1} a^{2/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 b^{4/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}-\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 b^{2/3} d \sqrt {b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac {4 \text {arctanh}\left (\frac {(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 b^{2/3} d \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}-\frac {\cos (c+d x)}{b d} \]

[In]

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

[Out]

(-2*(-1)^(2/3)*a^(2/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3
)]])/(3*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*b^(4/3)*d) + (2*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^
(2/3) - b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*a^(2/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/
2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(4/3)*d) - (4*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*
x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*Sqrt[a^(2/3) - b^(2/3)]*b^(2/3)*d) + (2*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/
3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*d) - (
2*(-1)^(1/3)*a^(2/3)*ArcTan[((-1)^(2/3)*b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]
])/(3*Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]*b^(4/3)*d) - (2*ArcTan[((-1)^(1/3)*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[
(c + d*x)/2]))/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]])/(3*a^(2/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) + (4*ArcT
anh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*Sqrt[-((-1)^(2/
3)*a^(2/3)) + b^(2/3)]*b^(2/3)*d) + (4*ArcTanh[(b^(1/3) + (-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)
*a^(2/3) + b^(2/3)]])/(3*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]*b^(2/3)*d) - Cos[c + d*x]/(b*d)

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 3292

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3305

Int[cos[(e_.) + (f_.)*(x_)]^(m_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_)), x_Symbol] :> Int[Expand[(1 - Sin
[e + f*x]^2)^(m/2)/(a + b*Sin[e + f*x]^n), x], x] /; FreeQ[{a, b, e, f}, x] && IGtQ[m/2, 0] && IntegerQ[(n - 1
)/2]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a+b \sin ^3(c+d x)}-\frac {2 \sin ^2(c+d x)}{a+b \sin ^3(c+d x)}+\frac {\sin ^4(c+d x)}{a+b \sin ^3(c+d x)}\right ) \, dx \\ & = -\left (2 \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx\right )+\int \frac {1}{a+b \sin ^3(c+d x)} \, dx+\int \frac {\sin ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx \\ & = -\left (2 \int \left (\frac {1}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac {1}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac {1}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx\right )+\int \left (-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx+\int \left (\frac {\sin (c+d x)}{b}-\frac {a \sin (c+d x)}{b \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx \\ & = -\frac {\int \frac {1}{-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{2/3}}-\frac {\int \frac {1}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{2/3}}-\frac {\int \frac {1}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{2/3}}+\frac {\int \sin (c+d x) \, dx}{b}-\frac {a \int \frac {\sin (c+d x)}{a+b \sin ^3(c+d x)} \, dx}{b}-\frac {2 \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{2/3}}-\frac {2 \int \frac {1}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{2/3}}-\frac {2 \int \frac {1}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{2/3}} \\ & = -\frac {\cos (c+d x)}{b d}-\frac {a \int \left (-\frac {1}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}-\frac {(-1)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}+\frac {\sqrt [3]{-1}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{b}-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [3]{a}-2 \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{2/3} d}-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [3]{a}+2 \sqrt [3]{-1} \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{2/3} d}-\frac {2 \text {Subst}\left (\int \frac {1}{-\sqrt [3]{a}-2 (-1)^{2/3} \sqrt [3]{b} x-\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{2/3} d}-\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d}-\frac {4 \text {Subst}\left (\int \frac {1}{-\sqrt [3]{-1} \sqrt [3]{a}+2 \sqrt [3]{b} x-\sqrt [3]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d}-\frac {4 \text {Subst}\left (\int \frac {1}{(-1)^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d} \\ & = -\frac {\cos (c+d x)}{b d}+\frac {a^{2/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{4/3}}-\frac {\left (\sqrt [3]{-1} a^{2/3}\right ) \int \frac {1}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{4/3}}+\frac {\left ((-1)^{2/3} a^{2/3}\right ) \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 b^{4/3}}+\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{2/3} d}+\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )-x^2} \, dx,x,-2 (-1)^{2/3} \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{2/3} d}+\frac {4 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{-1} \sqrt [3]{b}-2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{2/3} d}+\frac {8 \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d}+\frac {8 \text {Subst}\left (\int \frac {1}{-4 \left ((-1)^{2/3} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}-2 \sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d}+\frac {8 \text {Subst}\left (\int \frac {1}{4 \left (\sqrt [3]{-1} a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 (-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{2/3} d} \\ & = -\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {2 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-b^{2/3}} d}-\frac {4 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{2/3} d}+\frac {2 \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} b^{2/3} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} b^{2/3} d}-\frac {\cos (c+d x)}{b d}+\frac {\left (2 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{4/3} d}-\frac {\left (2 \sqrt [3]{-1} a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 (-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{4/3} d}+\frac {\left (2 (-1)^{2/3} a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-2 \sqrt [3]{-1} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{4/3} d} \\ & = -\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {2 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-b^{2/3}} d}-\frac {4 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{2/3} d}+\frac {2 \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} b^{2/3} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} b^{2/3} d}-\frac {\cos (c+d x)}{b d}-\frac {\left (4 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{4/3} d}+\frac {\left (4 \sqrt [3]{-1} a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )-x^2} \, dx,x,2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{4/3} d}-\frac {\left (4 (-1)^{2/3} a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 b^{4/3} d} \\ & = -\frac {2 \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 (-1)^{2/3} a^{2/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} b^{4/3} d}+\frac {2 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 a^{2/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{4/3} d}-\frac {4 \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 \sqrt {a^{2/3}-b^{2/3}} b^{2/3} d}+\frac {2 \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{2/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {2 \sqrt [3]{-1} a^{2/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} b^{4/3} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} b^{2/3} d}+\frac {4 \text {arctanh}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} b^{2/3} d}-\frac {\cos (c+d x)}{b d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.54 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.39 \[ \int \frac {\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {3 \cos (c+d x)+i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-2 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}-a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+2 i a \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3+a \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3+2 b \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i b \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{3 b d} \]

[In]

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

[Out]

-1/3*(3*Cos[c + d*x] + I*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)*b*#1^4 + I*b*#1^6 & , (2*b*ArcTan[Si
n[c + d*x]/(Cos[c + d*x] - #1)] - I*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (2*I)*a*ArcTan[Sin[c + d*x]/(Cos[c +
d*x] - #1)]*#1 - a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1 + (2*I)*a*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1
^3 + a*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^3 + 2*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*b*Log[1
- 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 + b*#1^5) & ])/(b*d)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.74 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.16

method	result	size
derivativedivides	\(\frac {-\frac {2}{b \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b -2 \textit {\_R}^{3} a -6 \textit {\_R}^{2} b -2 \textit {\_R} a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{3 b}}{d}\)	\(121\)
default	\(\frac {-\frac {2}{b \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} b -2 \textit {\_R}^{3} a -6 \textit {\_R}^{2} b -2 \textit {\_R} a +b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}}{3 b}}{d}\)	\(121\)
risch	\(-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 b d}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (729 a^{4} b^{8} d^{6} \textit {\_Z}^{6}-729 a^{4} b^{6} d^{4} \textit {\_Z}^{4}+\left (162 a^{4} b^{4} d^{2}+81 a^{2} b^{6} d^{2}\right ) \textit {\_Z}^{2}+a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {243 d^{5} b^{7} a^{6}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}-\frac {972 d^{5} b^{9} a^{4}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}\right ) \textit {\_R}^{5}+\left (\frac {81 i d^{4} b^{5} a^{7}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}-\frac {162 i d^{4} b^{7} a^{5}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}+\frac {81 i d^{4} b^{9} a^{3}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}\right ) \textit {\_R}^{4}+\left (-\frac {189 d^{3} b^{5} a^{6}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}+\frac {945 d^{3} b^{7} a^{4}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}-\frac {27 d^{3} b^{9} a^{2}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}\right ) \textit {\_R}^{3}+\left (-\frac {54 i d^{2} b^{3} a^{7}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}+\frac {189 i d^{2} b^{5} a^{5}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}-\frac {135 i d^{2} b^{7} a^{3}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}\right ) \textit {\_R}^{2}+\left (\frac {3 d b \,a^{8}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}+\frac {9 d \,b^{3} a^{6}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}-\frac {189 d \,b^{5} a^{4}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}-\frac {66 d \,b^{7} a^{2}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}\right ) \textit {\_R} +\frac {3 i a^{7} b}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}-\frac {27 i b^{3} a^{5}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}+\frac {18 i b^{5} a^{3}}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}+\frac {6 i b^{7} a}{a^{8}-4 a^{6} b^{2}-21 a^{4} b^{4}+23 a^{2} b^{6}+b^{8}}\right )\right )\)	\(976\)

[In]

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

[Out]

1/d*(-2/b/(1+tan(1/2*d*x+1/2*c)^2)+1/3/b*sum((_R^4*b-2*_R^3*a-6*_R^2*b-2*_R*a+b)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*
a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.71 (sec) , antiderivative size = 23437, normalized size of antiderivative = 30.68 \[ \int \frac {\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-(b*d*integrate(4*(3*a*b*cos(4*d*x + 4*c)^2 + 3*a*b*cos(2*d*x + 2*c)^2 + 3*a*b*sin(4*d*x + 4*c)^2 + 3*b^2*cos(
d*x + c)*sin(2*d*x + 2*c) + 3*a*b*sin(2*d*x + 2*c)^2 + b^2*sin(d*x + c) - (a*b*cos(4*d*x + 4*c) - a*b*cos(2*d*
x + 2*c) + b^2*sin(5*d*x + 5*c) + b^2*sin(d*x + c))*cos(6*d*x + 6*c) - (8*a*b*cos(3*d*x + 3*c) + 3*b^2*sin(4*d
*x + 4*c) - 3*b^2*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) - (6*a*b*cos(2*d*x + 2*c) + 8*a^2*sin(3*d*x + 3*c) - 3*b^
2*sin(d*x + c) - a*b)*cos(4*d*x + 4*c) - 8*(a*b*cos(d*x + c) + a^2*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) - (3*b^2
*sin(d*x + c) + a*b)*cos(2*d*x + 2*c) + (b^2*cos(5*d*x + 5*c) + b^2*cos(d*x + c) - a*b*sin(4*d*x + 4*c) + a*b*
sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + (3*b^2*cos(4*d*x + 4*c) - 3*b^2*cos(2*d*x + 2*c) - 8*a*b*sin(3*d*x + 3*c)
+ b^2)*sin(5*d*x + 5*c) + (8*a^2*cos(3*d*x + 3*c) - 3*b^2*cos(d*x + c) - 6*a*b*sin(2*d*x + 2*c))*sin(4*d*x +
4*c) + 8*(a^2*cos(2*d*x + 2*c) - a*b*sin(d*x + c))*sin(3*d*x + 3*c))/(b^3*cos(6*d*x + 6*c)^2 + 9*b^3*cos(4*d*x
+ 4*c)^2 + 64*a^2*b*cos(3*d*x + 3*c)^2 + 9*b^3*cos(2*d*x + 2*c)^2 + b^3*sin(6*d*x + 6*c)^2 + 9*b^3*sin(4*d*x
+ 4*c)^2 + 64*a^2*b*sin(3*d*x + 3*c)^2 - 48*a*b^2*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 9*b^3*sin(2*d*x + 2*c)^2
- 6*b^3*cos(2*d*x + 2*c) + b^3 - 2*(3*b^3*cos(4*d*x + 4*c) - 3*b^3*cos(2*d*x + 2*c) - 8*a*b^2*sin(3*d*x + 3*c
) + b^3)*cos(6*d*x + 6*c) - 6*(3*b^3*cos(2*d*x + 2*c) + 8*a*b^2*sin(3*d*x + 3*c) - b^3)*cos(4*d*x + 4*c) - 2*(
8*a*b^2*cos(3*d*x + 3*c) + 3*b^3*sin(4*d*x + 4*c) - 3*b^3*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(8*a*b^2*cos(
3*d*x + 3*c) - 3*b^3*sin(2*d*x + 2*c))*sin(4*d*x + 4*c) + 16*(3*a*b^2*cos(2*d*x + 2*c) - a*b^2)*sin(3*d*x + 3*
c)), x) + cos(d*x + c))/(b*d)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 17.29 (sec) , antiderivative size = 2338, normalized size of antiderivative = 3.06 \[ \int \frac {\cos ^4(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

symsum(log(172032*a^4*b^9 - 81920*a^2*b^11 - 98304*a^6*b^7 + 8192*a^8*b^5 - 319488*root(729*a^4*b^8*d^6 - 729*
a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)*a^2*b^12 - 688128*ro
ot(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d
, k)*a^4*b^10 + 344064*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 +
3*a^2*b^4 + a^6 - b^6, d, k)*a^6*b^8 - 98304*a*b^12*tan(c/2 + (d*x)/2) - 294912*root(729*a^4*b^8*d^6 - 729*a^
4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^2*a^2*b^13 - 1400832*r
oot(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6,
d, k)^2*a^4*b^11 + 1695744*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b
^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^2*a^6*b^9 + 5750784*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2
+ 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^3*a^4*b^12 - 3760128*root(729*a^4*b^8*d^6 - 729*a
^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^3*a^6*b^10 + 3317760*
root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6,
d, k)^4*a^4*b^13 - 3317760*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*
b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^4*a^6*b^11 - 7962624*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d
^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^5*a^4*b^14 + 5971968*root(729*a^4*b^8*d^6 - 729
*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^5*a^6*b^12 + 196608
*a^3*b^10*tan(c/2 + (d*x)/2) - 98304*a^5*b^8*tan(c/2 + (d*x)/2) - 1277952*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d
^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)*a^3*b^11*tan(c/2 + (d*x)/2) +
712704*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^
6 - b^6, d, k)*a^5*b^9*tan(c/2 + (d*x)/2) + 98304*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 8
1*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)*a^7*b^7*tan(c/2 + (d*x)/2) + 589824*root(729*a^4*b^8*
d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^2*a^3*b^12
*tan(c/2 + (d*x)/2) - 294912*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4
*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^2*a^5*b^10*tan(c/2 + (d*x)/2) - 294912*root(729*a^4*b^8*d^6 - 729*a^4*b^6*
d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^2*a^7*b^8*tan(c/2 + (d*x)/2)
+ 5308416*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 +
a^6 - b^6, d, k)^3*a^3*b^13*tan(c/2 + (d*x)/2) - 3538944*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4
*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^3*a^5*b^11*tan(c/2 + (d*x)/2) + 221184*root(7
29*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)
^3*a^7*b^9*tan(c/2 + (d*x)/2) - 5308416*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*
d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^4*a^3*b^14*tan(c/2 + (d*x)/2) + 5308416*root(729*a^4*b^8*d^6 -
729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)^4*a^5*b^12*tan(c
/2 + (d*x)/2) - 1990656*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2
+ 3*a^2*b^4 + a^6 - b^6, d, k)^5*a^5*b^13*tan(c/2 + (d*x)/2) - 196608*root(729*a^4*b^8*d^6 - 729*a^4*b^6*d^4 +
162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)*a*b^13*tan(c/2 + (d*x)/2))*root(7
29*a^4*b^8*d^6 - 729*a^4*b^6*d^4 + 162*a^4*b^4*d^2 + 81*a^2*b^6*d^2 - 3*a^4*b^2 + 3*a^2*b^4 + a^6 - b^6, d, k)
, k, 1, 6)/d - 2/(d*(b + b*tan(c/2 + (d*x)/2)^2))