3.3 \(\int \csc ^3(a+b x) \, dx\)

Optimal. Leaf size=34 \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\cot (a+b x) \csc (a+b x)}{2 b} \]

[Out]

-ArcTanh[Cos[a + b*x]]/(2*b) - (Cot[a + b*x]*Csc[a + b*x])/(2*b)

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Rubi [A]  time = 0.0138963, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3768, 3770} \[ -\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\cot (a+b x) \csc (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3,x]

[Out]

-ArcTanh[Cos[a + b*x]]/(2*b) - (Cot[a + b*x]*Csc[a + b*x])/(2*b)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc ^3(a+b x) \, dx &=-\frac{\cot (a+b x) \csc (a+b x)}{2 b}+\frac{1}{2} \int \csc (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\cot (a+b x) \csc (a+b x)}{2 b}\\ \end{align*}

Mathematica [B]  time = 0.0129009, size = 75, normalized size = 2.21 \[ -\frac{\csc ^2\left (\frac{1}{2} (a+b x)\right )}{8 b}+\frac{\sec ^2\left (\frac{1}{2} (a+b x)\right )}{8 b}+\frac{\log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{2 b}-\frac{\log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3,x]

[Out]

-Csc[(a + b*x)/2]^2/(8*b) - Log[Cos[(a + b*x)/2]]/(2*b) + Log[Sin[(a + b*x)/2]]/(2*b) + Sec[(a + b*x)/2]^2/(8*
b)

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Maple [A]  time = 0.022, size = 40, normalized size = 1.2 \begin{align*} -{\frac{\csc \left ( bx+a \right ) \cot \left ( bx+a \right ) }{2\,b}}+{\frac{\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^3,x)

[Out]

-1/2*cot(b*x+a)*csc(b*x+a)/b+1/2/b*ln(csc(b*x+a)-cot(b*x+a))

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Maxima [A]  time = 1.04182, size = 62, normalized size = 1.82 \begin{align*} \frac{\frac{2 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3,x, algorithm="maxima")

[Out]

1/4*(2*cos(b*x + a)/(cos(b*x + a)^2 - 1) - log(cos(b*x + a) + 1) + log(cos(b*x + a) - 1))/b

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Fricas [B]  time = 0.484921, size = 201, normalized size = 5.91 \begin{align*} -\frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) -{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (b x + a\right )}{4 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/4*((cos(b*x + a)^2 - 1)*log(1/2*cos(b*x + a) + 1/2) - (cos(b*x + a)^2 - 1)*log(-1/2*cos(b*x + a) + 1/2) - 2
*cos(b*x + a))/(b*cos(b*x + a)^2 - b)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{3}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)**3,x)

[Out]

Integral(csc(a + b*x)**3, x)

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Giac [B]  time = 1.29105, size = 124, normalized size = 3.65 \begin{align*} -\frac{\frac{{\left (\frac{2 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )}{\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3,x, algorithm="giac")

[Out]

-1/8*((2*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)*(cos(b*x + a) + 1)/(cos(b*x + a) - 1) + (cos(b*x + a) - 1)
/(cos(b*x + a) + 1) - 2*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)))/b