\(\int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx\) [493]

Optimal result

Integrand size = 25, antiderivative size = 165 \[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=-\frac {\left (8 a^2-8 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{3/2} f}+\frac {\left (8 a^2-8 a b-b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac {(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f} \]

[Out]

-1/8*(8*a^2-8*a*b-b^2)*arctanh((a+b*sin(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/f+1/8*(8*a+b)*csc(f*x+e)^2*(a+b*sin(f
*x+e)^2)^(3/2)/a^2/f-1/4*csc(f*x+e)^4*(a+b*sin(f*x+e)^2)^(3/2)/a/f+1/8*(8*a^2-8*a*b-b^2)*(a+b*sin(f*x+e)^2)^(1
/2)/a^2/f

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3273, 91, 79, 52, 65, 214} \[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\left (8 a^2-8 a b-b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac {(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac {\left (8 a^2-8 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{3/2} f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f} \]

[In]

Int[Cot[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

-1/8*((8*a^2 - 8*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]])/(a^(3/2)*f) + ((8*a^2 - 8*a*b - b^2)*
Sqrt[a + b*Sin[e + f*x]^2])/(8*a^2*f) + ((8*a + b)*Csc[e + f*x]^2*(a + b*Sin[e + f*x]^2)^(3/2))/(8*a^2*f) - (C
sc[e + f*x]^4*(a + b*Sin[e + f*x]^2)^(3/2))/(4*a*f)

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 214

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

Rule 3273

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m
+ 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(1-x)^2 \sqrt {a+b x}}{x^3} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = -\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} (-8 a-b)+2 a x\right ) \sqrt {a+b x}}{x^2} \, dx,x,\sin ^2(e+f x)\right )}{4 a f} \\ & = \frac {(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac {\left (8 a^2-8 a b-b^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{16 a^2 f} \\ & = \frac {\left (8 a^2-8 a b-b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac {(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac {\left (8 a^2-8 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{16 a f} \\ & = \frac {\left (8 a^2-8 a b-b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac {(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f}+\frac {\left (8 a^2-8 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{8 a b f} \\ & = -\frac {\left (8 a^2-8 a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{8 a^{3/2} f}+\frac {\left (8 a^2-8 a b-b^2\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a^2 f}+\frac {(8 a+b) \csc ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{8 a^2 f}-\frac {\csc ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.62 \[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\left (-8 a^2+8 a b+b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (8 a+(8 a-b) \csc ^2(e+f x)-2 a \csc ^4(e+f x)\right ) \sqrt {a+b \sin ^2(e+f x)}}{8 a^{3/2} f} \]

[In]

Integrate[Cot[e + f*x]^5*Sqrt[a + b*Sin[e + f*x]^2],x]

[Out]

((-8*a^2 + 8*a*b + b^2)*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]] + Sqrt[a]*(8*a + (8*a - b)*Csc[e + f*x]^2
- 2*a*Csc[e + f*x]^4)*Sqrt[a + b*Sin[e + f*x]^2])/(8*a^(3/2)*f)

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.28

method	result	size
default	\(\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}-\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{4 \sin \left (f x +e \right )^{4}}-\frac {b \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 a \sin \left (f x +e \right )^{2}}+\frac {b^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{8 a^{\frac {3}{2}}}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{\sqrt {a}}+\frac {\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )^{2}}}{f}\)	\(212\)

[In]

int(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((a+b*sin(f*x+e)^2)^(1/2)-1/4/sin(f*x+e)^4*(a+b*sin(f*x+e)^2)^(1/2)-1/8/a*b/sin(f*x+e)^2*(a+b*sin(f*x+e)^2)^(1
/2)+1/8/a^(3/2)*b^2*ln((2*a+2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/2))/sin(f*x+e))-a^(1/2)*ln((2*a+2*a^(1/2)*(a+b*sin
(f*x+e)^2)^(1/2))/sin(f*x+e))+1/a^(1/2)*b*ln((2*a+2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/2))/sin(f*x+e))+1/sin(f*x+e)
^2*(a+b*sin(f*x+e)^2)^(1/2))/f

Fricas [A] (verification not implemented)

none

Time = 1.29 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.52 \[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\left [-\frac {{\left ({\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} - 8 \, a b - b^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (8 \, a^{2} \cos \left (f x + e\right )^{4} - {\left (24 \, a^{2} - a b\right )} \cos \left (f x + e\right )^{2} + 14 \, a^{2} - a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{16 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f\right )}}, \frac {{\left ({\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (8 \, a^{2} - 8 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, a^{2} - 8 \, a b - b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) + {\left (8 \, a^{2} \cos \left (f x + e\right )^{4} - {\left (24 \, a^{2} - a b\right )} \cos \left (f x + e\right )^{2} + 14 \, a^{2} - a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{8 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f\right )}}\right ] \]

[In]

integrate(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/16*(((8*a^2 - 8*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 - 8*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 - 8*a*b - b^2)*
sqrt(a)*log(2*(b*cos(f*x + e)^2 - 2*sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(a) - 2*a - b)/(cos(f*x + e)^2 - 1)) -
2*(8*a^2*cos(f*x + e)^4 - (24*a^2 - a*b)*cos(f*x + e)^2 + 14*a^2 - a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2
*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), 1/8*(((8*a^2 - 8*a*b - b^2)*cos(f*x + e)^4 - 2*(8*a^2 - 8
*a*b - b^2)*cos(f*x + e)^2 + 8*a^2 - 8*a*b - b^2)*sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/a)
+ (8*a^2*cos(f*x + e)^4 - (24*a^2 - a*b)*cos(f*x + e)^2 + 14*a^2 - a*b)*sqrt(-b*cos(f*x + e)^2 + a + b))/(a^2*
f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f)]

Sympy [F]

\[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \cot ^{5}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**5*(a+b*sin(f*x+e)**2)**(1/2),x)

[Out]

Integral(sqrt(a + b*sin(e + f*x)**2)*cot(e + f*x)**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.30 \[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=-\frac {8 \, \sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - \frac {8 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{\sqrt {a}} - \frac {b^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} - 8 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} + \frac {8 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} b}{a} + \frac {\sqrt {b \sin \left (f x + e\right )^{2} + a} b^{2}}{a^{2}} - \frac {8 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right )^{2}} - \frac {{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b}{a^{2} \sin \left (f x + e\right )^{2}} + \frac {2 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right )^{4}}}{8 \, f} \]

[In]

integrate(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/8*(8*sqrt(a)*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e)))) - 8*b*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e))))/sqrt(a
) - b^2*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e))))/a^(3/2) - 8*sqrt(b*sin(f*x + e)^2 + a) + 8*sqrt(b*sin(f*x + e
)^2 + a)*b/a + sqrt(b*sin(f*x + e)^2 + a)*b^2/a^2 - 8*(b*sin(f*x + e)^2 + a)^(3/2)/(a*sin(f*x + e)^2) - (b*sin
(f*x + e)^2 + a)^(3/2)*b/(a^2*sin(f*x + e)^2) + 2*(b*sin(f*x + e)^2 + a)^(3/2)/(a*sin(f*x + e)^4))/f

Giac [F(-1)]

Timed out. \[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\text {Timed out} \]

[In]

integrate(cot(f*x+e)^5*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \cot ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^5\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x \]

[In]

int(cot(e + f*x)^5*(a + b*sin(e + f*x)^2)^(1/2),x)

[Out]

int(cot(e + f*x)^5*(a + b*sin(e + f*x)^2)^(1/2), x)