3.1.72 \(\int (e x)^{-1+n} (a+b \text {csch}(c+d x^n)) \, dx\) [72]

Optimal. Leaf size=45 \[ \frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 5549, 5545, 3855} \begin {gather*} \frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{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*} \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+n}+b (e x)^{-1+n} \text {csch}\left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^n}{e n}+b \int (e x)^{-1+n} \text {csch}\left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \int x^{-1+n} \text {csch}\left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^n}{e n}+\frac {\left (b x^{-n} (e x)^n\right ) \text {Subst}\left (\int \text {csch}(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^n}{e n}-\frac {b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {x^{-n} (e x)^n \left (a \left (c+d x^n\right )+b \log \left (\tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )}{d e n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.81, size = 155, normalized size = 3.44

method	result	size
risch	\(\frac {a x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{n}-\frac {2 \arctanh \left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} b \,{\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d e n}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/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*arctanh(exp(c+d*x^n))/d/e*e^n/n*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)))

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Maxima [A]
time = 0.29, size = 75, normalized size = 1.67 \begin {gather*} -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 {\left (e x\right )^{n} a}{e n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)) + (e*x)
^n*a/(e*n)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (45) = 90\).
time = 0.40, size = 153, normalized size = 3.40 \begin {gather*} \frac {{\left (a d \cosh \left (n - 1\right ) + a d \sinh \left (n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) - {\left (b \cosh \left (n - 1\right ) + b \sinh \left (n - 1\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 1\right ) + {\left (b \cosh \left (n - 1\right ) + b \sinh \left (n - 1\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - 1\right ) + {\left (a d \cosh \left (n - 1\right ) + a d \sinh \left (n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )}{d n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*d*cosh(n - 1) + a*d*sinh(n - 1))*cosh(n*log(x)) - (b*cosh(n - 1) + b*sinh(n - 1))*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) + (b*cosh(n - 1) + b*sinh(n - 1
))*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) + (a
*d*cosh(n - 1) + a*d*sinh(n - 1))*sinh(n*log(x)))/(d*n)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{n - 1} \left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.37, size = 112, normalized size = 2.49 \begin {gather*} \frac {a\,x\,{\left (e\,x\right )}^{n-1}}{n}-\frac {2\,\mathrm {atan}\left (\frac {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 {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {-d^2\,n^2\,x^{2\,n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x*(e*x)^(n - 1))/n - (2*atan((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*(b^2*x
^2*(e*x)^(2*n - 2))^(1/2)))*(b^2*x^2*(e*x)^(2*n - 2))^(1/2))/(-d^2*n^2*x^(2*n))^(1/2)

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