\(\int \csc ^5(a+b x) \, dx\) [5]

Optimal result

Integrand size = 8, antiderivative size = 55 \[ \int \csc ^5(a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b} \]

[Out]

-3/8*arctanh(cos(b*x+a))/b-3/8*cot(b*x+a)*csc(b*x+a)/b-1/4*cot(b*x+a)*csc(b*x+a)^3/b

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3855} \[ \int \csc ^5(a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b} \]

[In]

Int[Csc[a + b*x]^5,x]

[Out]

(-3*ArcTanh[Cos[a + b*x]])/(8*b) - (3*Cot[a + b*x]*Csc[a + b*x])/(8*b) - (Cot[a + b*x]*Csc[a + b*x]^3)/(4*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 ^3(a+b x)}{4 b}+\frac {3}{4} \int \csc ^3(a+b x) \, dx \\ & = -\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac {3}{8} \int \csc (a+b x) \, dx \\ & = -\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(55)=110\).

Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.05 \[ \int \csc ^5(a+b x) \, dx=-\frac {3 \csc ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {3 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \sec ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \]

[In]

Integrate[Csc[a + b*x]^5,x]

[Out]

(-3*Csc[(a + b*x)/2]^2)/(32*b) - Csc[(a + b*x)/2]^4/(64*b) - (3*Log[Cos[(a + b*x)/2]])/(8*b) + (3*Log[Sin[(a +
b*x)/2]])/(8*b) + (3*Sec[(a + b*x)/2]^2)/(32*b) + Sec[(a + b*x)/2]^4/(64*b)

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {\left (-\frac {\csc \left (x b +a \right )^{3}}{4}-\frac {3 \csc \left (x b +a \right )}{8}\right ) \cot \left (x b +a \right )+\frac {3 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{8}}{b}\)	\(50\)
default	\(\frac {\left (-\frac {\csc \left (x b +a \right )^{3}}{4}-\frac {3 \csc \left (x b +a \right )}{8}\right ) \cot \left (x b +a \right )+\frac {3 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{8}}{b}\)	\(50\)
parallelrisch	\(\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+8 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-8 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+24 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{64 b}\)	\(69\)
norman	\(\frac {-\frac {1}{64 b}-\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{64 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}+\frac {3 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{8 b}\)	\(83\)
risch	\(\frac {3 \,{\mathrm e}^{7 i \left (x b +a \right )}-11 \,{\mathrm e}^{5 i \left (x b +a \right )}-11 \,{\mathrm e}^{3 i \left (x b +a \right )}+3 \,{\mathrm e}^{i \left (x b +a \right )}}{4 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{8 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{8 b}\)	\(99\)

[In]

int(csc(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/b*((-1/4*csc(b*x+a)^3-3/8*csc(b*x+a))*cot(b*x+a)+3/8*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. 112 vs. \(2 (49) = 98\).

Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.04 \[ \int \csc ^5(a+b x) \, dx=\frac {6 \, \cos \left (b x + a\right )^{3} - 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 10 \, \cos \left (b x + a\right )}{16 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]

[In]

integrate(csc(b*x+a)^5,x, algorithm="fricas")

[Out]

1/16*(6*cos(b*x + a)^3 - 3*(cos(b*x + a)^4 - 2*cos(b*x + a)^2 + 1)*log(1/2*cos(b*x + a) + 1/2) + 3*(cos(b*x +
a)^4 - 2*cos(b*x + a)^2 + 1)*log(-1/2*cos(b*x + a) + 1/2) - 10*cos(b*x + a))/(b*cos(b*x + a)^4 - 2*b*cos(b*x +
a)^2 + b)

Sympy [F]

\[ \int \csc ^5(a+b x) \, dx=\int \csc ^{5}{\left (a + b x \right )}\, dx \]

[In]

integrate(csc(b*x+a)**5,x)

[Out]

Integral(csc(a + b*x)**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.29 \[ \int \csc ^5(a+b x) \, dx=\frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} - 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \]

[In]

integrate(csc(b*x+a)^5,x, algorithm="maxima")

[Out]

1/16*(2*(3*cos(b*x + a)^3 - 5*cos(b*x + a))/(cos(b*x + a)^4 - 2*cos(b*x + a)^2 + 1) - 3*log(cos(b*x + a) + 1)
+ 3*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. 138 vs. \(2 (49) = 98\).

Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.51 \[ \int \csc ^5(a+b x) \, dx=\frac {\frac {{\left (\frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {18 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 12 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \]

[In]

integrate(csc(b*x+a)^5,x, algorithm="giac")

[Out]

1/64*((8*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 18*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 - 1)*(cos(b*x +
a) + 1)^2/(cos(b*x + a) - 1)^2 - 8*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + (cos(b*x + a) - 1)^2/(cos(b*x + a)
+ 1)^2 + 12*log(abs(-cos(b*x + a) + 1)/abs(cos(b*x + a) + 1)))/b

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07 \[ \int \csc ^5(a+b x) \, dx=-\frac {3\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{8\,b}-\frac {\frac {5\,\cos \left (a+b\,x\right )}{8}-\frac {3\,{\cos \left (a+b\,x\right )}^3}{8}}{b\,\left ({\cos \left (a+b\,x\right )}^4-2\,{\cos \left (a+b\,x\right )}^2+1\right )} \]

[In]

int(1/sin(a + b*x)^5,x)

[Out]

- (3*atanh(cos(a + b*x)))/(8*b) - ((5*cos(a + b*x))/8 - (3*cos(a + b*x)^3)/8)/(b*(cos(a + b*x)^4 - 2*cos(a + b
*x)^2 + 1))