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

Optimal result

Integrand size = 8, antiderivative size = 34 \[ \int \csc ^3(a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, 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 = {3853, 3855} \[ \int \csc ^3(a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]

[In]

Int[Csc[a + b*x]^3,x]

[Out]

-1/2*ArcTanh[Cos[a + b*x]]/b - (Cot[a + b*x]*Csc[a + b*x])/(2*b)

Rule 3853

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 3855

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (a+b x) \csc (a+b x)}{2 b}+\frac {1}{2} \int \csc (a+b x) \, dx \\ & = -\frac {\text {arctanh}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(34)=68\).

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.21 \[ \int \csc ^3(a+b x) \, dx=-\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b} \]

[In]

Integrate[Csc[a + b*x]^3,x]

[Out]

-1/8*Csc[(a + b*x)/2]^2/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)

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\frac {-\frac {\csc \left (x b +a \right ) \cot \left (x b +a \right )}{2}+\frac {\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{2}}{b}\)	\(38\)
default	\(\frac {-\frac {\csc \left (x b +a \right ) \cot \left (x b +a \right )}{2}+\frac {\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{2}}{b}\)	\(38\)
parallelrisch	\(\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+4 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{8 b}\)	\(43\)
norman	\(\frac {-\frac {1}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{2 b}\)	\(51\)
risch	\(\frac {{\mathrm e}^{3 i \left (x b +a \right )}+{\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{2 b}\)	\(72\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (30) = 60\).

Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.12 \[ \int \csc ^3(a+b x) \, dx=-\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (b x + a\right )}{4 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]

[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)

Sympy [F]

\[ \int \csc ^3(a+b x) \, dx=\int \csc ^{3}{\left (a + b x \right )}\, dx \]

[In]

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

[Out]

Integral(csc(a + b*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \csc ^3(a+b x) \, dx=\frac {\frac {2 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{4 \, b} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (30) = 60\).

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.71 \[ \int \csc ^3(a+b x) \, dx=-\frac {\frac {{\left (\frac {2 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{8 \, b} \]

[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

Mupad [B] (verification not implemented)

Time = 21.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \csc ^3(a+b x) \, dx=\frac {\cos \left (a+b\,x\right )}{2\,b\,\left ({\cos \left (a+b\,x\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2\,b} \]

[In]

int(1/sin(a + b*x)^3,x)

[Out]

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