Integrand size = 12, antiderivative size = 90 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5 \text {arctanh}\left (\cosh \left (a+b x^2\right )\right )}{32 b}-\frac {5 \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{32 b}+\frac {5 \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{48 b}-\frac {\coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{12 b} \] Output:
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*coth(b*x^2+a)*csch(b*x^2+a)^5/b
Time = 0.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.86 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=-\frac {5 \text {csch}^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\text {csch}^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {csch}^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b}+\frac {5 \log \left (\cosh \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}-\frac {5 \log \left (\sinh \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}-\frac {5 \text {sech}^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {sech}^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\text {sech}^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b} \] Input:
Integrate[x*Csch[a + b*x^2]^7,x]
Output:
(-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[Cosh[(a + b*x^2)/2]])/(32*b) - (5*Log[Si nh[(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)
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.27, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {5960, 3042, 26, 4255, 26, 3042, 26, 4255, 26, 3042, 26, 4255, 26, 3042, 26, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \text {csch}^7\left (a+b x^2\right ) \, dx\) |
| \(\Big \downarrow \) 5960 |
| \(\displaystyle \frac {1}{2} \int \text {csch}^7\left (b x^2+a\right )dx^2\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \int -i \csc \left (i b x^2+i a\right )^7dx^2\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{2} i \int \csc \left (i b x^2+i a\right )^7dx^2\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \int -i \text {csch}^5\left (b x^2+a\right )dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{2} i \left (-\frac {5}{6} i \int \text {csch}^5\left (b x^2+a\right )dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} i \left (-\frac {5}{6} i \int i \csc \left (i b x^2+i a\right )^5dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \int \csc \left (i b x^2+i a\right )^5dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} \int i \text {csch}^3\left (b x^2+a\right )dx^2+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} i \int \text {csch}^3\left (b x^2+a\right )dx^2+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} i \int -i \csc \left (i b x^2+i a\right )^3dx^2+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} \int \csc \left (i b x^2+i a\right )^3dx^2+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int -i \text {csch}\left (b x^2+a\right )dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{2 b}\right )+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {1}{2} i \int \text {csch}\left (b x^2+a\right )dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{2 b}\right )+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {1}{2} i \int i \csc \left (i b x^2+i a\right )dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{2 b}\right )+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (i b x^2+i a\right )dx^2-\frac {i \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{2 b}\right )+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {1}{2} i \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {i \text {arctanh}\left (\cosh \left (a+b x^2\right )\right )}{2 b}-\frac {i \coth \left (a+b x^2\right ) \text {csch}\left (a+b x^2\right )}{2 b}\right )+\frac {i \coth \left (a+b x^2\right ) \text {csch}^3\left (a+b x^2\right )}{4 b}\right )-\frac {i \coth \left (a+b x^2\right ) \text {csch}^5\left (a+b x^2\right )}{6 b}\right )\) |
Input:
Int[x*Csch[a + b*x^2]^7,x]
Output:
(-1/2*I)*(((-1/6*I)*Coth[a + b*x^2]*Csch[a + b*x^2]^5)/b + (5*(((I/4)*Coth [a + b*x^2]*Csch[a + b*x^2]^3)/b + (3*(((I/2)*ArcTanh[Cosh[a + b*x^2]])/b - ((I/2)*Coth[a + b*x^2]*Csch[a + b*x^2])/b))/4))/6)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(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[Simplify[(m + 1)/n], 0] && IntegerQ[p]
Time = 0.46 (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 | \(\frac {\tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{6}-\coth \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{6}-9 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{4}+9 \coth \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{4}+45 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{2}-45 \coth \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{2}-120 \ln \left (\tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )\right )}{768 b}\) | \(109\) |
| risch | \(-\frac {{\mathrm e}^{b \,x^{2}+a} \left (15 \,{\mathrm e}^{10 b \,x^{2}+10 a}-85 \,{\mathrm e}^{8 b \,x^{2}+8 a}+198 \,{\mathrm e}^{6 b \,x^{2}+6 a}+198 \,{\mathrm e}^{4 b \,x^{2}+4 a}-85 \,{\mathrm e}^{2 b \,x^{2}+2 a}+15\right )}{48 b \left ({\mathrm e}^{2 b \,x^{2}+2 a}-1\right )^{6}}+\frac {5 \ln \left (1+{\mathrm e}^{b \,x^{2}+a}\right )}{32 b}-\frac {5 \ln \left ({\mathrm e}^{b \,x^{2}+a}-1\right )}{32 b}\) | \(129\) |
Input:
int(x*csch(b*x^2+a)^7,x,method=_RETURNVERBOSE)
Output:
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)))
Leaf count of result is larger than twice the leaf count of optimal. 2590 vs. \(2 (82) = 164\).
Time = 0.11 (sec) , antiderivative size = 2590, normalized size of antiderivative = 28.78 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate(x*csch(b*x^2+a)^7,x, algorithm="fricas")
Output:
-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))*sin h(b*x^2 + a)^8 + 36*(275*cosh(b*x^2 + a)^4 - 170*cosh(b*x^2 + a)^2 + 11)*s inh(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*cosh( 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*cosh(b*x^2 + a)^5 + 660*cosh(b*x^2 + a)^3 - 85*cosh(b*x^2 + a))*sinh(b*x^2 + a)^2 - 1 5*(cosh(b*x^2 + a)^12 + 12*cosh(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*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(...
\[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\int x \operatorname {csch}^{7}{\left (a + b x^{2} \right )}\, dx \] Input:
integrate(x*csch(b*x**2+a)**7,x)
Output:
Integral(x*csch(a + b*x**2)**7, x)
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (82) = 164\).
Time = 0.08 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.28 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5 \, \log \left (e^{\left (-b x^{2} - a\right )} + 1\right )}{32 \, b} - \frac {5 \, \log \left (e^{\left (-b x^{2} - a\right )} - 1\right )}{32 \, b} + \frac {15 \, e^{\left (-b x^{2} - a\right )} - 85 \, e^{\left (-3 \, b x^{2} - 3 \, a\right )} + 198 \, e^{\left (-5 \, b x^{2} - 5 \, a\right )} + 198 \, e^{\left (-7 \, b x^{2} - 7 \, a\right )} - 85 \, e^{\left (-9 \, b x^{2} - 9 \, a\right )} + 15 \, e^{\left (-11 \, b x^{2} - 11 \, a\right )}}{48 \, b {\left (6 \, e^{\left (-2 \, b x^{2} - 2 \, a\right )} - 15 \, e^{\left (-4 \, b x^{2} - 4 \, a\right )} + 20 \, e^{\left (-6 \, b x^{2} - 6 \, a\right )} - 15 \, e^{\left (-8 \, b x^{2} - 8 \, a\right )} + 6 \, e^{\left (-10 \, b x^{2} - 10 \, a\right )} - e^{\left (-12 \, b x^{2} - 12 \, a\right )} - 1\right )}} \] Input:
integrate(x*csch(b*x^2+a)^7,x, algorithm="maxima")
Output:
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))
Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.76 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5 \, \log \left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )} + 2\right )}{64 \, b} - \frac {5 \, \log \left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )} - 2\right )}{64 \, b} - \frac {15 \, {\left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )}\right )}^{5} - 160 \, {\left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )}\right )}^{3} + 528 \, e^{\left (b x^{2} + a\right )} + 528 \, e^{\left (-b x^{2} - a\right )}}{48 \, {\left ({\left (e^{\left (b x^{2} + a\right )} + e^{\left (-b x^{2} - a\right )}\right )}^{2} - 4\right )}^{3} b} \] Input:
integrate(x*csch(b*x^2+a)^7,x, algorithm="giac")
Output:
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)
Time = 2.40 (sec) , antiderivative size = 399, normalized size of antiderivative = 4.43 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^a\,{\mathrm {e}}^{b\,x^2}\,\sqrt {-b^2}}{b}\right )}{16\,\sqrt {-b^2}}-\frac {8\,{\mathrm {e}}^{3\,b\,x^2+3\,a}}{3\,b\,\left (5\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-10\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+10\,{\mathrm {e}}^{6\,b\,x^2+6\,a}-5\,{\mathrm {e}}^{8\,b\,x^2+8\,a}+{\mathrm {e}}^{10\,b\,x^2+10\,a}-1\right )}-\frac {{\mathrm {e}}^{b\,x^2+a}}{b\,\left (6\,{\mathrm {e}}^{4\,b\,x^2+4\,a}-4\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-4\,{\mathrm {e}}^{6\,b\,x^2+6\,a}+{\mathrm {e}}^{8\,b\,x^2+8\,a}+1\right )}+\frac {5\,{\mathrm {e}}^{b\,x^2+a}}{24\,b\,\left ({\mathrm {e}}^{4\,b\,x^2+4\,a}-2\,{\mathrm {e}}^{2\,b\,x^2+2\,a}+1\right )}-\frac {16\,{\mathrm {e}}^{5\,b\,x^2+5\,a}}{3\,b\,\left (15\,{\mathrm {e}}^{4\,b\,x^2+4\,a}-6\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-20\,{\mathrm {e}}^{6\,b\,x^2+6\,a}+15\,{\mathrm {e}}^{8\,b\,x^2+8\,a}-6\,{\mathrm {e}}^{10\,b\,x^2+10\,a}+{\mathrm {e}}^{12\,b\,x^2+12\,a}+1\right )}-\frac {{\mathrm {e}}^{b\,x^2+a}}{6\,b\,\left (3\,{\mathrm {e}}^{2\,b\,x^2+2\,a}-3\,{\mathrm {e}}^{4\,b\,x^2+4\,a}+{\mathrm {e}}^{6\,b\,x^2+6\,a}-1\right )}-\frac {5\,{\mathrm {e}}^{b\,x^2+a}}{16\,b\,\left ({\mathrm {e}}^{2\,b\,x^2+2\,a}-1\right )} \] Input:
int(x/sinh(a + b*x^2)^7,x)
Output:
(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))
Time = 0.23 (sec) , antiderivative size = 513, normalized size of antiderivative = 5.70 \[ \int x \text {csch}^7\left (a+b x^2\right ) \, dx=\frac {-15 e^{12 b \,x^{2}+12 a} \mathrm {log}\left (e^{b \,x^{2}+a}-1\right )+15 e^{12 b \,x^{2}+12 a} \mathrm {log}\left (e^{b \,x^{2}+a}+1\right )-30 e^{11 b \,x^{2}+11 a}+90 e^{10 b \,x^{2}+10 a} \mathrm {log}\left (e^{b \,x^{2}+a}-1\right )-90 e^{10 b \,x^{2}+10 a} \mathrm {log}\left (e^{b \,x^{2}+a}+1\right )+170 e^{9 b \,x^{2}+9 a}-225 e^{8 b \,x^{2}+8 a} \mathrm {log}\left (e^{b \,x^{2}+a}-1\right )+225 e^{8 b \,x^{2}+8 a} \mathrm {log}\left (e^{b \,x^{2}+a}+1\right )-396 e^{7 b \,x^{2}+7 a}+300 e^{6 b \,x^{2}+6 a} \mathrm {log}\left (e^{b \,x^{2}+a}-1\right )-300 e^{6 b \,x^{2}+6 a} \mathrm {log}\left (e^{b \,x^{2}+a}+1\right )-396 e^{5 b \,x^{2}+5 a}-225 e^{4 b \,x^{2}+4 a} \mathrm {log}\left (e^{b \,x^{2}+a}-1\right )+225 e^{4 b \,x^{2}+4 a} \mathrm {log}\left (e^{b \,x^{2}+a}+1\right )+170 e^{3 b \,x^{2}+3 a}+90 e^{2 b \,x^{2}+2 a} \mathrm {log}\left (e^{b \,x^{2}+a}-1\right )-90 e^{2 b \,x^{2}+2 a} \mathrm {log}\left (e^{b \,x^{2}+a}+1\right )-30 e^{b \,x^{2}+a}-15 \,\mathrm {log}\left (e^{b \,x^{2}+a}-1\right )+15 \,\mathrm {log}\left (e^{b \,x^{2}+a}+1\right )}{96 b \left (e^{12 b \,x^{2}+12 a}-6 e^{10 b \,x^{2}+10 a}+15 e^{8 b \,x^{2}+8 a}-20 e^{6 b \,x^{2}+6 a}+15 e^{4 b \,x^{2}+4 a}-6 e^{2 b \,x^{2}+2 a}+1\right )} \] Input:
int(x*csch(b*x^2+a)^7,x)
Output:
( - 15*e**(12*a + 12*b*x**2)*log(e**(a + b*x**2) - 1) + 15*e**(12*a + 12*b *x**2)*log(e**(a + b*x**2) + 1) - 30*e**(11*a + 11*b*x**2) + 90*e**(10*a + 10*b*x**2)*log(e**(a + b*x**2) - 1) - 90*e**(10*a + 10*b*x**2)*log(e**(a + b*x**2) + 1) + 170*e**(9*a + 9*b*x**2) - 225*e**(8*a + 8*b*x**2)*log(e** (a + b*x**2) - 1) + 225*e**(8*a + 8*b*x**2)*log(e**(a + b*x**2) + 1) - 396 *e**(7*a + 7*b*x**2) + 300*e**(6*a + 6*b*x**2)*log(e**(a + b*x**2) - 1) - 300*e**(6*a + 6*b*x**2)*log(e**(a + b*x**2) + 1) - 396*e**(5*a + 5*b*x**2) - 225*e**(4*a + 4*b*x**2)*log(e**(a + b*x**2) - 1) + 225*e**(4*a + 4*b*x* *2)*log(e**(a + b*x**2) + 1) + 170*e**(3*a + 3*b*x**2) + 90*e**(2*a + 2*b* x**2)*log(e**(a + b*x**2) - 1) - 90*e**(2*a + 2*b*x**2)*log(e**(a + b*x**2 ) + 1) - 30*e**(a + b*x**2) - 15*log(e**(a + b*x**2) - 1) + 15*log(e**(a + b*x**2) + 1))/(96*b*(e**(12*a + 12*b*x**2) - 6*e**(10*a + 10*b*x**2) + 15 *e**(8*a + 8*b*x**2) - 20*e**(6*a + 6*b*x**2) + 15*e**(4*a + 4*b*x**2) - 6 *e**(2*a + 2*b*x**2) + 1))