\(\int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx\) [61]

Optimal result

Integrand size = 19, antiderivative size = 47 \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {\csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{b} \]

[Out]

-csc(b*x+a)*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*sin(2*b*x+2*a)^
(1/2)*(d*tan(b*x+a))^(1/2)/b

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2681, 2653, 2720} \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\frac {\sqrt {\sin (2 a+2 b x)} \csc (a+b x) \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right ) \sqrt {d \tan (a+b x)}}{b} \]

[In]

Int[Csc[a + b*x]*Sqrt[d*Tan[a + b*x]],x]

[Out]

(Csc[a + b*x]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*a + 2*b*x]]*Sqrt[d*Tan[a + b*x]])/b

Rule 2653

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]

Rule 2681

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[Cos[e + f*x]
^n*((b*Tan[e + f*x])^n/(a*Sin[e + f*x])^n), Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(-1)]) || IntegersQ[m - 1/2, n -
1/2])

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{\sqrt {\sin (a+b x)}} \\ & = \left (\csc (a+b x) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = \frac {\csc (a+b x) \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right ) \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}{b} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.55 \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {2 \sqrt [4]{-1} \cos (a+b x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (a+b x)}\right ),-1\right ) \sqrt {\sec ^2(a+b x)} \sqrt {d \tan (a+b x)}}{b \sqrt {\tan (a+b x)}} \]

[In]

Integrate[Csc[a + b*x]*Sqrt[d*Tan[a + b*x]],x]

[Out]

(-2*(-1)^(1/4)*Cos[a + b*x]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[a + b*x]]], -1]*Sqrt[Sec[a + b*x]^2]*Sqrt[
d*Tan[a + b*x]])/(b*Sqrt[Tan[a + b*x]])

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.30

method	result	size
default	\(\frac {\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}\, \sqrt {-\csc \left (b x +a \right )+1+\cot \left (b x +a \right )}\, \sqrt {\cot \left (b x +a \right )-\csc \left (b x +a \right )}\, F\left (\sqrt {1+\csc \left (b x +a \right )-\cot \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {d \tan \left (b x +a \right )}\, \left (\cot \left (b x +a \right )+\csc \left (b x +a \right )\right ) \sqrt {2}}{b}\)	\(108\)

[In]

int(csc(b*x+a)*(d*tan(b*x+a))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/b*(1+csc(b*x+a)-cot(b*x+a))^(1/2)*(-csc(b*x+a)+1+cot(b*x+a))^(1/2)*(cot(b*x+a)-csc(b*x+a))^(1/2)*EllipticF((
1+csc(b*x+a)-cot(b*x+a))^(1/2),1/2*2^(1/2))*(d*tan(b*x+a))^(1/2)*(cot(b*x+a)+csc(b*x+a))*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.15 \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=-\frac {\sqrt {i \, d} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + \sqrt {-i \, d} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1)}{b} \]

[In]

integrate(csc(b*x+a)*(d*tan(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

-(sqrt(I*d)*elliptic_f(arcsin(cos(b*x + a) + I*sin(b*x + a)), -1) + sqrt(-I*d)*elliptic_f(arcsin(cos(b*x + a)
- I*sin(b*x + a)), -1))/b

Sympy [F]

\[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \sqrt {d \tan {\left (a + b x \right )}} \csc {\left (a + b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)*(d*tan(b*x+a))**(1/2),x)

[Out]

Integral(sqrt(d*tan(a + b*x))*csc(a + b*x), x)

Maxima [F]

\[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \csc \left (b x + a\right ) \,d x } \]

[In]

integrate(csc(b*x+a)*(d*tan(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*tan(b*x + a))*csc(b*x + a), x)

Giac [F]

\[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int { \sqrt {d \tan \left (b x + a\right )} \csc \left (b x + a\right ) \,d x } \]

[In]

integrate(csc(b*x+a)*(d*tan(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*tan(b*x + a))*csc(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int \csc (a+b x) \sqrt {d \tan (a+b x)} \, dx=\int \frac {\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sin \left (a+b\,x\right )} \,d x \]

[In]

int((d*tan(a + b*x))^(1/2)/sin(a + b*x),x)

[Out]

int((d*tan(a + b*x))^(1/2)/sin(a + b*x), x)