\(\int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx\) [25]

Optimal result

Integrand size = 11, antiderivative size = 73 \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2} \]

[Out]

-1/2*arctanh((b*cos(x)-a*sin(x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(3/2)+1/2*(-b*cos(x)+a*sin(x))/(a^2+b^2)/(a*cos(x)
+b*sin(x))^2

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3155, 3153, 212} \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {\text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)-a \sin (x)}{2 \left (a^2+b^2\right ) (a \cos (x)+b \sin (x))^2} \]

[In]

Int[(a*Cos[x] + b*Sin[x])^(-3),x]

[Out]

-1/2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(a^2 + b^2)^(3/2) - (b*Cos[x] - a*Sin[x])/(2*(a^2 + b^2)*(
a*Cos[x] + b*Sin[x])^2)

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 3153

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Dist[-d^(-1), Subst[Int
[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2,
0]

Rule 3155

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {\left (a^2+b^2\right ) (-b \cos (x)+a \sin (x))+2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (x)+b \sin (x))^2}{2 (a-i b)^2 (a+i b)^2 (a \cos (x)+b \sin (x))^2} \]

[In]

Integrate[(a*Cos[x] + b*Sin[x])^(-3),x]

[Out]

((a^2 + b^2)*(-(b*Cos[x]) + a*Sin[x]) + 2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]]*(a*Cos[x]
+ b*Sin[x])^2)/(2*(a - I*b)^2*(a + I*b)^2*(a*Cos[x] + b*Sin[x])^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(65)=130\).

Time = 0.53 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.15

method	result	size
default	\(-\frac {2 \left (-\frac {\left (a^{2}+2 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{3}}{2 \left (a^{2}+b^{2}\right ) a}-\frac {b \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}}{2 \left (a^{2}+b^{2}\right ) a^{2}}-\frac {\left (a^{2}-2 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{2 \left (a^{2}+b^{2}\right ) a}+\frac {b}{2 a^{2}+2 b^{2}}\right )}{\left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\)	\(157\)
risch	\(\frac {{\mathrm e}^{i x} \left (i a \,{\mathrm e}^{2 i x}+b \,{\mathrm e}^{2 i x}-i a +b \right )}{\left (b \,{\mathrm e}^{2 i x}+i a \,{\mathrm e}^{2 i x}-b +i a \right )^{2} \left (-i a +b \right ) \left (i a +b \right )}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left ({\mathrm e}^{i x}-\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\)	\(187\)

[In]

int(1/(a*cos(x)+b*sin(x))^3,x,method=_RETURNVERBOSE)

[Out]

-2*(-1/2*(a^2+2*b^2)/(a^2+b^2)/a*tan(1/2*x)^3-1/2*b*(a^2-2*b^2)/(a^2+b^2)/a^2*tan(1/2*x)^2-1/2*(a^2-2*b^2)/(a^
2+b^2)/a*tan(1/2*x)+1/2*b/(a^2+b^2))/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-a)^2+1/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*ta
n(1/2*x)-2*b)/(a^2+b^2)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (65) = 130\).

Time = 0.26 (sec) , antiderivative size = 225, normalized size of antiderivative = 3.08 \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {{\left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} \sin \left (x\right )}{4 \, {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (x\right ) \sin \left (x\right )\right )}} \]

[In]

integrate(1/(a*cos(x)+b*sin(x))^3,x, algorithm="fricas")

[Out]

1/4*((2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)*sqrt(a^2 + b^2)*log(-(2*a*b*cos(x)*sin(x) + (a^2 - b^2
)*cos(x)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^
2 + b^2)) - 2*(a^2*b + b^3)*cos(x) + 2*(a^3 + a*b^2)*sin(x))/(a^4*b^2 + 2*a^2*b^4 + b^6 + (a^6 + a^4*b^2 - a^2
*b^4 - b^6)*cos(x)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(x)*sin(x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(1/(a*cos(x)+b*sin(x))**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. 250 vs. \(2 (65) = 130\).

Time = 0.33 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.42 \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {a^{2} b - \frac {{\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{a^{6} + a^{4} b^{2} + \frac {4 \, {\left (a^{5} b + a^{3} b^{3}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, {\left (a^{6} - a^{4} b^{2} - 2 \, a^{2} b^{4}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {4 \, {\left (a^{5} b + a^{3} b^{3}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {{\left (a^{6} + a^{4} b^{2}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} - \frac {\log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(a*cos(x)+b*sin(x))^3,x, algorithm="maxima")

[Out]

-(a^2*b - (a^3 - 2*a*b^2)*sin(x)/(cos(x) + 1) - (a^2*b - 2*b^3)*sin(x)^2/(cos(x) + 1)^2 - (a^3 + 2*a*b^2)*sin(
x)^3/(cos(x) + 1)^3)/(a^6 + a^4*b^2 + 4*(a^5*b + a^3*b^3)*sin(x)/(cos(x) + 1) - 2*(a^6 - a^4*b^2 - 2*a^2*b^4)*
sin(x)^2/(cos(x) + 1)^2 - 4*(a^5*b + a^3*b^3)*sin(x)^3/(cos(x) + 1)^3 + (a^6 + a^4*b^2)*sin(x)^4/(cos(x) + 1)^
4) - 1/2*log((b - a*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(b - a*sin(x)/(cos(x) + 1) - sqrt(a^2 + b^2)))/(a^2
+ b^2)^(3/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (65) = 130\).

Time = 0.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.27 \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {\log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{3} + a^{2} b \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b^{3} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{3} \tan \left (\frac {1}{2} \, x\right ) - 2 \, a b^{2} \tan \left (\frac {1}{2} \, x\right ) - a^{2} b}{{\left (a^{4} + a^{2} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )}^{2}} \]

[In]

integrate(1/(a*cos(x)+b*sin(x))^3,x, algorithm="giac")

[Out]

-1/2*log(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*x) - 2*b + 2*sqrt(a^2 + b^2)))/(a^2 + b
^2)^(3/2) + (a^3*tan(1/2*x)^3 + 2*a*b^2*tan(1/2*x)^3 + a^2*b*tan(1/2*x)^2 - 2*b^3*tan(1/2*x)^2 + a^3*tan(1/2*x
) - 2*a*b^2*tan(1/2*x) - a^2*b)/((a^4 + a^2*b^2)*(a*tan(1/2*x)^2 - 2*b*tan(1/2*x) - a)^2)

Mupad [B] (verification not implemented)

Time = 21.69 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.96 \[ \int \frac {1}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (a^2+2\,b^2\right )}{a\,\left (a^2+b^2\right )}-\frac {b}{a^2+b^2}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2-2\,b^2\right )}{a\,\left (a^2+b^2\right )}+\frac {b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (a^2-2\,b^2\right )}{a^2\,\left (a^2+b^2\right )}}{a^2-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2-4\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )-4\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}-\frac {\mathrm {atanh}\left (-\frac {\left (2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )-\frac {2\,a^2\,b+2\,b^3}{a^2+b^2}\right )\,\left (\frac {a^2}{2}+\frac {b^2}{2}\right )}{{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}} \]

[In]

int(1/(a*cos(x) + b*sin(x))^3,x)

[Out]

((tan(x/2)^3*(a^2 + 2*b^2))/(a*(a^2 + b^2)) - b/(a^2 + b^2) + (tan(x/2)*(a^2 - 2*b^2))/(a*(a^2 + b^2)) + (b*ta
n(x/2)^2*(a^2 - 2*b^2))/(a^2*(a^2 + b^2)))/(a^2 - tan(x/2)^2*(2*a^2 - 4*b^2) + a^2*tan(x/2)^4 + 4*a*b*tan(x/2)
- 4*a*b*tan(x/2)^3) - atanh(-((2*a*tan(x/2) - (2*a^2*b + 2*b^3)/(a^2 + b^2))*(a^2/2 + b^2/2))/(a^2 + b^2)^(3/
2))/(a^2 + b^2)^(3/2)