Integrand size = 23, antiderivative size = 98 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^2 (a-3 b) \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}-\frac {3 a (a-b) b \tanh (c+d x)}{d}-\frac {(3 a-b) b^2 \tanh ^3(c+d x)}{3 d}-\frac {b^3 \tanh ^5(c+d x)}{5 d} \] Output:
a^2*(a-3*b)*coth(d*x+c)/d-1/3*a^3*coth(d*x+c)^3/d-3*a*(a-b)*b*tanh(d*x+c)/ d-1/3*(3*a-b)*b^2*tanh(d*x+c)^3/d-1/5*b^3*tanh(d*x+c)^5/d
Time = 2.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-5 a^2 \coth (c+d x) \left (-2 a+9 b+a \text {csch}^2(c+d x)\right )+b \left (-45 a^2+30 a b+2 b^2+b (15 a+b) \text {sech}^2(c+d x)-3 b^2 \text {sech}^4(c+d x)\right ) \tanh (c+d x)}{15 d} \] Input:
Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(-5*a^2*Coth[c + d*x]*(-2*a + 9*b + a*Csch[c + d*x]^2) + b*(-45*a^2 + 30*a *b + 2*b^2 + b*(15*a + b)*Sech[c + d*x]^2 - 3*b^2*Sech[c + d*x]^4)*Tanh[c + d*x])/(15*d)
Time = 0.51 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4146, 355, 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.
| \(\displaystyle \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \tan (i c+i d x)^2\right )^3}{\sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 355 |
| \(\displaystyle \frac {\int \left (a^3 \coth ^4(c+d x)-a^2 (a-3 b) \coth ^2(c+d x)-b^3 \tanh ^4(c+d x)-(3 a-b) b^2 \tanh ^2(c+d x)-3 a (a-b) b\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{3} a^3 \coth ^3(c+d x)+a^2 (a-3 b) \coth (c+d x)-\frac {1}{3} b^2 (3 a-b) \tanh ^3(c+d x)-3 a b (a-b) \tanh (c+d x)-\frac {1}{5} b^3 \tanh ^5(c+d x)}{d}\) |
Input:
Int[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^2)^3,x]
Output:
(a^2*(a - 3*b)*Coth[c + d*x] - (a^3*Coth[c + d*x]^3)/3 - 3*a*(a - b)*b*Tan h[c + d*x] - ((3*a - b)*b^2*Tanh[c + d*x]^3)/3 - (b^3*Tanh[c + d*x]^5)/5)/ d
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Time = 37.96 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.39
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )+3 b^{2} a \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )}{d}\) | \(136\) |
| default | \(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )+3 b^{2} a \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )}{d}\) | \(136\) |
| risch | \(-\frac {4 \left (65 b^{3} {\mathrm e}^{8 d x +8 c}-44 b^{3} {\mathrm e}^{6 d x +6 c}+17 b^{3} {\mathrm e}^{4 d x +4 c}-10 \,{\mathrm e}^{2 d x +2 c} a^{3}-15 b^{2} a -5 a^{3}+45 a^{2} b +45 a^{2} b \,{\mathrm e}^{12 d x +12 c}+90 a^{2} b \,{\mathrm e}^{10 d x +10 c}-30 a \,b^{2} {\mathrm e}^{10 d x +10 c}-45 a^{2} b \,{\mathrm e}^{8 d x +8 c}-105 a \,b^{2} {\mathrm e}^{8 d x +8 c}-180 a^{2} b \,{\mathrm e}^{6 d x +6 c}+60 a \,b^{2} {\mathrm e}^{6 d x +6 c}-45 a^{2} b \,{\mathrm e}^{4 d x +4 c}+75 a \,b^{2} {\mathrm e}^{4 d x +4 c}+90 a^{2} b \,{\mathrm e}^{2 d x +2 c}-30 a \,b^{2} {\mathrm e}^{2 d x +2 c}-b^{3}+100 a^{3} {\mathrm e}^{6 d x +6 c}-50 b^{3} {\mathrm e}^{10 d x +10 c}+125 a^{3} {\mathrm e}^{8 d x +8 c}+15 b^{3} {\mathrm e}^{12 d x +12 c}-2 b^{3} {\mathrm e}^{2 d x +2 c}+45 a \,b^{2} {\mathrm e}^{12 d x +12 c}+15 a^{3} {\mathrm e}^{12 d x +12 c}+25 \,{\mathrm e}^{4 d x +4 c} a^{3}+70 \,{\mathrm e}^{10 d x +10 c} a^{3}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\) | \(403\) |
Input:
int(csch(d*x+c)^4*(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(-1/sinh(d*x+c)/cosh( d*x+c)-2*tanh(d*x+c))+3*b^2*a*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)+b^3*(-1/ 4*sinh(d*x+c)/cosh(d*x+c)^5+1/4*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2 )*tanh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (92) = 184\).
Time = 0.09 (sec) , antiderivative size = 925, normalized size of antiderivative = 9.44 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")
Output:
-8/15*((5*a^3 + 45*a^2*b + 15*a*b^2 + 7*b^3)*cosh(d*x + c)^6 + 12*(5*a^3 + 15*a*b^2 + 4*b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (5*a^3 + 45*a^2*b + 15* a*b^2 + 7*b^3)*sinh(d*x + c)^6 + 2*(15*a^3 + 45*a^2*b - 15*a*b^2 - 13*b^3) *cosh(d*x + c)^4 + (30*a^3 + 90*a^2*b - 30*a*b^2 - 26*b^3 + 15*(5*a^3 + 45 *a^2*b + 15*a*b^2 + 7*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(5*(5*a^3 + 15*a*b^2 + 4*b^3)*cosh(d*x + c)^3 + 4*(5*a^3 - 3*b^3)*cosh(d*x + c))*sin h(d*x + c)^3 + 50*a^3 - 90*a^2*b + 30*a*b^2 - 22*b^3 + (75*a^3 - 45*a^2*b - 15*a*b^2 + 41*b^3)*cosh(d*x + c)^2 + (15*(5*a^3 + 45*a^2*b + 15*a*b^2 + 7*b^3)*cosh(d*x + c)^4 + 75*a^3 - 45*a^2*b - 15*a*b^2 + 41*b^3 + 12*(15*a^ 3 + 45*a^2*b - 15*a*b^2 - 13*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(3* (5*a^3 + 15*a*b^2 + 4*b^3)*cosh(d*x + c)^5 + 8*(5*a^3 - 3*b^3)*cosh(d*x + c)^3 + (25*a^3 - 45*a*b^2 + 12*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh( d*x + c)^10 + 10*d*cosh(d*x + c)*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 + 2* d*cosh(d*x + c)^8 + (45*d*cosh(d*x + c)^2 + 2*d)*sinh(d*x + c)^8 + 8*(15*d *cosh(d*x + c)^3 + 2*d*cosh(d*x + c))*sinh(d*x + c)^7 - 3*d*cosh(d*x + c)^ 6 + (210*d*cosh(d*x + c)^4 + 56*d*cosh(d*x + c)^2 - 3*d)*sinh(d*x + c)^6 + 2*(126*d*cosh(d*x + c)^5 + 56*d*cosh(d*x + c)^3 - 3*d*cosh(d*x + c))*sinh (d*x + c)^5 - 8*d*cosh(d*x + c)^4 + (210*d*cosh(d*x + c)^6 + 140*d*cosh(d* x + c)^4 - 45*d*cosh(d*x + c)^2 - 8*d)*sinh(d*x + c)^4 + 4*(30*d*cosh(d*x + c)^7 + 28*d*cosh(d*x + c)^5 - 5*d*cosh(d*x + c)^3 - 4*d*cosh(d*x + c)...
\[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)**4*(a+b*tanh(d*x+c)**2)**3,x)
Output:
Integral((a + b*tanh(c + d*x)**2)**3*csch(c + d*x)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (92) = 184\).
Time = 0.05 (sec) , antiderivative size = 493, normalized size of antiderivative = 5.03 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {4}{15} \, b^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} - \frac {5 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {15 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}} + \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + 4 \, a b^{2} {\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}{3} \, a^{3} {\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 {12 \, a^{2} b}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")
Output:
4/15*b^3*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) - 5* e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d *x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 15*e^(-6*d*x - 6*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10* d*x - 10*c) + 1))) + 4*a*b^2*(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))) + 4/3*a^3*(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))) + 12* a^2*b/(d*(e^(-4*d*x - 4*c) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (92) = 184\).
Time = 0.25 (sec) , antiderivative size = 257, normalized size of antiderivative = 2.62 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {2 \, {\left (\frac {5 \, {\left (9 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 18 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{3} + 9 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac {45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 90 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 30 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 210 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 10 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 150 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 10 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b - 30 \, a b^{2} - 2 \, b^{3}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}\right )}}{15 \, d} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")
Output:
-2/15*(5*(9*a^2*b*e^(4*d*x + 4*c) + 6*a^3*e^(2*d*x + 2*c) - 18*a^2*b*e^(2* d*x + 2*c) - 2*a^3 + 9*a^2*b)/(e^(2*d*x + 2*c) - 1)^3 - (45*a^2*b*e^(8*d*x + 8*c) + 180*a^2*b*e^(6*d*x + 6*c) - 90*a*b^2*e^(6*d*x + 6*c) - 30*b^3*e^ (6*d*x + 6*c) + 270*a^2*b*e^(4*d*x + 4*c) - 210*a*b^2*e^(4*d*x + 4*c) + 10 *b^3*e^(4*d*x + 4*c) + 180*a^2*b*e^(2*d*x + 2*c) - 150*a*b^2*e^(2*d*x + 2* c) - 10*b^3*e^(2*d*x + 2*c) + 45*a^2*b - 30*a*b^2 - 2*b^3)/(e^(2*d*x + 2*c ) + 1)^5)/d
Time = 1.54 (sec) , antiderivative size = 622, normalized size of antiderivative = 6.35 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\frac {2\,\left (9\,a^2\,b-12\,a\,b^2+4\,b^3\right )}{15\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {\frac {2\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+4\,b^3\right )}{5\,d}-\frac {6\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {\frac {6\,a^2\,b}{5\,d}-\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+4\,b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {\frac {2\,\left (-3\,a^2\,b+3\,a\,b^2+b^3\right )}{5\,d}-\frac {6\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {4\,a^3}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a^3}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {6\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}+\frac {6\,a^2\,b}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int((a + b*tanh(c + d*x)^2)^3/sinh(c + d*x)^4,x)
Output:
((2*(9*a^2*b - 12*a*b^2 + 4*b^3))/(15*d) - (4*exp(2*c + 2*d*x)*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) + (6*a^2*b*exp(4*c + 4*d*x))/(5*d))/(3*exp(2*c + 2*d *x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - ((2*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) + (6*exp(4*c + 4*d*x)*(3*a*b^2 - 3*a^2*b + b^3))/(5*d) - (2* exp(2*c + 2*d*x)*(9*a^2*b - 12*a*b^2 + 4*b^3))/(5*d) - (6*a^2*b*exp(6*c + 6*d*x))/(5*d))/(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d* x) + exp(8*c + 8*d*x) + 1) + ((6*a^2*b)/(5*d) - (8*exp(6*c + 6*d*x)*(3*a*b ^2 - 3*a^2*b + b^3))/(5*d) - (8*exp(2*c + 2*d*x)*(3*a*b^2 - 3*a^2*b + b^3) )/(5*d) + (4*exp(4*c + 4*d*x)*(9*a^2*b - 12*a*b^2 + 4*b^3))/(5*d) + (6*a^2 *b*exp(8*c + 8*d*x))/(5*d))/(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10 *exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1) - ((2*(3* a*b^2 - 3*a^2*b + b^3))/(5*d) - (6*a^2*b*exp(2*c + 2*d*x))/(5*d))/(2*exp(2 *c + 2*d*x) + exp(4*c + 4*d*x) + 1) - (4*a^3)/(d*(exp(4*c + 4*d*x) - 2*exp (2*c + 2*d*x) + 1)) - (8*a^3)/(3*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x ) + exp(6*c + 6*d*x) - 1)) - (6*a^2*b)/(d*(exp(2*c + 2*d*x) - 1)) + (6*a^2 *b)/(5*d*(exp(2*c + 2*d*x) + 1))
Time = 0.26 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.94 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-20 e^{4 d x +4 c} a \,b^{2}+\frac {4 b^{3}}{15}-24 e^{10 d x +10 c} a^{2} b +8 e^{10 d x +10 c} a \,b^{2}+12 e^{8 d x +8 c} a^{2} b +28 e^{8 d x +8 c} a \,b^{2}+48 e^{6 d x +6 c} a^{2} b -16 e^{6 d x +6 c} a \,b^{2}+12 e^{4 d x +4 c} a^{2} b +4 a \,b^{2}-\frac {68 e^{4 d x +4 c} b^{3}}{15}+8 e^{2 d x +2 c} a \,b^{2}+\frac {4 a^{3}}{3}+\frac {8 e^{2 d x +2 c} b^{3}}{15}-12 e^{12 d x +12 c} a^{2} b -4 e^{12 d x +12 c} a^{3}-4 e^{12 d x +12 c} b^{3}-24 e^{2 d x +2 c} a^{2} b -12 e^{12 d x +12 c} a \,b^{2}-12 a^{2} b -\frac {56 e^{10 d x +10 c} a^{3}}{3}+\frac {40 e^{10 d x +10 c} b^{3}}{3}-\frac {100 e^{8 d x +8 c} a^{3}}{3}-\frac {52 e^{8 d x +8 c} b^{3}}{3}-\frac {80 e^{6 d x +6 c} a^{3}}{3}+\frac {176 e^{6 d x +6 c} b^{3}}{15}-\frac {20 e^{4 d x +4 c} a^{3}}{3}+\frac {8 e^{2 d x +2 c} a^{3}}{3}}{d \left (e^{16 d x +16 c}+2 e^{14 d x +14 c}-2 e^{12 d x +12 c}-6 e^{10 d x +10 c}+6 e^{6 d x +6 c}+2 e^{4 d x +4 c}-2 e^{2 d x +2 c}-1\right )} \] Input:
int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^2)^3,x)
Output:
(4*( - 15*e**(12*c + 12*d*x)*a**3 - 45*e**(12*c + 12*d*x)*a**2*b - 45*e**( 12*c + 12*d*x)*a*b**2 - 15*e**(12*c + 12*d*x)*b**3 - 70*e**(10*c + 10*d*x) *a**3 - 90*e**(10*c + 10*d*x)*a**2*b + 30*e**(10*c + 10*d*x)*a*b**2 + 50*e **(10*c + 10*d*x)*b**3 - 125*e**(8*c + 8*d*x)*a**3 + 45*e**(8*c + 8*d*x)*a **2*b + 105*e**(8*c + 8*d*x)*a*b**2 - 65*e**(8*c + 8*d*x)*b**3 - 100*e**(6 *c + 6*d*x)*a**3 + 180*e**(6*c + 6*d*x)*a**2*b - 60*e**(6*c + 6*d*x)*a*b** 2 + 44*e**(6*c + 6*d*x)*b**3 - 25*e**(4*c + 4*d*x)*a**3 + 45*e**(4*c + 4*d *x)*a**2*b - 75*e**(4*c + 4*d*x)*a*b**2 - 17*e**(4*c + 4*d*x)*b**3 + 10*e* *(2*c + 2*d*x)*a**3 - 90*e**(2*c + 2*d*x)*a**2*b + 30*e**(2*c + 2*d*x)*a*b **2 + 2*e**(2*c + 2*d*x)*b**3 + 5*a**3 - 45*a**2*b + 15*a*b**2 + b**3))/(1 5*d*(e**(16*c + 16*d*x) + 2*e**(14*c + 14*d*x) - 2*e**(12*c + 12*d*x) - 6* e**(10*c + 10*d*x) + 6*e**(6*c + 6*d*x) + 2*e**(4*c + 4*d*x) - 2*e**(2*c + 2*d*x) - 1))