\(\int \csc ^2(a+b x) \, dx\) [2]

Optimal result

Integrand size = 8, antiderivative size = 11 \[ \int \csc ^2(a+b x) \, dx=-\frac {\cot (a+b x)}{b} \]

[Out]

-cot(b*x+a)/b

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852, 8} \[ \int \csc ^2(a+b x) \, dx=-\frac {\cot (a+b x)}{b} \]

[In]

Int[Csc[a + b*x]^2,x]

[Out]

-(Cot[a + b*x]/b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}(\int 1 \, dx,x,\cot (a+b x))}{b} \\ & = -\frac {\cot (a+b x)}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \csc ^2(a+b x) \, dx=-\frac {\cot (a+b x)}{b} \]

[In]

Integrate[Csc[a + b*x]^2,x]

[Out]

-(Cot[a + b*x]/b)

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(-\frac {\cot \left (x b +a \right )}{b}\)	\(12\)
default	\(-\frac {\cot \left (x b +a \right )}{b}\)	\(12\)
risch	\(-\frac {2 i}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}\)	\(20\)
parallelrisch	\(\frac {-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{2 b}\)	\(27\)
norman	\(\frac {-\frac {1}{2 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\)	\(35\)

[In]

int(csc(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-cot(b*x+a)/b

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \csc ^2(a+b x) \, dx=-\frac {\cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \]

[In]

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

[Out]

-cos(b*x + a)/(b*sin(b*x + a))

Sympy [F]

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

[In]

integrate(csc(b*x+a)**2,x)

[Out]

Integral(csc(a + b*x)**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \csc ^2(a+b x) \, dx=-\frac {1}{b \tan \left (b x + a\right )} \]

[In]

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

[Out]

-1/(b*tan(b*x + a))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \csc ^2(a+b x) \, dx=-\frac {1}{b \tan \left (b x + a\right )} \]

[In]

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

[Out]

-1/(b*tan(b*x + a))

Mupad [B] (verification not implemented)

Time = 21.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \csc ^2(a+b x) \, dx=-\frac {\mathrm {cot}\left (a+b\,x\right )}{b} \]

[In]

int(1/sin(a + b*x)^2,x)

[Out]

-cot(a + b*x)/b