\(\int (e x)^{-1+n} (a+b \text {csch}(c+d x^n))^2 \, dx\) [75]

Optimal result

Integrand size = 22, antiderivative size = 80 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n} \]

[Out]

a^2*(e*x)^n/e/n-2*a*b*(e*x)^n*arctanh(cosh(c+d*x^n))/d/e/n/(x^n)-b^2*(e*x)^n*coth(c+d*x^n)/d/e/n/(x^n)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5549, 5545, 3858, 3855, 3852, 8} \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n} \]

[In]

Int[(e*x)^(-1 + n)*(a + b*Csch[c + d*x^n])^2,x]

[Out]

(a^2*(e*x)^n)/(e*n) - (2*a*b*(e*x)^n*ArcTanh[Cosh[c + d*x^n]])/(d*e*n*x^n) - (b^2*(e*x)^n*Coth[c + d*x^n])/(d*
e*n*x^n)

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]

Rule 3855

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

Rule 3858

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Dist[2*a*b, Int[Csc[c + d*x], x],
x] + Dist[b^2, Int[Csc[c + d*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x]

Rule 5545

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 5549

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x_Symbol] :> Dist[e^IntPart[m]*((e*
x)^FracPart[m]/x^FracPart[m]), Int[x^m*(a + b*Csch[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx}{e} \\ & = \frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int (a+b \text {csch}(c+d x))^2 \, dx,x,x^n\right )}{e n} \\ & = \frac {a^2 (e x)^n}{e n}+\frac {\left (2 a b x^{-n} (e x)^n\right ) \text {Subst}\left (\int \text {csch}(c+d x) \, dx,x,x^n\right )}{e n}+\frac {\left (b^2 x^{-n} (e x)^n\right ) \text {Subst}\left (\int \text {csch}^2(c+d x) \, dx,x,x^n\right )}{e n} \\ & = \frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {\left (i b^2 x^{-n} (e x)^n\right ) \text {Subst}\left (\int 1 \, dx,x,-i \coth \left (c+d x^n\right )\right )}{d e n} \\ & = \frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.29 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=\frac {x^{-n} (e x)^n \left (-b^2 \coth \left (\frac {1}{2} \left (c+d x^n\right )\right )+2 a \left (a c+a d x^n-2 b \log \left (\cosh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )+2 b \log \left (\sinh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )-b^2 \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{2 d e n} \]

[In]

Integrate[(e*x)^(-1 + n)*(a + b*Csch[c + d*x^n])^2,x]

[Out]

((e*x)^n*(-(b^2*Coth[(c + d*x^n)/2]) + 2*a*(a*c + a*d*x^n - 2*b*Log[Cosh[(c + d*x^n)/2]] + 2*b*Log[Sinh[(c + d
*x^n)/2]]) - b^2*Tanh[(c + d*x^n)/2]))/(2*d*e*n*x^n)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.90 (sec) , antiderivative size = 271, normalized size of antiderivative = 3.39

method	result	size
risch	\(\frac {a^{2} x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{n}-\frac {2 x \,x^{-n} {\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}} b^{2}}{d n \left ({\mathrm e}^{2 c +2 d \,x^{n}}-1\right )}-\frac {4 \,\operatorname {arctanh}\left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} a b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{d e n}\)	\(271\)

[In]

int((e*x)^(-1+n)*(a+b*csch(c+d*x^n))^2,x,method=_RETURNVERBOSE)

[Out]

a^2/n*x*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*csgn(I*e)*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(I*
e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln(e)+2*ln(x)))-2/d/n*x/(x^n)*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x
)*Pi+I*csgn(I*e)*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln(e)+2*ln(x)))*b^2/(exp(2
*c+2*d*x^n)-1)-4*arctanh(exp(c+d*x^n))/d/e*e^n/n*a*b*exp(1/2*I*Pi*csgn(I*e*x)*(-1+n)*(csgn(I*e*x)-csgn(I*x))*(
-csgn(I*e*x)+csgn(I*e)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (80) = 160\).

Time = 0.28 (sec) , antiderivative size = 854, normalized size of antiderivative = 10.68 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((e*x)^(-1+n)*(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")

[Out]

-(a^2*d*cosh((n - 1)*log(e))*cosh(n*log(x)) - (a^2*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + a^2*d*cosh(n*log(x)
)*sinh((n - 1)*log(e)) + (a^2*d*cosh((n - 1)*log(e)) + a^2*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh
(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*b^2*cosh((n - 1)*log(e)) - 2*(a^2*d*cosh((n - 1)*log(e))*cosh(n*log(x
)) + a^2*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + (a^2*d*cosh((n - 1)*log(e)) + a^2*d*sinh((n - 1)*log(e)))*sin
h(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^
2*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + a^2*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + (a^2*d*cosh((n - 1)*log(
e)) + a^2*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((a*b*co
sh((n - 1)*log(e)) + a*b*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*cosh((n -
1)*log(e)) + 2*(a*b*cosh((n - 1)*log(e)) + a*b*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x))
+ c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*b*cosh((n - 1)*log(e)) + a*b*sinh((n - 1)*log(e)))*si
nh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*sinh((n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n
*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) - 2*((a*b*cosh((n - 1)*log(e)) + a*b*sinh((
n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*cosh((n - 1)*log(e)) + 2*(a*b*cosh((n -
1)*log(e)) + a*b*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d
*sinh(n*log(x)) + c) + (a*b*cosh((n - 1)*log(e)) + a*b*sinh((n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*
log(x)) + c)^2 - a*b*sinh((n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*l
og(x)) + d*sinh(n*log(x)) + c) - 1) + (a^2*d*cosh(n*log(x)) + 2*b^2)*sinh((n - 1)*log(e)) + (a^2*d*cosh((n - 1
)*log(e)) + a^2*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))/(d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 +
2*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + d*n*sinh(
d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - d*n)

Sympy [F]

\[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=\int \left (e x\right )^{n - 1} \left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}\, dx \]

[In]

integrate((e*x)**(-1+n)*(a+b*csch(c+d*x**n))**2,x)

[Out]

Integral((e*x)**(n - 1)*(a + b*csch(c + d*x**n))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=-2 \, a b {\left (\frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{d n} - \frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{d n}\right )} - \frac {2 \, b^{2} e^{n}}{d e n e^{\left (2 \, d x^{n} + 2 \, c\right )} - d e n} + \frac {\left (e x\right )^{n} a^{2}}{e n} \]

[In]

integrate((e*x)^(-1+n)*(a+b*csch(c+d*x^n))^2,x, algorithm="maxima")

[Out]

-2*a*b*(e^(n - 1)*log((e^(d*x^n + c) + 1)*e^(-c))/(d*n) - e^(n - 1)*log((e^(d*x^n + c) - 1)*e^(-c))/(d*n)) - 2
*b^2*e^n/(d*e*n*e^(2*d*x^n + 2*c) - d*e*n) + (e*x)^n*a^2/(e*n)

Giac [F]

\[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{n - 1} \,d x } \]

[In]

integrate((e*x)^(-1+n)*(a+b*csch(c+d*x^n))^2,x, algorithm="giac")

[Out]

integrate((b*csch(d*x^n + c) + a)^2*(e*x)^(n - 1), x)

Mupad [B] (verification not implemented)

Time = 2.32 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.00 \[ \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx=\frac {a^2\,x\,{\left (e\,x\right )}^{n-1}}{n}-\frac {4\,\mathrm {atan}\left (\frac {a\,b\,x\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,{\left (e\,x\right )}^{n-1}\,\sqrt {-d^2\,n^2\,x^{2\,n}}}{d\,n\,x^n\,\sqrt {a^2\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {a^2\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {-d^2\,n^2\,x^{2\,n}}}-\frac {2\,b^2\,x\,{\left (e\,x\right )}^{n-1}}{d\,n\,x^n\,\left ({\mathrm {e}}^{2\,c+2\,d\,x^n}-1\right )} \]

[In]

int((a + b/sinh(c + d*x^n))^2*(e*x)^(n - 1),x)

[Out]

(a^2*x*(e*x)^(n - 1))/n - (4*atan((a*b*x*exp(d*x^n)*exp(c)*(e*x)^(n - 1)*(-d^2*n^2*x^(2*n))^(1/2))/(d*n*x^n*(a
^2*b^2*x^2*(e*x)^(2*n - 2))^(1/2)))*(a^2*b^2*x^2*(e*x)^(2*n - 2))^(1/2))/(-d^2*n^2*x^(2*n))^(1/2) - (2*b^2*x*(
e*x)^(n - 1))/(d*n*x^n*(exp(2*c + 2*d*x^n) - 1))