Integrand size = 12, antiderivative size = 109 \[ \int (a+b \text {csch}(c+d x))^4 \, dx=a^4 x-\frac {2 a b \left (2 a^2-b^2\right ) \text {arctanh}(\cosh (c+d x))}{d}-\frac {b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{3 d}-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{3 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d} \] Output:
a^4*x-2*a*b*(2*a^2-b^2)*arctanh(cosh(d*x+c))/d-1/3*b^2*(17*a^2-2*b^2)*coth (d*x+c)/d-4/3*a*b^3*coth(d*x+c)*csch(d*x+c)/d-1/3*b^2*coth(d*x+c)*(a+b*csc h(d*x+c))^2/d
Leaf count is larger than twice the leaf count of optimal. \(567\) vs. \(2(109)=218\).
Time = 11.55 (sec) , antiderivative size = 567, normalized size of antiderivative = 5.20 \[ \int (a+b \text {csch}(c+d x))^4 \, dx=\frac {a^4 (c+d x) (a+b \text {csch}(c+d x))^4 \sinh ^4(c+d x)}{d (b+a \sinh (c+d x))^4}+\frac {\left (-9 a^2 b^2 \cosh \left (\frac {1}{2} (c+d x)\right )+b^4 \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right ) (a+b \text {csch}(c+d x))^4 \sinh ^4(c+d x)}{3 d (b+a \sinh (c+d x))^4}-\frac {a b^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) (a+b \text {csch}(c+d x))^4 \sinh ^4(c+d x)}{2 d (b+a \sinh (c+d x))^4}-\frac {b^4 \coth \left (\frac {1}{2} (c+d x)\right ) \text {csch}^2\left (\frac {1}{2} (c+d x)\right ) (a+b \text {csch}(c+d x))^4 \sinh ^4(c+d x)}{24 d (b+a \sinh (c+d x))^4}+\frac {2 \left (-2 a^3 b+a b^3\right ) (a+b \text {csch}(c+d x))^4 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right ) \sinh ^4(c+d x)}{d (b+a \sinh (c+d x))^4}-\frac {2 \left (-2 a^3 b+a b^3\right ) (a+b \text {csch}(c+d x))^4 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sinh ^4(c+d x)}{d (b+a \sinh (c+d x))^4}-\frac {a b^3 (a+b \text {csch}(c+d x))^4 \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) \sinh ^4(c+d x)}{2 d (b+a \sinh (c+d x))^4}+\frac {(a+b \text {csch}(c+d x))^4 \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-9 a^2 b^2 \sinh \left (\frac {1}{2} (c+d x)\right )+b^4 \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sinh ^4(c+d x)}{3 d (b+a \sinh (c+d x))^4}+\frac {b^4 (a+b \text {csch}(c+d x))^4 \text {sech}^2\left (\frac {1}{2} (c+d x)\right ) \sinh ^4(c+d x) \tanh \left (\frac {1}{2} (c+d x)\right )}{24 d (b+a \sinh (c+d x))^4} \] Input:
Integrate[(a + b*Csch[c + d*x])^4,x]
Output:
(a^4*(c + d*x)*(a + b*Csch[c + d*x])^4*Sinh[c + d*x]^4)/(d*(b + a*Sinh[c + d*x])^4) + ((-9*a^2*b^2*Cosh[(c + d*x)/2] + b^4*Cosh[(c + d*x)/2])*Csch[( c + d*x)/2]*(a + b*Csch[c + d*x])^4*Sinh[c + d*x]^4)/(3*d*(b + a*Sinh[c + d*x])^4) - (a*b^3*Csch[(c + d*x)/2]^2*(a + b*Csch[c + d*x])^4*Sinh[c + d*x ]^4)/(2*d*(b + a*Sinh[c + d*x])^4) - (b^4*Coth[(c + d*x)/2]*Csch[(c + d*x) /2]^2*(a + b*Csch[c + d*x])^4*Sinh[c + d*x]^4)/(24*d*(b + a*Sinh[c + d*x]) ^4) + (2*(-2*a^3*b + a*b^3)*(a + b*Csch[c + d*x])^4*Log[Cosh[(c + d*x)/2]] *Sinh[c + d*x]^4)/(d*(b + a*Sinh[c + d*x])^4) - (2*(-2*a^3*b + a*b^3)*(a + b*Csch[c + d*x])^4*Log[Sinh[(c + d*x)/2]]*Sinh[c + d*x]^4)/(d*(b + a*Sinh [c + d*x])^4) - (a*b^3*(a + b*Csch[c + d*x])^4*Sech[(c + d*x)/2]^2*Sinh[c + d*x]^4)/(2*d*(b + a*Sinh[c + d*x])^4) + ((a + b*Csch[c + d*x])^4*Sech[(c + d*x)/2]*(-9*a^2*b^2*Sinh[(c + d*x)/2] + b^4*Sinh[(c + d*x)/2])*Sinh[c + d*x]^4)/(3*d*(b + a*Sinh[c + d*x])^4) + (b^4*(a + b*Csch[c + d*x])^4*Sech [(c + d*x)/2]^2*Sinh[c + d*x]^4*Tanh[(c + d*x)/2])/(24*d*(b + a*Sinh[c + d *x])^4)
Time = 0.55 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3042, 4269, 3042, 4536, 2009}
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.
| \(\displaystyle \int (a+b \text {csch}(c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i b \csc (i c+i d x))^4dx\) |
| \(\Big \downarrow \) 4269 |
| \(\displaystyle \frac {1}{3} \int (a+b \text {csch}(c+d x)) \left (3 a^3+8 b^2 \text {csch}^2(c+d x) a+b \left (9 a^2-2 b^2\right ) \text {csch}(c+d x)\right )dx-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}+\frac {1}{3} \int (a+i b \csc (i c+i d x)) \left (3 a^3-8 b^2 \csc (i c+i d x)^2 a+i b \left (9 a^2-2 b^2\right ) \csc (i c+i d x)\right )dx\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \int \left (6 a^4+12 b \left (2 a^2-b^2\right ) \text {csch}(c+d x) a+2 b^2 \left (17 a^2-2 b^2\right ) \text {csch}^2(c+d x)\right )dx-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{d}\right )-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {1}{2} \left (6 a^4 x-\frac {12 a b \left (2 a^2-b^2\right ) \text {arctanh}(\cosh (c+d x))}{d}-\frac {2 b^2 \left (17 a^2-2 b^2\right ) \coth (c+d x)}{d}\right )-\frac {4 a b^3 \coth (c+d x) \text {csch}(c+d x)}{d}\right )-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))^2}{3 d}\) |
Input:
Int[(a + b*Csch[c + d*x])^4,x]
Output:
-1/3*(b^2*Coth[c + d*x]*(a + b*Csch[c + d*x])^2)/d + ((6*a^4*x - (12*a*b*( 2*a^2 - b^2)*ArcTanh[Cosh[c + d*x]])/d - (2*b^2*(17*a^2 - 2*b^2)*Coth[c + d*x])/d)/2 - (4*a*b^3*Coth[c + d*x]*Csch[c + d*x])/d)/3
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[1/(n - 1) Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3* a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] / ; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Simp[1/2 Int[Simp[2*A*a + (2*B*a + b*(2* A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, e, f, A, B, C}, x]
Time = 0.72 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {a^{4} \left (d x +c \right )-8 a^{3} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )-6 a^{2} b^{2} \coth \left (d x +c \right )+4 a \,b^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{4} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}\) | \(92\) |
| default | \(\frac {a^{4} \left (d x +c \right )-8 a^{3} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )-6 a^{2} b^{2} \coth \left (d x +c \right )+4 a \,b^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{4} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}\) | \(92\) |
| parts | \(a^{4} x +\frac {b^{4} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}-\frac {6 a^{2} b^{2} \coth \left (d x +c \right )}{d}+\frac {4 a^{3} b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(99\) |
| parallelrisch | \(\frac {48 \left (2 a^{3} b -a \,b^{3}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{4}-12 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{3}+9 \left (-8 a^{2} b^{2}+b^{4}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{4}+12 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{3}+9 \left (-8 a^{2} b^{2}+b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+24 a^{4} d x}{24 d}\) | \(151\) |
| risch | \(a^{4} x -\frac {4 b^{2} \left (3 a b \,{\mathrm e}^{5 d x +5 c}+9 a^{2} {\mathrm e}^{4 d x +4 c}-18 a^{2} {\mathrm e}^{2 d x +2 c}+3 b^{2} {\mathrm e}^{2 d x +2 c}-3 a b \,{\mathrm e}^{d x +c}+9 a^{2}-b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}+\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {2 a \,b^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}\) | \(176\) |
Input:
int((a+csch(d*x+c)*b)^4,x,method=_RETURNVERBOSE)
Output:
1/d*(a^4*(d*x+c)-8*a^3*b*arctanh(exp(d*x+c))-6*a^2*b^2*coth(d*x+c)+4*a*b^3 *(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+b^4*(2/3-1/3*csch(d*x+ c)^2)*coth(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 1440 vs. \(2 (103) = 206\).
Time = 0.11 (sec) , antiderivative size = 1440, normalized size of antiderivative = 13.21 \[ \int (a+b \text {csch}(c+d x))^4 \, dx=\text {Too large to display} \] Input:
integrate((a+b*csch(d*x+c))^4,x, algorithm="fricas")
Output:
1/3*(3*a^4*d*x*cosh(d*x + c)^6 + 3*a^4*d*x*sinh(d*x + c)^6 - 12*a*b^3*cosh (d*x + c)^5 - 3*a^4*d*x + 6*(3*a^4*d*x*cosh(d*x + c) - 2*a*b^3)*sinh(d*x + c)^5 + 12*a*b^3*cosh(d*x + c) - 9*(a^4*d*x + 4*a^2*b^2)*cosh(d*x + c)^4 + 3*(15*a^4*d*x*cosh(d*x + c)^2 - 3*a^4*d*x - 20*a*b^3*cosh(d*x + c) - 12*a ^2*b^2)*sinh(d*x + c)^4 - 36*a^2*b^2 + 4*b^4 + 12*(5*a^4*d*x*cosh(d*x + c) ^3 - 10*a*b^3*cosh(d*x + c)^2 - 3*(a^4*d*x + 4*a^2*b^2)*cosh(d*x + c))*sin h(d*x + c)^3 + 3*(3*a^4*d*x + 24*a^2*b^2 - 4*b^4)*cosh(d*x + c)^2 + 3*(15* a^4*d*x*cosh(d*x + c)^4 - 40*a*b^3*cosh(d*x + c)^3 + 3*a^4*d*x + 24*a^2*b^ 2 - 4*b^4 - 18*(a^4*d*x + 4*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 6* ((2*a^3*b - a*b^3)*cosh(d*x + c)^6 + 6*(2*a^3*b - a*b^3)*cosh(d*x + c)*sin h(d*x + c)^5 + (2*a^3*b - a*b^3)*sinh(d*x + c)^6 - 3*(2*a^3*b - a*b^3)*cos h(d*x + c)^4 - 3*(2*a^3*b - a*b^3 - 5*(2*a^3*b - a*b^3)*cosh(d*x + c)^2)*s inh(d*x + c)^4 - 2*a^3*b + a*b^3 + 4*(5*(2*a^3*b - a*b^3)*cosh(d*x + c)^3 - 3*(2*a^3*b - a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(2*a^3*b - a*b^3) *cosh(d*x + c)^2 + 3*(5*(2*a^3*b - a*b^3)*cosh(d*x + c)^4 + 2*a^3*b - a*b^ 3 - 6*(2*a^3*b - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 6*((2*a^3*b - a *b^3)*cosh(d*x + c)^5 - 2*(2*a^3*b - a*b^3)*cosh(d*x + c)^3 + (2*a^3*b - a *b^3)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 6*((2*a^3*b - a*b^3)*cosh(d*x + c)^6 + 6*(2*a^3*b - a*b^3)*cosh(d*x + c )*sinh(d*x + c)^5 + (2*a^3*b - a*b^3)*sinh(d*x + c)^6 - 3*(2*a^3*b - a*...
\[ \int (a+b \text {csch}(c+d x))^4 \, dx=\int \left (a + b \operatorname {csch}{\left (c + d x \right )}\right )^{4}\, dx \] Input:
integrate((a+b*csch(d*x+c))**4,x)
Output:
Integral((a + b*csch(c + d*x))**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (103) = 206\).
Time = 0.04 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.15 \[ \int (a+b \text {csch}(c+d x))^4 \, dx=a^{4} x + 2 \, a b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac {4}{3} \, b^{4} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {4 \, a^{3} b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {12 \, a^{2} b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \] Input:
integrate((a+b*csch(d*x+c))^4,x, algorithm="maxima")
Output:
a^4*x + 2*a*b^3*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^ (-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) + 4/3*b^4*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1))) + 4*a^3*b*log(tanh(1/2*d*x + 1/2*c))/d + 12 *a^2*b^2/(d*(e^(-2*d*x - 2*c) - 1))
Time = 0.12 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.55 \[ \int (a+b \text {csch}(c+d x))^4 \, dx=\frac {3 \, {\left (d x + c\right )} a^{4} - 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) + 6 \, {\left (2 \, a^{3} b - a b^{3}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {4 \, {\left (3 \, a b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 9 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, a b^{3} e^{\left (d x + c\right )} + 9 \, a^{2} b^{2} - b^{4}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \] Input:
integrate((a+b*csch(d*x+c))^4,x, algorithm="giac")
Output:
1/3*(3*(d*x + c)*a^4 - 6*(2*a^3*b - a*b^3)*log(e^(d*x + c) + 1) + 6*(2*a^3 *b - a*b^3)*log(abs(e^(d*x + c) - 1)) - 4*(3*a*b^3*e^(5*d*x + 5*c) + 9*a^2 *b^2*e^(4*d*x + 4*c) - 18*a^2*b^2*e^(2*d*x + 2*c) + 3*b^4*e^(2*d*x + 2*c) - 3*a*b^3*e^(d*x + c) + 9*a^2*b^2 - b^4)/(e^(2*d*x + 2*c) - 1)^3)/d
Time = 0.20 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.19 \[ \int (a+b \text {csch}(c+d x))^4 \, dx=a^4\,x-\frac {\frac {12\,a^2\,b^2}{d}+\frac {4\,a\,b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {\frac {4\,b^4}{d}+\frac {8\,a\,b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {8\,b^4}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}+\frac {4\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,b^3\,\sqrt {-d^2}-2\,a^3\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {4\,a^6\,b^2-4\,a^4\,b^4+a^2\,b^6}}\right )\,\sqrt {4\,a^6\,b^2-4\,a^4\,b^4+a^2\,b^6}}{\sqrt {-d^2}} \] Input:
int((a + b/sinh(c + d*x))^4,x)
Output:
a^4*x - ((12*a^2*b^2)/d + (4*a*b^3*exp(c + d*x))/d)/(exp(2*c + 2*d*x) - 1) - ((4*b^4)/d + (8*a*b^3*exp(c + d*x))/d)/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - (8*b^4)/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp( 6*c + 6*d*x) - 1)) + (4*atan((exp(d*x)*exp(c)*(a*b^3*(-d^2)^(1/2) - 2*a^3* b*(-d^2)^(1/2)))/(d*(a^2*b^6 - 4*a^4*b^4 + 4*a^6*b^2)^(1/2)))*(a^2*b^6 - 4 *a^4*b^4 + 4*a^6*b^2)^(1/2))/(-d^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 571, normalized size of antiderivative = 5.24 \[ \int (a+b \text {csch}(c+d x))^4 \, dx=\frac {-3 a^{4} d x +4 b^{4}-12 \,\mathrm {log}\left (e^{d x +c}-1\right ) a^{3} b +12 \,\mathrm {log}\left (e^{d x +c}+1\right ) a^{3} b -36 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3} b +36 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3} b +12 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3} b -6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a \,b^{3}-12 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3} b +6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a \,b^{3}+18 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a \,b^{3}-18 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a \,b^{3}-18 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a \,b^{3}+18 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a \,b^{3}-12 e^{6 d x +6 c} a^{2} b^{2}-12 e^{5 d x +5 c} a \,b^{3}+36 e^{2 d x +2 c} a^{2} b^{2}+12 e^{d x +c} a \,b^{3}-24 a^{2} b^{2}+3 e^{6 d x +6 c} a^{4} d x -9 e^{4 d x +4 c} a^{4} d x +9 e^{2 d x +2 c} a^{4} d x -12 e^{2 d x +2 c} b^{4}+36 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3} b -36 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3} b +6 \,\mathrm {log}\left (e^{d x +c}-1\right ) a \,b^{3}-6 \,\mathrm {log}\left (e^{d x +c}+1\right ) a \,b^{3}}{3 d \left (e^{6 d x +6 c}-3 e^{4 d x +4 c}+3 e^{2 d x +2 c}-1\right )} \] Input:
int((a+b*csch(d*x+c))^4,x)
Output:
(12*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)*a**3*b - 6*e**(6*c + 6*d*x)*log (e**(c + d*x) - 1)*a*b**3 - 12*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**3 *b + 6*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a*b**3 + 3*e**(6*c + 6*d*x)* a**4*d*x - 12*e**(6*c + 6*d*x)*a**2*b**2 - 12*e**(5*c + 5*d*x)*a*b**3 - 36 *e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a**3*b + 18*e**(4*c + 4*d*x)*log(e **(c + d*x) - 1)*a*b**3 + 36*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a**3*b - 18*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a*b**3 - 9*e**(4*c + 4*d*x)*a **4*d*x + 36*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a**3*b - 18*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a*b**3 - 36*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a**3*b + 18*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a*b**3 + 9*e**(2* c + 2*d*x)*a**4*d*x + 36*e**(2*c + 2*d*x)*a**2*b**2 - 12*e**(2*c + 2*d*x)* b**4 + 12*e**(c + d*x)*a*b**3 - 12*log(e**(c + d*x) - 1)*a**3*b + 6*log(e* *(c + d*x) - 1)*a*b**3 + 12*log(e**(c + d*x) + 1)*a**3*b - 6*log(e**(c + d *x) + 1)*a*b**3 - 3*a**4*d*x - 24*a**2*b**2 + 4*b**4)/(3*d*(e**(6*c + 6*d* x) - 3*e**(4*c + 4*d*x) + 3*e**(2*c + 2*d*x) - 1))