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

Optimal. Leaf size=80 \[ \frac{a^2 (e x)^n}{e n}-\frac{2 a b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac{b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n} \]

[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)

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Rubi [A]  time = 0.102105, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {5441, 5437, 3773, 3770, 3767, 8} \[ \frac{a^2 (e x)^n}{e n}-\frac{2 a b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac{b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n} \]

Antiderivative was successfully verified.

[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 5441

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]

Rule 5437

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 3773

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 3770

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

Rule 3767

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (e x)^{-1+n} \left (a+b \text{csch}\left (c+d x^n\right )\right )^2 \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac{\left (i b^2 x^{-n} (e x)^n\right ) \operatorname{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 \tanh ^{-1}\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]  time = 0.367261, size = 87, normalized size = 1.09 \[ \frac{x^{-n} (e x)^n \left (2 a \left (a c+a d x^n+2 b \log \left (\tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )\right )\right )-b^2 \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )+b^2 \left (-\coth \left (\frac{1}{2} \left (c+d x^n\right )\right )\right )\right )}{2 d e n} \]

Antiderivative was successfully verified.

[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[Tanh[(c + d*x^n)/2]]) - b^2*Tanh[(c + d*x^
n)/2]))/(2*d*e*n*x^n)

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Maple [C]  time = 0.115, size = 271, normalized size = 3.4 \begin{align*}{\frac{{a}^{2}x}{n}{{\rm e}^{{\frac{ \left ( -1+n \right ) \left ( -i\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) +i\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}-i \left ({\it csgn} \left ( iex \right ) \right ) ^{3}\pi +2\,\ln \left ( x \right ) +2\,\ln \left ( e \right ) \right ) }{2}}}}}-2\,{\frac{x{b}^{2}{{\rm e}^{1/2\, \left ( -1+n \right ) \left ( -i\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) +i\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}+i\pi \,{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}-i \left ({\it csgn} \left ( iex \right ) \right ) ^{3}\pi +2\,\ln \left ( x \right ) +2\,\ln \left ( e \right ) \right ) }}}{d{x}^{n}n \left ({{\rm e}^{2\,c+2\,d{x}^{n}}}-1 \right ) }}-4\,{\frac{ba{e}^{n}{\it Artanh} \left ({{\rm e}^{c+d{x}^{n}}} \right ){{\rm e}^{-i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -1+n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}}{ned}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.46273, size = 2890, normalized size = 36.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{n - 1} \left (a + b \operatorname{csch}{\left (c + d x^{n} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{csch}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)