\(\int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\) [257]

Optimal result

Integrand size = 21, antiderivative size = 51 \[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 \sqrt {a+a \sin (c+d x)}}{d} \]

[Out]

-2*arctanh((a+a*sin(d*x+c))^(1/2)/a^(1/2))*a^(1/2)/d+2*(a+a*sin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2786, 52, 65, 213} \[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {2 \sqrt {a \sin (c+d x)+a}}{d}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {a}}\right )}{d} \]

[In]

Int[Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-2*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/Sqrt[a]])/d + (2*Sqrt[a + a*Sin[c + d*x]])/d

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2786

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.94 \[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {2 \left (-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {a}}\right )+\sqrt {a (1+\sin (c+d x))}\right )}{d} \]

[In]

Integrate[Cot[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(2*(-(Sqrt[a]*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/Sqrt[a]]) + Sqrt[a*(1 + Sin[c + d*x])]))/d

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(43)=86\).

Time = 0.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.61

method	result	size
derivativedivides	\(-\frac {\sqrt {\frac {a \left (\csc \left (d x +c \right )+1\right )}{\csc \left (d x +c \right )}}\, \left (\ln \left (\frac {1}{2}+\csc \left (d x +c \right )+\sqrt {\csc ^{2}\left (d x +c \right )+\csc \left (d x +c \right )}\right ) \left (\csc ^{2}\left (d x +c \right )\right )-2 \left (\csc ^{2}\left (d x +c \right )+\csc \left (d x +c \right )\right )^{\frac {3}{2}}+2 \sqrt {\csc ^{2}\left (d x +c \right )+\csc \left (d x +c \right )}\, \left (\csc ^{2}\left (d x +c \right )\right )\right )}{d \csc \left (d x +c \right ) \sqrt {\csc \left (d x +c \right ) \left (\csc \left (d x +c \right )+1\right )}}\)	\(133\)
default	\(-\frac {\sqrt {\frac {a \left (\csc \left (d x +c \right )+1\right )}{\csc \left (d x +c \right )}}\, \left (\ln \left (\frac {1}{2}+\csc \left (d x +c \right )+\sqrt {\csc ^{2}\left (d x +c \right )+\csc \left (d x +c \right )}\right ) \left (\csc ^{2}\left (d x +c \right )\right )-2 \left (\csc ^{2}\left (d x +c \right )+\csc \left (d x +c \right )\right )^{\frac {3}{2}}+2 \sqrt {\csc ^{2}\left (d x +c \right )+\csc \left (d x +c \right )}\, \left (\csc ^{2}\left (d x +c \right )\right )\right )}{d \csc \left (d x +c \right ) \sqrt {\csc \left (d x +c \right ) \left (\csc \left (d x +c \right )+1\right )}}\)	\(133\)

[In]

int(cos(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/d*(a*(csc(d*x+c)+1)/csc(d*x+c))^(1/2)/csc(d*x+c)*(ln(1/2+csc(d*x+c)+(csc(d*x+c)^2+csc(d*x+c))^(1/2))*csc(d*
x+c)^2-2*(csc(d*x+c)^2+csc(d*x+c))^(3/2)+2*(csc(d*x+c)^2+csc(d*x+c))^(1/2)*csc(d*x+c)^2)/(csc(d*x+c)*(csc(d*x+
c)+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.69 \[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{2} + 4 \, \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\sin \left (d x + c\right ) + 2\right )} - 8 \, a \sin \left (d x + c\right ) - 9 \, a}{\cos \left (d x + c\right )^{2} - 1}\right ) + 4 \, \sqrt {a \sin \left (d x + c\right ) + a}}{2 \, d} \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/2*(sqrt(a)*log((a*cos(d*x + c)^2 + 4*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(sin(d*x + c) + 2) - 8*a*sin(d*x + c)
- 9*a)/(cos(d*x + c)^2 - 1)) + 4*sqrt(a*sin(d*x + c) + a))/d

Sympy [F]

\[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a*(sin(c + d*x) + 1))*cos(c + d*x)*csc(c + d*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20 \[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {a} \log \left (\frac {\sqrt {a \sin \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {a \sin \left (d x + c\right ) + a} + \sqrt {a}}\right ) + 2 \, \sqrt {a \sin \left (d x + c\right ) + a}}{d} \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

(sqrt(a)*log((sqrt(a*sin(d*x + c) + a) - sqrt(a))/(sqrt(a*sin(d*x + c) + a) + sqrt(a))) + 2*sqrt(a*sin(d*x + c
) + a))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (43) = 86\).

Time = 0.47 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.75 \[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) + 4 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{2 \, d} \]

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*(sqrt(2)*log(abs(-2*sqrt(2) + 4*cos(-1/4*pi + 1/2*d*x + 1/2*c))/abs(2*sqrt(2) + 4*cos(-1/4*pi + 1/
2*d*x + 1/2*c))) + 4*cos(-1/4*pi + 1/2*d*x + 1/2*c))*sqrt(a)*sgn(cos(-1/4*pi + 1/2*d*x + 1/2*c))/d

Mupad [B] (verification not implemented)

Time = 10.58 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {2\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{d}-\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\sqrt {a}}\right )}{d} \]

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x))^(1/2))/sin(c + d*x),x)

[Out]

(2*(a + a*sin(c + d*x))^(1/2))/d - (2*a^(1/2)*atanh((a + a*sin(c + d*x))^(1/2)/a^(1/2)))/d