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

Optimal. Leaf size=11 \[ -\frac {\cot (a+b x)}{b} \]

[Out]

-cot(b*x+a)/b

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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852, 8} \begin {gather*} -\frac {\cot (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \csc ^2(a+b x) \, dx &=-\frac {\text {Subst}(\int 1 \, dx,x,\cot (a+b x))}{b}\\ &=-\frac {\cot (a+b x)}{b}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\cot (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^2,x]

[Out]

-(Cot[a + b*x]/b)

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Maple [A]
time = 0.03, size = 12, normalized size = 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	\(-\frac {2 i}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}\)	\(20\)
norman	\(\frac {-\frac {1}{2 b}+\frac {\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )}{2 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cot(b*x+a)/b

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Maxima [A]
time = 0.28, size = 13, normalized size = 1.18 \begin {gather*} -\frac {1}{b \tan \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*tan(b*x + a))

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Fricas [A]
time = 3.03, size = 19, normalized size = 1.73 \begin {gather*} -\frac {\cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(b*x + a)/(b*sin(b*x + a))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \csc ^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(a + b*x)**2, x)

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Giac [A]
time = 0.45, size = 13, normalized size = 1.18 \begin {gather*} -\frac {1}{b \tan \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*tan(b*x + a))

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Mupad [B]
time = 0.10, size = 11, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {cot}\left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(a + b*x)^2,x)

[Out]

-cot(a + b*x)/b

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