\(\int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx\) [132]

Optimal result

Integrand size = 28, antiderivative size = 122 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {a \left (a^2-3 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {2 a b}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3165, 3564, 3610, 3612, 3611} \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=-\frac {2 a b}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {a x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3} \]

[In]

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

[Out]

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

Rule 3165

Int[cos[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symb
ol] :> Int[(a + b*Tan[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b
^2, 0]

Rule 3564

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*
(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.97 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.26 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {\frac {2 a \left (a^2-3 b^2\right ) (c+d x)}{\left (a^2+b^2\right )^3}-\frac {2 b \left (-3 a^2+b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3}-\frac {b^3}{(a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}+\frac {6 b^2 \sin (c+d x)}{\left (a^2+b^2\right )^2 (a \cos (c+d x)+b \sin (c+d x))}}{2 d} \]

[In]

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

[Out]

((2*a*(a^2 - 3*b^2)*(c + d*x))/(a^2 + b^2)^3 - (2*b*(-3*a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2
+ b^2)^3 - b^3/((a - I*b)^2*(a + I*b)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (6*b^2*Sin[c + d*x])/((a^2 + b^
2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])))/(2*d)

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{3}-3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {b}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(140\)
default	\(\frac {\frac {\frac {\left (-3 a^{2} b +b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (a^{3}-3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {b}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 a b}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(140\)
parallelrisch	\(\frac {6 \left (\left (a^{2}-b^{2}\right ) \cos \left (2 d x +2 c \right )+2 a b \sin \left (2 d x +2 c \right )+a^{2}+b^{2}\right ) b \left (a^{2}-2 b^{2}\right ) \left (a^{2}-\frac {b^{2}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )-6 \left (\left (a^{2}-b^{2}\right ) \cos \left (2 d x +2 c \right )+2 a b \sin \left (2 d x +2 c \right )+a^{2}+b^{2}\right ) b \left (a^{2}-2 b^{2}\right ) \left (a^{2}-\frac {b^{2}}{3}\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (2 x \,a^{7} d -12 x \,a^{5} b^{2} d +22 x \,a^{3} b^{4} d -12 x a \,b^{6} d +5 a^{4} b^{3}+6 a^{2} b^{5}+b^{7}\right ) \cos \left (2 d x +2 c \right )+4 b \left (a^{5} d x -5 a^{3} b^{2} d x +6 a \,b^{4} d x +\frac {3}{2} a^{4} b +4 a^{2} b^{3}+\frac {5}{2} b^{5}\right ) a \sin \left (2 d x +2 c \right )+2 \left (a^{5} d x -5 a^{3} b^{2} d x +6 a \,b^{4} d x +\frac {15}{2} a^{2} b^{3}+\frac {3}{2} b^{5}\right ) \left (a^{2}+b^{2}\right )}{2 \left (\left (a^{2}-b^{2}\right ) \cos \left (2 d x +2 c \right )+2 a b \sin \left (2 d x +2 c \right )+a^{2}+b^{2}\right ) \left (a^{2}+b^{2}\right )^{3} \left (a^{2}-2 b^{2}\right ) d}\)	\(399\)
risch	\(-\frac {x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 a \,b^{2}}-\frac {6 i b \,a^{2} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 i b^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 i b \,a^{2} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 i b^{3} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {2 b^{2} \left (-2 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+3 i b a +3 a^{2}\right )}{\left (-i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} d \left (i a +b \right )^{3}}+\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{2}}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\)	\(403\)
norman	\(\text {Expression too large to display}\)	\(1096\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (120) = 240\).

Time = 0.29 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.80 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {5 \, a^{2} b^{3} - b^{5} + 2 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} d x - 2 \, {\left (6 \, a^{2} b^{3} - {\left (a^{5} - 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} - 3 \, a b^{4} + 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} d x\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (3 \, a^{2} b^{3} - b^{5} + {\left (3 \, a^{4} b - 4 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )}{2 \, {\left ({\left (a^{8} + 2 \, a^{6} b^{2} - 2 \, a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d\right )}} \]

[In]

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

[Out]

1/2*(5*a^2*b^3 - b^5 + 2*(a^3*b^2 - 3*a*b^4)*d*x - 2*(6*a^2*b^3 - (a^5 - 4*a^3*b^2 + 3*a*b^4)*d*x)*cos(d*x + c
)^2 + 2*(3*a^3*b^2 - 3*a*b^4 + 2*(a^4*b - 3*a^2*b^3)*d*x)*cos(d*x + c)*sin(d*x + c) + (3*a^2*b^3 - b^5 + (3*a^
4*b - 4*a^2*b^3 + b^5)*cos(d*x + c)^2 + 2*(3*a^3*b^2 - a*b^4)*cos(d*x + c)*sin(d*x + c))*log(2*a*b*cos(d*x + c
)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2))/((a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d*cos(d*x + c)^2 + 2*
(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*cos(d*x + c)*sin(d*x + c) + (a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*
d)

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (120) = 240\).

Time = 0.33 (sec) , antiderivative size = 481, normalized size of antiderivative = 3.94 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (-a - \frac {2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (\frac {{\left (3 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {{\left (5 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (3 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + \frac {4 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, {\left (a^{8} - 3 \, a^{4} b^{4} - 2 \, a^{2} b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {{\left (a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}}{d} \]

[In]

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

[Out]

(2*(a^3 - 3*a*b^2)*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*a^2*b - b^
3)*log(-a - 2*b*sin(d*x + c)/(cos(d*x + c) + 1) + a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a^6 + 3*a^4*b^2 + 3*
a^2*b^4 + b^6) - (3*a^2*b - b^3)*log(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b
^6) + 2*((3*a^3*b^2 + a*b^4)*sin(d*x + c)/(cos(d*x + c) + 1) + (5*a^2*b^3 + b^5)*sin(d*x + c)^2/(cos(d*x + c)
+ 1)^2 - (3*a^3*b^2 + a*b^4)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^8 + 2*a^6*b^2 + a^4*b^4 + 4*(a^7*b + 2*a^
5*b^3 + a^3*b^5)*sin(d*x + c)/(cos(d*x + c) + 1) - 2*(a^8 - 3*a^4*b^4 - 2*a^2*b^6)*sin(d*x + c)^2/(cos(d*x + c
) + 1)^2 - 4*(a^7*b + 2*a^5*b^3 + a^3*b^5)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + (a^8 + 2*a^6*b^2 + a^4*b^4)*s
in(d*x + c)^4/(cos(d*x + c) + 1)^4))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (120) = 240\).

Time = 0.39 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.17 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {9 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} - 3 \, b^{5} \tan \left (d x + c\right )^{2} + 22 \, a^{3} b^{2} \tan \left (d x + c\right ) - 2 \, a b^{4} \tan \left (d x + c\right ) + 14 \, a^{4} b + 3 \, a^{2} b^{3} + b^{5}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]

[In]

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

[Out]

1/2*(2*(a^3 - 3*a*b^2)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b - b^3)*log(tan(d*x + c)^2 + 1)
/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(3*a^2*b^2 - b^4)*log(abs(b*tan(d*x + c) + a))/(a^6*b + 3*a^4*b^3 + 3
*a^2*b^5 + b^7) - (9*a^2*b^3*tan(d*x + c)^2 - 3*b^5*tan(d*x + c)^2 + 22*a^3*b^2*tan(d*x + c) - 2*a*b^4*tan(d*x
+ c) + 14*a^4*b + 3*a^2*b^3 + b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(b*tan(d*x + c) + a)^2))/d

Mupad [B] (verification not implemented)

Time = 32.44 (sec) , antiderivative size = 6190, normalized size of antiderivative = 50.74 \[ \int \frac {\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*tan(c/2 + (d*x)/2)^2*(b^5 + 5*a^2*b^3))/(a^2*(a^4 + b^4 + 2*a^2*b^2)) + (2*b*tan(c/2 + (d*x)/2)*(3*a^2*b +
b^3))/(a*(a^4 + b^4 + 2*a^2*b^2)) - (2*b*tan(c/2 + (d*x)/2)^3*(3*a^2*b + b^3))/(a*(a^4 + b^4 + 2*a^2*b^2)))/(
d*(a^2*tan(c/2 + (d*x)/2)^4 - tan(c/2 + (d*x)/2)^2*(2*a^2 - 4*b^2) + a^2 - 4*a*b*tan(c/2 + (d*x)/2)^3 + 4*a*b*
tan(c/2 + (d*x)/2))) - (log((((-(a^2*(a^2 - 3*b^2)^2)/(a^2 + b^2)^6)^(1/2) + (3*a^2*b - b^3)/(a^2 + b^2)^3)*((
(-(a^2*(a^2 - 3*b^2)^2)/(a^2 + b^2)^6)^(1/2) + (3*a^2*b - b^3)/(a^2 + b^2)^3)*((32*a*b*tan(c/2 + (d*x)/2)*(b^4
- 8*a^4 + 5*a^2*b^2))/(a^2 + b^2)^2 - (32*a^2*(a^4 + 4*b^4 - 7*a^2*b^2))/(a^2 + b^2)^2 + 96*a*b*(a + b*tan(c/
2 + (d*x)/2))*((-(a^2*(a^2 - 3*b^2)^2)/(a^2 + b^2)^6)^(1/2) + (3*a^2*b - b^3)/(a^2 + b^2)^3)*(a^2 + b^2)) - (3
2*a^2*b*(5*a^2 - 3*b^2))/(a^2 + b^2)^3 + (32*a*tan(c/2 + (d*x)/2)*(a^6 - 3*b^6 + 27*a^2*b^4 - 17*a^4*b^2))/(a^
2 + b^2)^4) - (64*a^2*b^2*(3*a^4 + b^4 - 4*a^2*b^2))/(a^2 + b^2)^6 + (32*a*b*tan(c/2 + (d*x)/2)*(3*a^6 - b^6 -
3*a^2*b^4 + 17*a^4*b^2))/(a^2 + b^2)^6)*(((-(a^2*(a^2 - 3*b^2)^2)/(a^2 + b^2)^6)^(1/2) - (3*a^2*b - b^3)/(a^2
+ b^2)^3)*(((-(a^2*(a^2 - 3*b^2)^2)/(a^2 + b^2)^6)^(1/2) - (3*a^2*b - b^3)/(a^2 + b^2)^3)*((32*a^2*(a^4 + 4*b
^4 - 7*a^2*b^2))/(a^2 + b^2)^2 - (32*a*b*tan(c/2 + (d*x)/2)*(b^4 - 8*a^4 + 5*a^2*b^2))/(a^2 + b^2)^2 + 96*a*b*
(a + b*tan(c/2 + (d*x)/2))*((-(a^2*(a^2 - 3*b^2)^2)/(a^2 + b^2)^6)^(1/2) - (3*a^2*b - b^3)/(a^2 + b^2)^3)*(a^2
+ b^2)) - (32*a^2*b*(5*a^2 - 3*b^2))/(a^2 + b^2)^3 + (32*a*tan(c/2 + (d*x)/2)*(a^6 - 3*b^6 + 27*a^2*b^4 - 17*
a^4*b^2))/(a^2 + b^2)^4) + (64*a^2*b^2*(3*a^4 + b^4 - 4*a^2*b^2))/(a^2 + b^2)^6 - (32*a*b*tan(c/2 + (d*x)/2)*(
3*a^6 - b^6 - 3*a^2*b^4 + 17*a^4*b^2))/(a^2 + b^2)^6))*(6*a^2*b - 2*b^3))/(2*d*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*
b^2)) + (log(a + 2*b*tan(c/2 + (d*x)/2) - a*tan(c/2 + (d*x)/2)^2)*(3*a^2*b - b^3))/(d*(a^6 + b^6 + 3*a^2*b^4 +
3*a^4*b^2)) - (2*a*atan((tan(c/2 + (d*x)/2)*((((a*(a^2 - 3*b^2)*((32*(3*a*b^10 - a^11 - 21*a^3*b^8 - 34*a^5*b
^6 + 6*a^7*b^4 + 15*a^9*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) -
((6*a^2*b - 2*b^3)*((32*(a*b^13 - 8*a^13*b + 9*a^3*b^11 + 18*a^5*b^9 + 2*a^7*b^7 - 27*a^9*b^5 - 27*a^11*b^3))
/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a*b
^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6
+ b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))
))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - ((6*a^2*b - 2*b^3)*((a*(a^2
- 3*b^2)*((32*(a*b^13 - 8*a^13*b + 9*a^3*b^11 + 18*a^5*b^9 + 2*a^7*b^7 - 27*a^9*b^5 - 27*a^11*b^3))/(a^12 + b
^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a*b^16 + 21*a
^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*
a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(a^6 +
b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*a*(6*a^2*b - 2*b^3)*(a^2 - 3*b^2)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 1
05*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a
^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*
a^4*b^2)) + (32*a^3*(a^2 - 3*b^2)^3*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^
11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*
b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(a^8 + 4*b^8 - 61*a^2*b^6 + 155*a^4*b^4 - 67*a^6*b^2))/(a^8 + 4*
b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*a^6*b^2)^2 + (2*a*b*(7*a^6 - 10*b^6 + 59*a^2*b^4 - 68*a^4*b^2)*((32*(a*b^7
- 3*a^7*b + 3*a^3*b^5 - 17*a^5*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10
*b^2) + ((6*a^2*b - 2*b^3)*((32*(3*a*b^10 - a^11 - 21*a^3*b^8 - 34*a^5*b^6 + 6*a^7*b^4 + 15*a^9*b^2))/(a^12 +
b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - ((6*a^2*b - 2*b^3)*((32*(a*b^13 - 8*a
^13*b + 9*a^3*b^11 + 18*a^5*b^9 + 2*a^7*b^7 - 27*a^9*b^5 - 27*a^11*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^
8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*
a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 +
b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b
^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*(a^2 - 3*b^2)*((a*(a^2 - 3*b^2)*((32*(a*b^13 - 8*a^13*b +
9*a^3*b^11 + 18*a^5*b^9 + 2*a^7*b^7 - 27*a^9*b^5 - 27*a^11*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a
^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10
+ 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 +
6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*a
*(6*a^2*b - 2*b^3)*(a^2 - 3*b^2)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*
b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8
+ 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*a^2*(6*a^2*b - 2*b^3)*(a
^2 - 3*b^2)^2*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 +
3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*
a^8*b^4 + 6*a^10*b^2))))/(a^8 + 4*b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*a^6*b^2)^2)*(a^16 + b^16 + 8*a^2*b^14 + 2
8*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 56*a^10*b^6 + 28*a^12*b^4 + 8*a^14*b^2))/(32*a^4 - 96*a^2*b^2) + (((a*
(a^2 - 3*b^2)*((32*(5*a^10*b - 3*a^2*b^9 - 4*a^4*b^7 + 6*a^6*b^5 + 12*a^8*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15
*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - ((6*a^2*b - 2*b^3)*((32*(3*a^6*b^8 - 4*a^2*b^12 - 9*a^4*b^1
0 - a^14 + 22*a^8*b^6 + 18*a^10*b^4 + 3*a^12*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^
8*b^4 + 6*a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 +
105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 1
5*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(a^6 + b^6 + 3*
a^2*b^4 + 3*a^4*b^2) - ((6*a^2*b - 2*b^3)*((a*(a^2 - 3*b^2)*((32*(3*a^6*b^8 - 4*a^2*b^12 - 9*a^4*b^10 - a^14 +
22*a^8*b^6 + 18*a^10*b^4 + 3*a^12*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*
a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*
b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8
+ 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*a*(6*a^2*b - 2*b^3)*(a^2
- 3*b^2)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^
14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b
^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (32*a^3*(a^2 - 3*b^2)^3*(3*a^16*b + 3*a^2*b^15 +
21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4
+ 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(a^8 + 4*b^8 -
61*a^2*b^6 + 155*a^4*b^4 - 67*a^6*b^2)*(a^16 + b^16 + 8*a^2*b^14 + 28*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 5
6*a^10*b^6 + 28*a^12*b^4 + 8*a^14*b^2))/((32*a^4 - 96*a^2*b^2)*(a^8 + 4*b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*a^6
*b^2)^2) + (2*a*b*((32*(2*a^2*b^6 - 8*a^4*b^4 + 6*a^6*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^
6 + 15*a^8*b^4 + 6*a^10*b^2) + ((6*a^2*b - 2*b^3)*((32*(5*a^10*b - 3*a^2*b^9 - 4*a^4*b^7 + 6*a^6*b^5 + 12*a^8*
b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - ((6*a^2*b - 2*b^3)*((32
*(3*a^6*b^8 - 4*a^2*b^12 - 9*a^4*b^10 - a^14 + 22*a^8*b^6 + 18*a^10*b^4 + 3*a^12*b^2))/(a^12 + b^12 + 6*a^2*b^
10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a^16*b + 3*a^2*b^15 + 21*a^
4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^
4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2
*b^4 + 3*a^4*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*(a^2 - 3*b^2)*((a*(a^2 - 3*b^2)*((32*(3*a^6*
b^8 - 4*a^2*b^12 - 9*a^4*b^10 - a^14 + 22*a^8*b^6 + 18*a^10*b^4 + 3*a^12*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*
a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a^2*b - 2*b^3)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 +
63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(
a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^
4*b^2) + (16*a*(6*a^2*b - 2*b^3)*(a^2 - 3*b^2)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^
9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^1
0 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*a^2*(6*a^2
*b - 2*b^3)*(a^2 - 3*b^2)^2*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 +
63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20
*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(7*a^6 - 10*b^6 + 59*a^2*b^4 - 68*a^4*b^2)*(a^16 + b^16 + 8*a^2*b^14 + 2
8*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 56*a^10*b^6 + 28*a^12*b^4 + 8*a^14*b^2))/((32*a^4 - 96*a^2*b^2)*(a^8 +
4*b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*a^6*b^2)^2))*(a^2 - 3*b^2))/(d*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))