\(\int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\) [86]

Optimal result

Integrand size = 21, antiderivative size = 20 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]

[Out]

-2/3*d/b/(d*tan(b*x+a))^(3/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2671, 30} \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]

[In]

Int[Csc[a + b*x]^2/Sqrt[d*Tan[a + b*x]],x]

[Out]

(-2*d)/(3*b*(d*Tan[a + b*x])^(3/2))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2671

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[b*(ff/f), Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff
)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {d \text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,d \tan (a+b x)\right )}{b} \\ & = -\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2 d}{3 b (d \tan (a+b x))^{3/2}} \]

[In]

Integrate[Csc[a + b*x]^2/Sqrt[d*Tan[a + b*x]],x]

[Out]

(-2*d)/(3*b*(d*Tan[a + b*x])^(3/2))

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(-\frac {2 d}{3 b \left (d \tan \left (b x +a \right )\right )^{\frac {3}{2}}}\)	\(17\)
default	\(-\frac {2 d}{3 b \left (d \tan \left (b x +a \right )\right )^{\frac {3}{2}}}\)	\(17\)

[In]

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

[Out]

-2/3*d/b/(d*tan(b*x+a))^(3/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (16) = 32\).

Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.30 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\frac {2 \, \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, {\left (b d \cos \left (b x + a\right )^{2} - b d\right )}} \]

[In]

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

[Out]

2/3*sqrt(d*sin(b*x + a)/cos(b*x + a))*cos(b*x + a)^2/(b*d*cos(b*x + a)^2 - b*d)

Sympy [F]

\[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )}}{\sqrt {d \tan {\left (a + b x \right )}}}\, dx \]

[In]

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

[Out]

Integral(csc(a + b*x)**2/sqrt(d*tan(a + b*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \]

[In]

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

[Out]

-2/3/(sqrt(d*tan(b*x + a))*b*tan(b*x + a))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2}{3 \, \sqrt {d \tan \left (b x + a\right )} b \tan \left (b x + a\right )} \]

[In]

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

[Out]

-2/3/(sqrt(d*tan(b*x + a))*b*tan(b*x + a))

Mupad [B] (verification not implemented)

Time = 3.43 (sec) , antiderivative size = 102, normalized size of antiderivative = 5.10 \[ \int \frac {\csc ^2(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx=-\frac {2\,\sqrt {\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{\cos \left (2\,a+2\,b\,x\right )+1}}\,\left (\cos \left (2\,a+2\,b\,x\right )+2\,\cos \left (4\,a+4\,b\,x\right )-\cos \left (6\,a+6\,b\,x\right )-2\right )}{3\,b\,d\,\left (15\,\cos \left (2\,a+2\,b\,x\right )-6\,\cos \left (4\,a+4\,b\,x\right )+\cos \left (6\,a+6\,b\,x\right )-10\right )} \]

[In]

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

[Out]

-(2*((d*sin(2*a + 2*b*x))/(cos(2*a + 2*b*x) + 1))^(1/2)*(cos(2*a + 2*b*x) + 2*cos(4*a + 4*b*x) - cos(6*a + 6*b
*x) - 2))/(3*b*d*(15*cos(2*a + 2*b*x) - 6*cos(4*a + 4*b*x) + cos(6*a + 6*b*x) - 10))