\(\int x \text {csch}^7(a+b x^2) \, dx\) [15]

Optimal result

Integrand size = 12, antiderivative size = 90 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5 \text {arctanh}\left (\cosh \left (a+b x^2\right )\right )}{32 b}-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b} \]

[Out]

5/32*arctanh(cosh(b*x^2+a))/b-5/32*coth(b*x^2+a)*csch(b*x^2+a)/b+5/48*coth(b*x^2+a)*csch(b*x^2+a)^3/b-1/12*cot
h(b*x^2+a)*csch(b*x^2+a)^5/b

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5545, 3853, 3855} \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5 \text {arctanh}\left (\cosh \left (a+b x^2\right )\right )}{32 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b} \]

[In]

Int[x*Csch[a + b*x^2]^7,x]

[Out]

(5*ArcTanh[Cosh[a + b*x^2]])/(32*b) - (5*Coth[a + b*x^2]*Csch[a + b*x^2])/(32*b) + (5*Coth[a + b*x^2]*Csch[a +
b*x^2]^3)/(48*b) - (Coth[a + b*x^2]*Csch[a + b*x^2]^5)/(12*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]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \text {csch}^7(a+b x) \, dx,x,x^2\right ) \\ & = -\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}-\frac {5}{12} \text {Subst}\left (\int \text {csch}^5(a+b x) \, dx,x,x^2\right ) \\ & = \frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}+\frac {5}{16} \text {Subst}\left (\int \text {csch}^3(a+b x) \, dx,x,x^2\right ) \\ & = -\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b}-\frac {5}{32} \text {Subst}\left (\int \text {csch}(a+b x) \, dx,x,x^2\right ) \\ & = \frac {5 \text {arctanh}\left (\cosh \left (a+b x^2\right )\right )}{32 b}-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.86 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=-\frac {5 \text {csch}^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\text {csch}^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {csch}^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b}+\frac {5 \log \left (\cosh \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}-\frac {5 \log \left (\sinh \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}-\frac {5 \text {sech}^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {sech}^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {sech}^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b} \]

[In]

Integrate[x*Csch[a + b*x^2]^7,x]

[Out]

(-5*Csch[(a + b*x^2)/2]^2)/(128*b) + Csch[(a + b*x^2)/2]^4/(128*b) - Csch[(a + b*x^2)/2]^6/(768*b) + (5*Log[Co
sh[(a + b*x^2)/2]])/(32*b) - (5*Log[Sinh[(a + b*x^2)/2]])/(32*b) - (5*Sech[(a + b*x^2)/2]^2)/(128*b) - Sech[(a
+ b*x^2)/2]^4/(128*b) - Sech[(a + b*x^2)/2]^6/(768*b)

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.69

method	result	size
derivativedivides	\(\frac {\left (-\frac {\operatorname {csch}\left (b \,x^{2}+a \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (b \,x^{2}+a \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (b \,x^{2}+a \right )}{16}\right ) \coth \left (b \,x^{2}+a \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{b \,x^{2}+a}\right )}{8}}{2 b}\)	\(62\)
default	\(\frac {\left (-\frac {\operatorname {csch}\left (b \,x^{2}+a \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (b \,x^{2}+a \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (b \,x^{2}+a \right )}{16}\right ) \coth \left (b \,x^{2}+a \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{b \,x^{2}+a}\right )}{8}}{2 b}\)	\(62\)
parallelrisch	\(\frac {-\coth \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{6}+\tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{6}+9 \coth \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{4}-9 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{4}-45 \coth \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{2}+45 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{2}-120 \ln \left (\tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )\right )}{768 b}\)	\(109\)
risch	\(-\frac {{\mathrm e}^{b \,x^{2}+a} \left (15 \,{\mathrm e}^{10 b \,x^{2}+10 a}-85 \,{\mathrm e}^{8 b \,x^{2}+8 a}+198 \,{\mathrm e}^{6 b \,x^{2}+6 a}+198 \,{\mathrm e}^{4 b \,x^{2}+4 a}-85 \,{\mathrm e}^{2 b \,x^{2}+2 a}+15\right )}{48 b \left ({\mathrm e}^{2 b \,x^{2}+2 a}-1\right )^{6}}+\frac {5 \ln \left ({\mathrm e}^{b \,x^{2}+a}+1\right )}{32 b}-\frac {5 \ln \left ({\mathrm e}^{b \,x^{2}+a}-1\right )}{32 b}\)	\(129\)

[In]

int(x*csch(b*x^2+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/2/b*((-1/6*csch(b*x^2+a)^5+5/24*csch(b*x^2+a)^3-5/16*csch(b*x^2+a))*coth(b*x^2+a)+5/8*arctanh(exp(b*x^2+a)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2590 vs. \(2 (82) = 164\).

Time = 0.27 (sec) , antiderivative size = 2590, normalized size of antiderivative = 28.78 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate(x*csch(b*x^2+a)^7,x, algorithm="fricas")

[Out]

-1/96*(30*cosh(b*x^2 + a)^11 + 330*cosh(b*x^2 + a)*sinh(b*x^2 + a)^10 + 30*sinh(b*x^2 + a)^11 + 10*(165*cosh(b
*x^2 + a)^2 - 17)*sinh(b*x^2 + a)^9 - 170*cosh(b*x^2 + a)^9 + 90*(55*cosh(b*x^2 + a)^3 - 17*cosh(b*x^2 + a))*s
inh(b*x^2 + a)^8 + 36*(275*cosh(b*x^2 + a)^4 - 170*cosh(b*x^2 + a)^2 + 11)*sinh(b*x^2 + a)^7 + 396*cosh(b*x^2
+ a)^7 + 84*(165*cosh(b*x^2 + a)^5 - 170*cosh(b*x^2 + a)^3 + 33*cosh(b*x^2 + a))*sinh(b*x^2 + a)^6 + 36*(385*c
osh(b*x^2 + a)^6 - 595*cosh(b*x^2 + a)^4 + 231*cosh(b*x^2 + a)^2 + 11)*sinh(b*x^2 + a)^5 + 396*cosh(b*x^2 + a)
^5 + 180*(55*cosh(b*x^2 + a)^7 - 119*cosh(b*x^2 + a)^5 + 77*cosh(b*x^2 + a)^3 + 11*cosh(b*x^2 + a))*sinh(b*x^2
+ a)^4 + 10*(495*cosh(b*x^2 + a)^8 - 1428*cosh(b*x^2 + a)^6 + 1386*cosh(b*x^2 + a)^4 + 396*cosh(b*x^2 + a)^2
- 17)*sinh(b*x^2 + a)^3 - 170*cosh(b*x^2 + a)^3 + 6*(275*cosh(b*x^2 + a)^9 - 1020*cosh(b*x^2 + a)^7 + 1386*cos
h(b*x^2 + a)^5 + 660*cosh(b*x^2 + a)^3 - 85*cosh(b*x^2 + a))*sinh(b*x^2 + a)^2 - 15*(cosh(b*x^2 + a)^12 + 12*c
osh(b*x^2 + a)*sinh(b*x^2 + a)^11 + sinh(b*x^2 + a)^12 + 6*(11*cosh(b*x^2 + a)^2 - 1)*sinh(b*x^2 + a)^10 - 6*c
osh(b*x^2 + a)^10 + 20*(11*cosh(b*x^2 + a)^3 - 3*cosh(b*x^2 + a))*sinh(b*x^2 + a)^9 + 15*(33*cosh(b*x^2 + a)^4
- 18*cosh(b*x^2 + a)^2 + 1)*sinh(b*x^2 + a)^8 + 15*cosh(b*x^2 + a)^8 + 24*(33*cosh(b*x^2 + a)^5 - 30*cosh(b*x
^2 + a)^3 + 5*cosh(b*x^2 + a))*sinh(b*x^2 + a)^7 + 4*(231*cosh(b*x^2 + a)^6 - 315*cosh(b*x^2 + a)^4 + 105*cosh
(b*x^2 + a)^2 - 5)*sinh(b*x^2 + a)^6 - 20*cosh(b*x^2 + a)^6 + 24*(33*cosh(b*x^2 + a)^7 - 63*cosh(b*x^2 + a)^5
+ 35*cosh(b*x^2 + a)^3 - 5*cosh(b*x^2 + a))*sinh(b*x^2 + a)^5 + 15*(33*cosh(b*x^2 + a)^8 - 84*cosh(b*x^2 + a)^
6 + 70*cosh(b*x^2 + a)^4 - 20*cosh(b*x^2 + a)^2 + 1)*sinh(b*x^2 + a)^4 + 15*cosh(b*x^2 + a)^4 + 20*(11*cosh(b*
x^2 + a)^9 - 36*cosh(b*x^2 + a)^7 + 42*cosh(b*x^2 + a)^5 - 20*cosh(b*x^2 + a)^3 + 3*cosh(b*x^2 + a))*sinh(b*x^
2 + a)^3 + 6*(11*cosh(b*x^2 + a)^10 - 45*cosh(b*x^2 + a)^8 + 70*cosh(b*x^2 + a)^6 - 50*cosh(b*x^2 + a)^4 + 15*
cosh(b*x^2 + a)^2 - 1)*sinh(b*x^2 + a)^2 - 6*cosh(b*x^2 + a)^2 + 12*(cosh(b*x^2 + a)^11 - 5*cosh(b*x^2 + a)^9
+ 10*cosh(b*x^2 + a)^7 - 10*cosh(b*x^2 + a)^5 + 5*cosh(b*x^2 + a)^3 - cosh(b*x^2 + a))*sinh(b*x^2 + a) + 1)*lo
g(cosh(b*x^2 + a) + sinh(b*x^2 + a) + 1) + 15*(cosh(b*x^2 + a)^12 + 12*cosh(b*x^2 + a)*sinh(b*x^2 + a)^11 + si
nh(b*x^2 + a)^12 + 6*(11*cosh(b*x^2 + a)^2 - 1)*sinh(b*x^2 + a)^10 - 6*cosh(b*x^2 + a)^10 + 20*(11*cosh(b*x^2
+ a)^3 - 3*cosh(b*x^2 + a))*sinh(b*x^2 + a)^9 + 15*(33*cosh(b*x^2 + a)^4 - 18*cosh(b*x^2 + a)^2 + 1)*sinh(b*x^
2 + a)^8 + 15*cosh(b*x^2 + a)^8 + 24*(33*cosh(b*x^2 + a)^5 - 30*cosh(b*x^2 + a)^3 + 5*cosh(b*x^2 + a))*sinh(b*
x^2 + a)^7 + 4*(231*cosh(b*x^2 + a)^6 - 315*cosh(b*x^2 + a)^4 + 105*cosh(b*x^2 + a)^2 - 5)*sinh(b*x^2 + a)^6 -
20*cosh(b*x^2 + a)^6 + 24*(33*cosh(b*x^2 + a)^7 - 63*cosh(b*x^2 + a)^5 + 35*cosh(b*x^2 + a)^3 - 5*cosh(b*x^2
+ a))*sinh(b*x^2 + a)^5 + 15*(33*cosh(b*x^2 + a)^8 - 84*cosh(b*x^2 + a)^6 + 70*cosh(b*x^2 + a)^4 - 20*cosh(b*x
^2 + a)^2 + 1)*sinh(b*x^2 + a)^4 + 15*cosh(b*x^2 + a)^4 + 20*(11*cosh(b*x^2 + a)^9 - 36*cosh(b*x^2 + a)^7 + 42
*cosh(b*x^2 + a)^5 - 20*cosh(b*x^2 + a)^3 + 3*cosh(b*x^2 + a))*sinh(b*x^2 + a)^3 + 6*(11*cosh(b*x^2 + a)^10 -
45*cosh(b*x^2 + a)^8 + 70*cosh(b*x^2 + a)^6 - 50*cosh(b*x^2 + a)^4 + 15*cosh(b*x^2 + a)^2 - 1)*sinh(b*x^2 + a)
^2 - 6*cosh(b*x^2 + a)^2 + 12*(cosh(b*x^2 + a)^11 - 5*cosh(b*x^2 + a)^9 + 10*cosh(b*x^2 + a)^7 - 10*cosh(b*x^2
+ a)^5 + 5*cosh(b*x^2 + a)^3 - cosh(b*x^2 + a))*sinh(b*x^2 + a) + 1)*log(cosh(b*x^2 + a) + sinh(b*x^2 + a) -
1) + 6*(55*cosh(b*x^2 + a)^10 - 255*cosh(b*x^2 + a)^8 + 462*cosh(b*x^2 + a)^6 + 330*cosh(b*x^2 + a)^4 - 85*cos
h(b*x^2 + a)^2 + 5)*sinh(b*x^2 + a) + 30*cosh(b*x^2 + a))/(b*cosh(b*x^2 + a)^12 + 12*b*cosh(b*x^2 + a)*sinh(b*
x^2 + a)^11 + b*sinh(b*x^2 + a)^12 - 6*b*cosh(b*x^2 + a)^10 + 6*(11*b*cosh(b*x^2 + a)^2 - b)*sinh(b*x^2 + a)^1
0 + 20*(11*b*cosh(b*x^2 + a)^3 - 3*b*cosh(b*x^2 + a))*sinh(b*x^2 + a)^9 + 15*b*cosh(b*x^2 + a)^8 + 15*(33*b*co
sh(b*x^2 + a)^4 - 18*b*cosh(b*x^2 + a)^2 + b)*sinh(b*x^2 + a)^8 + 24*(33*b*cosh(b*x^2 + a)^5 - 30*b*cosh(b*x^2
+ a)^3 + 5*b*cosh(b*x^2 + a))*sinh(b*x^2 + a)^7 - 20*b*cosh(b*x^2 + a)^6 + 4*(231*b*cosh(b*x^2 + a)^6 - 315*b
*cosh(b*x^2 + a)^4 + 105*b*cosh(b*x^2 + a)^2 - 5*b)*sinh(b*x^2 + a)^6 + 24*(33*b*cosh(b*x^2 + a)^7 - 63*b*cosh
(b*x^2 + a)^5 + 35*b*cosh(b*x^2 + a)^3 - 5*b*cosh(b*x^2 + a))*sinh(b*x^2 + a)^5 + 15*b*cosh(b*x^2 + a)^4 + 15*
(33*b*cosh(b*x^2 + a)^8 - 84*b*cosh(b*x^2 + a)^6 + 70*b*cosh(b*x^2 + a)^4 - 20*b*cosh(b*x^2 + a)^2 + b)*sinh(b
*x^2 + a)^4 + 20*(11*b*cosh(b*x^2 + a)^9 - 36*b*cosh(b*x^2 + a)^7 + 42*b*cosh(b*x^2 + a)^5 - 20*b*cosh(b*x^2 +
a)^3 + 3*b*cosh(b*x^2 + a))*sinh(b*x^2 + a)^3 - 6*b*cosh(b*x^2 + a)^2 + 6*(11*b*cosh(b*x^2 + a)^10 - 45*b*cos
h(b*x^2 + a)^8 + 70*b*cosh(b*x^2 + a)^6 - 50*b*cosh(b*x^2 + a)^4 + 15*b*cosh(b*x^2 + a)^2 - b)*sinh(b*x^2 + a)
^2 + 12*(b*cosh(b*x^2 + a)^11 - 5*b*cosh(b*x^2 + a)^9 + 10*b*cosh(b*x^2 + a)^7 - 10*b*cosh(b*x^2 + a)^5 + 5*b*
cosh(b*x^2 + a)^3 - b*cosh(b*x^2 + a))*sinh(b*x^2 + a) + b)

Sympy [F]

\[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\int x \operatorname {csch}^{7}{\left (a + b x^{2} \right )}\, dx \]

[In]

integrate(x*csch(b*x**2+a)**7,x)

[Out]

Integral(x*csch(a + b*x**2)**7, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (82) = 164\).

Time = 0.23 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.28 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5 \, \log \left (e^{\left (-b x^{2} - a\right )} + 1\right )}{32 \, b} - \frac {5 \, \log \left (e^{\left (-b x^{2} - a\right )} - 1\right )}{32 \, b} + \frac {15 \, e^{\left (-b x^{2} - a\right )} - 85 \, e^{\left (-3 \, b x^{2} - 3 \, a\right )} + 198 \, e^{\left (-5 \, b x^{2} - 5 \, a\right )} + 198 \, e^{\left (-7 \, b x^{2} - 7 \, a\right )} - 85 \, e^{\left (-9 \, b x^{2} - 9 \, a\right )} + 15 \, e^{\left (-11 \, b x^{2} - 11 \, a\right )}}{48 \, b {\left (6 \, e^{\left (-2 \, b x^{2} - 2 \, a\right )} - 15 \, e^{\left (-4 \, b x^{2} - 4 \, a\right )} + 20 \, e^{\left (-6 \, b x^{2} - 6 \, a\right )} - 15 \, e^{\left (-8 \, b x^{2} - 8 \, a\right )} + 6 \, e^{\left (-10 \, b x^{2} - 10 \, a\right )} - e^{\left (-12 \, b x^{2} - 12 \, a\right )} - 1\right )}} \]

[In]

integrate(x*csch(b*x^2+a)^7,x, algorithm="maxima")

[Out]

5/32*log(e^(-b*x^2 - a) + 1)/b - 5/32*log(e^(-b*x^2 - a) - 1)/b + 1/48*(15*e^(-b*x^2 - a) - 85*e^(-3*b*x^2 - 3
*a) + 198*e^(-5*b*x^2 - 5*a) + 198*e^(-7*b*x^2 - 7*a) - 85*e^(-9*b*x^2 - 9*a) + 15*e^(-11*b*x^2 - 11*a))/(b*(6
*e^(-2*b*x^2 - 2*a) - 15*e^(-4*b*x^2 - 4*a) + 20*e^(-6*b*x^2 - 6*a) - 15*e^(-8*b*x^2 - 8*a) + 6*e^(-10*b*x^2 -
10*a) - e^(-12*b*x^2 - 12*a) - 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.76 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5 \, \log \left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )} + 2\right )}{64 \, b} - \frac {5 \, \log \left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )} - 2\right )}{64 \, b} - \frac {15 \, {\left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )}\right )}^{5} - 160 \, {\left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )}\right )}^{3} + 528 \, e^{\left (b x^{2} + a\right )} + 528 \, e^{\left (-b x^{2} - a\right )}}{48 \, {\left ({\left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )}\right )}^{2} - 4\right )}^{3} b} \]

[In]

integrate(x*csch(b*x^2+a)^7,x, algorithm="giac")

[Out]

5/64*log(e^(b*x^2 + a) + e^(-b*x^2 - a) + 2)/b - 5/64*log(e^(b*x^2 + a) + e^(-b*x^2 - a) - 2)/b - 1/48*(15*(e^
(b*x^2 + a) + e^(-b*x^2 - a))^5 - 160*(e^(b*x^2 + a) + e^(-b*x^2 - a))^3 + 528*e^(b*x^2 + a) + 528*e^(-b*x^2 -
a))/(((e^(b*x^2 + a) + e^(-b*x^2 - a))^2 - 4)^3*b)

Mupad [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 399, normalized size of antiderivative = 4.43 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^a\,{\mathrm {e}}^{b\,x^2}\,\sqrt {-b^2}}{b}\right )}{16\,\sqrt {-b^2}}-\frac {8\,{\mathrm {e}}^{3\,b\,x^2+3\,a}}{3\,b\,\left (5\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-10\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+10\,{\mathrm {e}}^{6\,b\,x^2+6\,a}-5\,{\mathrm {e}}^{8\,b\,x^2+8\,a}+{\mathrm {e}}^{10\,b\,x^2+10\,a}-1\right )}-\frac {{\mathrm {e}}^{b\,x^2+a}}{b\,\left (6\,{\mathrm {e}}^{4\,b\,x^2+4\,a}-4\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-4\,{\mathrm {e}}^{6\,b\,x^2+6\,a}+{\mathrm {e}}^{8\,b\,x^2+8\,a}+1\right )}+\frac {5\,{\mathrm {e}}^{b\,x^2+a}}{24\,b\,\left ({\mathrm {e}}^{4\,b\,x^2+4\,a}-2\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+1\right )}-\frac {16\,{\mathrm {e}}^{5\,b\,x^2+5\,a}}{3\,b\,\left (15\,{\mathrm {e}}^{4\,b\,x^2+4\,a}-6\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-20\,{\mathrm {e}}^{6\,b\,x^2+6\,a}+15\,{\mathrm {e}}^{8\,b\,x^2+8\,a}-6\,{\mathrm {e}}^{10\,b\,x^2+10\,a}+{\mathrm {e}}^{12\,b\,x^2+12\,a}+1\right )}-\frac {{\mathrm {e}}^{b\,x^2+a}}{6\,b\,\left (3\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-3\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+{\mathrm {e}}^{6\,b\,x^2+6\,a}-1\right )}-\frac {5\,{\mathrm {e}}^{b\,x^2+a}}{16\,b\,\left ({\mathrm {e}}^{2\,b\,x^2+2\,a}-1\right )} \]

[In]

int(x/sinh(a + b*x^2)^7,x)

[Out]

(5*atan((exp(a)*exp(b*x^2)*(-b^2)^(1/2))/b))/(16*(-b^2)^(1/2)) - (8*exp(3*a + 3*b*x^2))/(3*b*(5*exp(2*a + 2*b*
x^2) - 10*exp(4*a + 4*b*x^2) + 10*exp(6*a + 6*b*x^2) - 5*exp(8*a + 8*b*x^2) + exp(10*a + 10*b*x^2) - 1)) - exp
(a + b*x^2)/(b*(6*exp(4*a + 4*b*x^2) - 4*exp(2*a + 2*b*x^2) - 4*exp(6*a + 6*b*x^2) + exp(8*a + 8*b*x^2) + 1))
+ (5*exp(a + b*x^2))/(24*b*(exp(4*a + 4*b*x^2) - 2*exp(2*a + 2*b*x^2) + 1)) - (16*exp(5*a + 5*b*x^2))/(3*b*(15
*exp(4*a + 4*b*x^2) - 6*exp(2*a + 2*b*x^2) - 20*exp(6*a + 6*b*x^2) + 15*exp(8*a + 8*b*x^2) - 6*exp(10*a + 10*b
*x^2) + exp(12*a + 12*b*x^2) + 1)) - exp(a + b*x^2)/(6*b*(3*exp(2*a + 2*b*x^2) - 3*exp(4*a + 4*b*x^2) + exp(6*
a + 6*b*x^2) - 1)) - (5*exp(a + b*x^2))/(16*b*(exp(2*a + 2*b*x^2) - 1))