\(\int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx\) [131]

Optimal result

Integrand size = 28, antiderivative size = 216 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {3 b^2 \left (4 a^2-b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2} d}+\frac {b \left (3 a^2-b^2\right ) \cos (c+d x)}{\left (a^2+b^2\right )^3 d}+\frac {a \left (a^2-3 b^2\right ) \sin (c+d x)}{\left (a^2+b^2\right )^3 d}+\frac {b^4 \sin (c+d x)}{2 a \left (a^2+b^2\right )^2 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {b^3 \left (8 a^2+b^2\right )}{2 a \left (a^2+b^2\right )^3 d (a \cos (c+d x)+b \sin (c+d x))} \]

[Out]

-3*b^2*(4*a^2-b^2)*arctanh((b-a*tan(1/2*d*x+1/2*c))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2)/d+b*(3*a^2-b^2)*cos(d*x+c
)/(a^2+b^2)^3/d+a*(a^2-3*b^2)*sin(d*x+c)/(a^2+b^2)^3/d+1/2*b^4*sin(d*x+c)/a/(a^2+b^2)^2/d/(a*cos(d*x+c)+b*sin(
d*x+c))^2-1/2*b^3*(8*a^2+b^2)/a/(a^2+b^2)^3/d/(a*cos(d*x+c)+b*sin(d*x+c))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(492\) vs. \(2(216)=432\).

Time = 2.04 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.28, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6874, 653, 209, 652, 632, 212, 628} \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {3 b^4 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{7/2}}+\frac {4 b^4 \left (3 a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{7/2}}+\frac {2 \left (a \left (a^2-3 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )+b \left (3 a^2-b^2\right )\right )}{d \left (a^2+b^2\right )^3 \left (\tan ^2\left (\frac {1}{2} (c+d x)\right )+1\right )}-\frac {2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d \left (a^2+b^2\right )^{7/2}}-\frac {3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d \left (a^2+b^2\right )^3 \left (-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+2 b \tan \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 b^4 \left (\left (a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )+a b\right )}{a^3 d \left (a^2+b^2\right )^2 \left (-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+2 b \tan \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {4 b^3 \left (2 a^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )-b^4\right )}{a^3 d \left (a^2+b^2\right )^3 \left (-a \tan ^2\left (\frac {1}{2} (c+d x)\right )+a+2 b \tan \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

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

[Out]

(-3*b^4*(a^2 + 2*b^2)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(7/2)*d) + (4*b^4*(3
*a^2 + 2*b^2)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(7/2)*d) - (2*b^2*(6*a^4 + 3
*a^2*b^2 + b^4)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(7/2)*d) + (2*(b*(3*a^2 -
b^2) + a*(a^2 - 3*b^2)*Tan[(c + d*x)/2]))/((a^2 + b^2)^3*d*(1 + Tan[(c + d*x)/2]^2)) + (2*b^4*(a*b + (a^2 + 2*
b^2)*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)^2*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2)^2) - (3*b^4*(a^
2 + 2*b^2)*(b - a*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)^3*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2)) -
(4*b^3*(2*a^4 - b^4 + a*b*(3*a^2 + 2*b^2)*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)^3*d*(a + 2*b*Tan[(c + d*x)/2] -
a*Tan[(c + d*x)/2]^2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

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 652

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b*x + c*x^2)^(p + 1), x] - Dist[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 -
4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{\left (1+x^2\right )^2 \left (a+2 b x-a x^2\right )^3} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {2 \left (a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) x\right )}{\left (a^2+b^2\right )^3 \left (1+x^2\right )^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 \left (1+x^2\right )}+\frac {4 b^3 \left (-b \left (a^2+b^2\right )-a \left (2 a^2+b^2\right ) x\right )}{a^3 \left (a^2+b^2\right )^2 \left (a+2 b x-a x^2\right )^2}-\frac {4 b^4 (a+2 b x)}{a^3 \left (a^2+b^2\right ) \left (-a-2 b x+a x^2\right )^3}-\frac {b^2 \left (6 a^4+3 a^2 b^2+b^4\right )}{a^2 \left (a^2+b^2\right )^3 \left (-a-2 b x+a x^2\right )}\right ) \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{d} \\ & = \frac {4 \text {Subst}\left (\int \frac {a \left (a^2-3 b^2\right )-b \left (3 a^2-b^2\right ) x}{\left (1+x^2\right )^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d}-\frac {\left (2 a \left (a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left (8 b^3\right ) \text {Subst}\left (\int \frac {-b \left (a^2+b^2\right )-a \left (2 a^2+b^2\right ) x}{\left (a+2 b x-a x^2\right )^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {\left (8 b^4\right ) \text {Subst}\left (\int \frac {a+2 b x}{\left (-a-2 b x+a x^2\right )^3} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {\left (2 b^2 \left (6 a^4+3 a^2 b^2+b^4\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 )}{a^2 \left (a^2+b^2\right )^3 d} \\ & = -\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\left (2 a \left (a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d}+\frac {\left (6 b^4 \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (-a-2 b x+a x^2\right )^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d}-\frac {\left (4 b^4 \left (3 a^2+2 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 )}{a^2 \left (a^2+b^2\right )^3 d}+\frac {\left (4 b^2 \left (6 a^4+3 a^2 b^2+b^4\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 )}{a^2 \left (a^2+b^2\right )^3 d} \\ & = -\frac {2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac {2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}-\frac {4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}-\frac {\left (3 b^4 \left (a^2+2 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 )}{a^2 \left (a^2+b^2\right )^3 d}+\frac {\left (8 b^4 \left (3 a^2+2 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 )}{a^2 \left (a^2+b^2\right )^3 d} \\ & = \frac {4 b^4 \left (3 a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}-\frac {2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac {2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}-\frac {4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {\left (6 b^4 \left (a^2+2 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 )}{a^2 \left (a^2+b^2\right )^3 d} \\ & = -\frac {3 b^4 \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac {4 b^4 \left (3 a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}-\frac {2 b^2 \left (6 a^4+3 a^2 b^2+b^4\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{7/2} d}+\frac {2 \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2+b^2\right )^3 d \left (1+\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 b^4 \left (a b+\left (a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^2 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {3 b^4 \left (a^2+2 b^2\right ) \left (b-a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}-\frac {4 b^3 \left (2 a^4-b^4+a b \left (3 a^2+2 b^2\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 \left (a^2+b^2\right )^3 d \left (a+2 b \tan \left (\frac {1}{2} (c+d x)\right )-a \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.97 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {-\frac {6 b^2 \left (-4 a^2+b^2\right ) \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {2 b \left (-3 a^2+b^2\right ) \cos (c+d x)}{\left (a^2+b^2\right )^3}+\frac {2 a \left (a^2-3 b^2\right ) \sin (c+d x)}{\left (a^2+b^2\right )^3}+\frac {b^4 \sin (c+d x)}{a (a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac {b^3 \left (8 a^2+b^2\right )}{a \left (a^2+b^2\right )^3 (a \cos (c+d x)+b \sin (c+d x))}}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(\frac {-\frac {2 b^{2} \left (\frac {-\frac {b^{2} \left (9 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {b \left (8 a^{4}-15 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}+\frac {b^{2} \left (23 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+4 a^{2} b +\frac {b^{3}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {3 \left (4 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\left (-a^{3}+3 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 a^{2} b +b^{3}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\)	\(283\)
default	\(\frac {-\frac {2 b^{2} \left (\frac {-\frac {b^{2} \left (9 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}-\frac {b \left (8 a^{4}-15 a^{2} b^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}+\frac {b^{2} \left (23 a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+4 a^{2} b +\frac {b^{3}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}-\frac {3 \left (4 a^{2}-b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 \left (\left (-a^{3}+3 a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3 a^{2} b +b^{3}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}}{d}\)	\(283\)
risch	\(-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 \left (3 i b \,a^{2}-i b^{3}+a^{3}-3 a \,b^{2}\right ) d}+\frac {b^{3} {\mathrm e}^{i \left (d x +c \right )} \left (-7 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+8 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+7 i b a +8 a^{2}+b^{2}\right )}{\left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} d \left (i a +b \right )^{3}}+\frac {6 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{2}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {6 b^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{2}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}+\frac {3 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 a^{2} b^{5}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}\)	\(611\)

[In]

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

[Out]

1/d*(-2*b^2/(a^2+b^2)^3*((-1/2*b^2*(9*a^2+2*b^2)/a*tan(1/2*d*x+1/2*c)^3-1/2*b*(8*a^4-15*a^2*b^2-2*b^4)/a^2*tan
(1/2*d*x+1/2*c)^2+1/2*b^2*(23*a^2+2*b^2)/a*tan(1/2*d*x+1/2*c)+4*a^2*b+1/2*b^3)/(tan(1/2*d*x+1/2*c)^2*a-2*b*tan
(1/2*d*x+1/2*c)-a)^2-3/2*(4*a^2-b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))
)-2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*((-a^3+3*a*b^2)*tan(1/2*d*x+1/2*c)-3*a^2*b+b^3)/(1+tan(1/2*d*x+1/2*c)^2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (209) = 418\).

Time = 0.29 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.22 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {4 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (4 \, a^{2} b^{4} - b^{6} + {\left (4 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (4 \, a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 2 \, {\left (4 \, a^{6} b - 10 \, a^{4} b^{3} - 17 \, a^{2} b^{5} - 3 \, b^{7}\right )} \cos \left (d x + c\right ) + 2 \, {\left (2 \, a^{5} b^{2} - 11 \, a^{3} b^{4} - 13 \, a b^{6} + 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{10} + 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} - 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \]

[In]

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

[Out]

1/4*(4*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^3 - 3*(4*a^2*b^4 - b^6 + (4*a^4*b^2 - 5*a^2*b^4 + b^
6)*cos(d*x + c)^2 + 2*(4*a^3*b^3 - a*b^5)*cos(d*x + c)*sin(d*x + c))*sqrt(a^2 + b^2)*log((2*a*b*cos(d*x + c)*s
in(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 - 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/
(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) + 2*(4*a^6*b - 10*a^4*b^3 - 17*a^2*b^5 -
3*b^7)*cos(d*x + c) + 2*(2*a^5*b^2 - 11*a^3*b^4 - 13*a*b^6 + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(d*x
+ c)^2)*sin(d*x + c))/((a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*d*cos(d*x + c)^2 + 2*(a^9
*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*d*cos(d*x + c)*sin(d*x + c) + (a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6
+ 4*a^2*b^8 + b^10)*d)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (209) = 418\).

Time = 0.33 (sec) , antiderivative size = 658, normalized size of antiderivative = 3.05 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (6 \, a^{6} b - 10 \, a^{4} b^{3} - a^{2} b^{5} + \frac {{\left (2 \, a^{7} + 18 \, a^{5} b^{2} - 31 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, {\left (2 \, a^{6} b - 2 \, a^{4} b^{3} + 12 \, a^{2} b^{5} + b^{7}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (2 \, a^{7} + 2 \, a^{5} b^{2} + 15 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {{\left (2 \, a^{6} b - 30 \, a^{4} b^{3} + 15 \, a^{2} b^{5} + 2 \, b^{7}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (2 \, a^{7} - 6 \, a^{5} b^{2} + 9 \, a^{3} b^{4} + 2 \, a b^{6}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6} + \frac {4 \, {\left (a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {{\left (a^{10} - a^{8} b^{2} - 9 \, a^{6} b^{4} - 11 \, a^{4} b^{6} - 4 \, a^{2} b^{8}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (a^{10} - a^{8} b^{2} - 9 \, a^{6} b^{4} - 11 \, a^{4} b^{6} - 4 \, a^{2} b^{8}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {4 \, {\left (a^{9} b + 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} + a^{3} b^{7}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}}{2 \, d} \]

[In]

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

[Out]

-1/2*(3*(4*a^2*b^2 - b^4)*log((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sqrt(a^2 + b^2))/(b - a*sin(d*x + c)/(c
os(d*x + c) + 1) - sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 2*(6*a^6*b - 10*a
^4*b^3 - a^2*b^5 + (2*a^7 + 18*a^5*b^2 - 31*a^3*b^4 - 2*a*b^6)*sin(d*x + c)/(cos(d*x + c) + 1) - 2*(2*a^6*b -
2*a^4*b^3 + 12*a^2*b^5 + b^7)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(2*a^7 + 2*a^5*b^2 + 15*a^3*b^4)*sin(d*x
+ c)^3/(cos(d*x + c) + 1)^3 - (2*a^6*b - 30*a^4*b^3 + 15*a^2*b^5 + 2*b^7)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4
+ (2*a^7 - 6*a^5*b^2 + 9*a^3*b^4 + 2*a*b^6)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(a^10 + 3*a^8*b^2 + 3*a^6*b^
4 + a^4*b^6 + 4*(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7)*sin(d*x + c)/(cos(d*x + c) + 1) - (a^10 - a^8*b^2 -
9*a^6*b^4 - 11*a^4*b^6 - 4*a^2*b^8)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - (a^10 - a^8*b^2 - 9*a^6*b^4 - 11*a^4
*b^6 - 4*a^2*b^8)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 4*(a^9*b + 3*a^7*b^3 + 3*a^5*b^5 + a^3*b^7)*sin(d*x +
c)^5/(cos(d*x + c) + 1)^5 + (a^10 + 3*a^8*b^2 + 3*a^6*b^4 + a^4*b^6)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6))/d

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.85 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {4 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2} b - b^{3}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}} - \frac {2 \, {\left (9 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 23 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{4} b^{3} - a^{2} b^{5}\right )}}{{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{2}}}{2 \, d} \]

[In]

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

[Out]

-1/2*(3*(4*a^2*b^2 - b^4)*log(abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*d*x + 1/
2*c) - 2*b + 2*sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 4*(a^3*tan(1/2*d*x +
1/2*c) - 3*a*b^2*tan(1/2*d*x + 1/2*c) + 3*a^2*b - b^3)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(1/2*d*x + 1/2
*c)^2 + 1)) - 2*(9*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 2*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 8*a^4*b^3*tan(1/2*d*x + 1
/2*c)^2 - 15*a^2*b^5*tan(1/2*d*x + 1/2*c)^2 - 2*b^7*tan(1/2*d*x + 1/2*c)^2 - 23*a^3*b^4*tan(1/2*d*x + 1/2*c) -
2*a*b^6*tan(1/2*d*x + 1/2*c) - 8*a^4*b^3 - a^2*b^5)/((a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*(a*tan(1/2*d*x +
1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c) - a)^2))/d

Mupad [B] (verification not implemented)

Time = 26.66 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.82 \[ \int \frac {\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {\frac {-6\,a^4\,b+10\,a^2\,b^3+b^5}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^5+2\,a^3\,b^2+15\,a\,b^4\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^6\,b-30\,a^4\,b^3+15\,a^2\,b^5+2\,b^7\right )}{a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^6+18\,a^4\,b^2-31\,a^2\,b^4-2\,b^6\right )}{a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (2\,a^6-6\,a^4\,b^2+9\,a^2\,b^4+2\,b^6\right )}{a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^6\,b-2\,a^4\,b^3+12\,a^2\,b^5+b^7\right )}{a^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+a^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^2-4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2-4\,b^2\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\mathrm {atan}\left (\frac {-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^7+a^6\,b\,1{}\mathrm {i}-3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^5\,b^2+a^4\,b^3\,3{}\mathrm {i}-3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^3\,b^4+a^2\,b^5\,3{}\mathrm {i}-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b^6+b^7\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{7/2}}\right )\,\left (3\,b^4-12\,a^2\,b^2\right )\,1{}\mathrm {i}}{d\,{\left (a^2+b^2\right )}^{7/2}} \]

[In]

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

[Out]

- ((b^5 - 6*a^4*b + 10*a^2*b^3)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (2*tan(c/2 + (d*x)/2)^3*(15*a*b^4 + 2*a^
5 + 2*a^3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (tan(c/2 + (d*x)/2)^4*(2*a^6*b + 2*b^7 + 15*a^2*b^5 - 30
*a^4*b^3))/(a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c/2 + (d*x)/2)*(2*a^6 - 2*b^6 - 31*a^2*b^4 + 18*a^
4*b^2))/(a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c/2 + (d*x)/2)^5*(2*a^6 + 2*b^6 + 9*a^2*b^4 - 6*a^4*b^2
))/(a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*tan(c/2 + (d*x)/2)^2*(2*a^6*b + b^7 + 12*a^2*b^5 - 2*a^4*b^3))
/(a^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^2*tan(c/2 + (d*x)/2)^6 + a^2 - tan(c/2 + (d*x)/2)^2*(a^2 - 4
*b^2) - tan(c/2 + (d*x)/2)^4*(a^2 - 4*b^2) - 4*a*b*tan(c/2 + (d*x)/2)^5 + 4*a*b*tan(c/2 + (d*x)/2))) - (atan((
a^6*b*1i + b^7*1i + a^2*b^5*3i + a^4*b^3*3i - a^7*tan(c/2 + (d*x)/2)*1i - a*b^6*tan(c/2 + (d*x)/2)*1i - a^3*b^
4*tan(c/2 + (d*x)/2)*3i - a^5*b^2*tan(c/2 + (d*x)/2)*3i)/(a^2 + b^2)^(7/2))*(3*b^4 - 12*a^2*b^2)*1i)/(d*(a^2 +
b^2)^(7/2))