Integrand size = 23, antiderivative size = 54 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a+b \sin ^2(e+f x)}}{f} \] Output:
-a^(1/2)*arctanh((a+b*sin(f*x+e)^2)^(1/2)/a^(1/2))/f+(a+b*sin(f*x+e)^2)^(1 /2)/f
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.94 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a+b \sin ^2(e+f x)}}{f} \] Input:
Integrate[Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]
Output:
(-(Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]]) + Sqrt[a + b*Sin[e + f*x]^2])/f
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3673, 60, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)^2}}{\tan (e+f x)}dx\) |
\(\Big \downarrow \) 3673 |
\(\displaystyle \frac {\int \csc ^2(e+f x) \sqrt {b \sin ^2(e+f x)+a}d\sin ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {a \int \frac {\csc ^2(e+f x)}{\sqrt {b \sin ^2(e+f x)+a}}d\sin ^2(e+f x)+2 \sqrt {a+b \sin ^2(e+f x)}}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 a \int \frac {1}{\frac {\sin ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \sin ^2(e+f x)+a}}{b}+2 \sqrt {a+b \sin ^2(e+f x)}}{2 f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \sqrt {a+b \sin ^2(e+f x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}\) |
Input:
Int[Cot[e + f*x]*Sqrt[a + b*Sin[e + f*x]^2],x]
Output:
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[a]] + 2*Sqrt[a + b*Sin [e + f*x]^2])/(2*f)
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[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] && Integ erQ[(m - 1)/2]
Time = 0.47 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {\sqrt {a +b \sin \left (f x +e \right )^{2}}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \sin \left (f x +e \right )^{2}}}{\sin \left (f x +e \right )}\right )}{f}\) | \(58\) |
Input:
int(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
((a+b*sin(f*x+e)^2)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(a+b*sin(f*x+e)^2)^(1/ 2))/sin(f*x+e)))/f
Time = 0.47 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.85 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{2 \, f}, -\frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{b \cos \left (f x + e\right )^{2} - a - b}\right ) - \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{f}\right ] \] Input:
integrate(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")
Output:
[1/2*(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*sqrt(-b*cos(f*x + e)^2 + a + b))/f, -(sqrt(-a)*arctan(sqrt(-b*cos(f*x + e)^2 + a + b)*sqrt(-a)/(b*cos(f *x + e)^2 - a - b)) - sqrt(-b*cos(f*x + e)^2 + a + b))/f]
\[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \cot {\left (e + f x \right )}\, dx \] Input:
integrate(cot(f*x+e)*(a+b*sin(f*x+e)**2)**(1/2),x)
Output:
Integral(sqrt(a + b*sin(e + f*x)**2)*cot(e + f*x), x)
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=-\frac {\sqrt {a} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - \sqrt {b \sin \left (f x + e\right )^{2} + a}}{f} \] Input:
integrate(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")
Output:
-(sqrt(a)*arcsinh(a/(sqrt(a*b)*abs(sin(f*x + e)))) - sqrt(b*sin(f*x + e)^2 + a))/f
Timed out. \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\text {Timed out} \] Input:
integrate(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \mathrm {cot}\left (e+f\,x\right )\,\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a} \,d x \] Input:
int(cot(e + f*x)*(a + b*sin(e + f*x)^2)^(1/2),x)
Output:
int(cot(e + f*x)*(a + b*sin(e + f*x)^2)^(1/2), x)
\[ \int \cot (e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {\sin \left (f x +e \right )^{2} b +a}\, \cot \left (f x +e \right )d x \] Input:
int(cot(f*x+e)*(a+b*sin(f*x+e)^2)^(1/2),x)
Output:
int(sqrt(sin(e + f*x)**2*b + a)*cot(e + f*x),x)