Integrand size = 23, antiderivative size = 59 \[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{2 d}+\frac {\sqrt {a+b \sin ^4(c+d x)}}{2 d} \] Output:
-1/2*a^(1/2)*arctanh((a+b*sin(d*x+c)^4)^(1/2)/a^(1/2))/d+1/2*(a+b*sin(d*x+ c)^4)^(1/2)/d
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.92 \[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\frac {-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )+\sqrt {a+b \sin ^4(c+d x)}}{2 d} \] Input:
Integrate[Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]
Output:
(-(Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]]) + Sqrt[a + b*Sin[c + d*x]^4])/(2*d)
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3708, 243, 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.
\(\displaystyle \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+b \sin (c+d x)^4}}{\tan (c+d x)}dx\) |
\(\Big \downarrow \) 3708 |
\(\displaystyle \frac {\int \csc ^2(c+d x) \sqrt {b \sin ^4(c+d x)+a}d\sin ^2(c+d x)}{2 d}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \csc ^2(c+d x) \sqrt {b \sin ^4(c+d x)+a}d\sin ^4(c+d x)}{4 d}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {a \int \frac {\csc ^2(c+d x)}{\sqrt {b \sin ^4(c+d x)+a}}d\sin ^4(c+d x)+2 \sqrt {a+b \sin ^4(c+d x)}}{4 d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 a \int \frac {1}{\frac {\sqrt {b \sin ^4(c+d x)+a}}{b}-\frac {a}{b}}d\sqrt {b \sin ^4(c+d x)+a}}{b}+2 \sqrt {a+b \sin ^4(c+d x)}}{4 d}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \sqrt {a+b \sin ^4(c+d x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^4(c+d x)}}{\sqrt {a}}\right )}{4 d}\) |
Input:
Int[Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]^4],x]
Output:
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sin[c + d*x]^4]/Sqrt[a]] + 2*Sqrt[a + b*Sin [c + d*x]^4])/(4*d)
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(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^(n/2)*x^(n/2))^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] && IntegerQ[n/2]
Time = 0.53 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\frac {\sqrt {a +b \sin \left (d x +c \right )^{4}}}{2}-\frac {\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \sin \left (d x +c \right )^{4}}}{\sin \left (d x +c \right )^{2}}\right )}{2}}{d}\) | \(60\) |
Input:
int(cot(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/d*(1/2*(a+b*sin(d*x+c)^4)^(1/2)-1/2*a^(1/2)*ln((2*a+2*a^(1/2)*(a+b*sin(d *x+c)^4)^(1/2))/sin(d*x+c)^2))
Time = 0.17 (sec) , antiderivative size = 192, normalized size of antiderivative = 3.25 \[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\left [\frac {\sqrt {a} \log \left (\frac {8 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b} \sqrt {a} + 2 \, a + b\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}{4 \, d}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}\right ) + \sqrt {b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b}}{2 \, d}\right ] \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="fricas")
Output:
[1/4*(sqrt(a)*log(8*(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 - 2*sqrt(b*cos( d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)*sqrt(a) + 2*a + b)/(cos(d*x + c)^ 4 - 2*cos(d*x + c)^2 + 1)) + 2*sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b))/d, 1/2*(sqrt(-a)*arctan(sqrt(-a)/sqrt(b*cos(d*x + c)^4 - 2*b*cos (d*x + c)^2 + a + b)) + sqrt(b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b ))/d]
\[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \sqrt {a + b \sin ^{4}{\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)**4)**(1/2),x)
Output:
Integral(sqrt(a + b*sin(c + d*x)**4)*cot(c + d*x), x)
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15 \[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\frac {\sqrt {a} \log \left (\frac {\sqrt {b \sin \left (d x + c\right )^{4} + a} - \sqrt {a}}{\sqrt {b \sin \left (d x + c\right )^{4} + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b \sin \left (d x + c\right )^{4} + a}}{4 \, d} \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="maxima")
Output:
1/4*(sqrt(a)*log((sqrt(b*sin(d*x + c)^4 + a) - sqrt(a))/(sqrt(b*sin(d*x + c)^4 + a) + sqrt(a))) + 2*sqrt(b*sin(d*x + c)^4 + a))/d
Timed out. \[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \mathrm {cot}\left (c+d\,x\right )\,\sqrt {b\,{\sin \left (c+d\,x\right )}^4+a} \,d x \] Input:
int(cot(c + d*x)*(a + b*sin(c + d*x)^4)^(1/2),x)
Output:
int(cot(c + d*x)*(a + b*sin(c + d*x)^4)^(1/2), x)
\[ \int \cot (c+d x) \sqrt {a+b \sin ^4(c+d x)} \, dx=\int \sqrt {\sin \left (d x +c \right )^{4} b +a}\, \cot \left (d x +c \right )d x \] Input:
int(cot(d*x+c)*(a+b*sin(d*x+c)^4)^(1/2),x)
Output:
int(sqrt(sin(c + d*x)**4*b + a)*cot(c + d*x),x)