\(\int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx\) [375]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f} \]

[Out]

arctan((b*sec(f*x+e))^(1/2)/b^(1/2))*b^(1/2)/f-arctanh((b*sec(f*x+e))^(1/2)/b^(1/2))*b^(1/2)/f

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2702, 335, 304, 209, 212} \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f} \]

[In]

Int[Csc[e + f*x]*Sqrt[b*Sec[e + f*x]],x]

[Out]

(Sqrt[b]*ArcTan[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f - (Sqrt[b]*ArcTanh[Sqrt[b*Sec[e + f*x]]/Sqrt[b]])/f

Rule 209

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

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 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {x}}{-1+\frac {x^2}{b^2}} \, dx,x,b \sec (e+f x)\right )}{b f} \\ & = \frac {2 \text {Subst}\left (\int \frac {x^2}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{b f} \\ & = -\frac {b \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{f}+\frac {b \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sec (e+f x)}\right )}{f} \\ & = \frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b \sec (e+f x)}}{\sqrt {b}}\right )}{f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {\left (2 \arctan \left (\sqrt {\sec (e+f x)}\right )+\log \left (1-\sqrt {\sec (e+f x)}\right )-\log \left (1+\sqrt {\sec (e+f x)}\right )\right ) \sqrt {b \sec (e+f x)}}{2 f \sqrt {\sec (e+f x)}} \]

[In]

Integrate[Csc[e + f*x]*Sqrt[b*Sec[e + f*x]],x]

[Out]

((2*ArcTan[Sqrt[Sec[e + f*x]]] + Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[b*Sec[e + f*x
]])/(2*f*Sqrt[Sec[e + f*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(46)=92\).

Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.60

method	result	size
default	\(\frac {\sqrt {b \sec \left (f x +e \right )}\, \left (\arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\right )-\ln \left (\frac {4 \cos \left (f x +e \right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-2 \cos \left (f x +e \right )+2}{\cos \left (f x +e \right )+1}\right )\right ) \cos \left (f x +e \right )}{2 f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\)	\(151\)

[In]

int(csc(f*x+e)*(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/f*(b*sec(f*x+e))^(1/2)*(arctan(1/2/(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))-ln(2*(2*cos(f*x+e)*(-cos(f*x+e)/(
cos(f*x+e)+1)^2)^(1/2)+2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)+1)/(cos(f*x+e)+1)))*cos(f*x+e)/(cos(f
*x+e)+1)/(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (46) = 92\).

Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 4.26 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\left [\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) + \sqrt {-b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, f}, -\frac {2 \, \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}} {\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {b}}\right ) - \sqrt {b} \log \left (\frac {b \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right )}{4 \, f}\right ] \]

[In]

integrate(csc(f*x+e)*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[1/4*(2*sqrt(-b)*arctan(1/2*sqrt(-b)*sqrt(b/cos(f*x + e))*(cos(f*x + e) + 1)/b) + sqrt(-b)*log((b*cos(f*x + e)
^2 - 4*(cos(f*x + e)^2 - cos(f*x + e))*sqrt(-b)*sqrt(b/cos(f*x + e)) - 6*b*cos(f*x + e) + b)/(cos(f*x + e)^2 +
2*cos(f*x + e) + 1)))/f, -1/4*(2*sqrt(b)*arctan(1/2*sqrt(b/cos(f*x + e))*(cos(f*x + e) - 1)/sqrt(b)) - sqrt(b
)*log((b*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + cos(f*x + e))*sqrt(b)*sqrt(b/cos(f*x + e)) + 6*b*cos(f*x + e) +
b)/(cos(f*x + e)^2 - 2*cos(f*x + e) + 1)))/f]

Sympy [F]

\[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \sqrt {b \sec {\left (e + f x \right )}} \csc {\left (e + f x \right )}\, dx \]

[In]

integrate(csc(f*x+e)*(b*sec(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(b*sec(e + f*x))*csc(e + f*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.24 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {b {\left (\frac {2 \, \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b}}\right )}{\sqrt {b}} + \frac {\log \left (-\frac {\sqrt {b} - \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{\sqrt {b} + \sqrt {\frac {b}{\cos \left (f x + e\right )}}}\right )}{\sqrt {b}}\right )}}{2 \, f} \]

[In]

integrate(csc(f*x+e)*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

1/2*b*(2*arctan(sqrt(b/cos(f*x + e))/sqrt(b))/sqrt(b) + log(-(sqrt(b) - sqrt(b/cos(f*x + e)))/(sqrt(b) + sqrt(
b/cos(f*x + e))))/sqrt(b))/f

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05 \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\frac {b^{2} {\left (\frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {\arctan \left (\frac {\sqrt {b \cos \left (f x + e\right )}}{\sqrt {b}}\right )}{b^{\frac {3}{2}}}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{f} \]

[In]

integrate(csc(f*x+e)*(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

b^2*(arctan(sqrt(b*cos(f*x + e))/sqrt(-b))/(sqrt(-b)*b) - arctan(sqrt(b*cos(f*x + e))/sqrt(b))/b^(3/2))*sgn(co
s(f*x + e))/f

Mupad [F(-1)]

Timed out. \[ \int \csc (e+f x) \sqrt {b \sec (e+f x)} \, dx=\int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{\sin \left (e+f\,x\right )} \,d x \]

[In]

int((b/cos(e + f*x))^(1/2)/sin(e + f*x),x)

[Out]

int((b/cos(e + f*x))^(1/2)/sin(e + f*x), x)