Integrand size = 15, antiderivative size = 61 \[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=-\frac {\cosh (a-c) \text {csch}^3(c+b x)}{3 b}+\frac {1}{2} \left (\frac {\text {arctanh}(\cosh (c+b x))}{b}-\frac {\coth (c+b x) \text {csch}(c+b x)}{b}\right ) \sinh (a-c) \] Output:
-1/3*cosh(a-c)*csch(b*x+c)^3/b+1/2*(arctanh(cosh(b*x+c))/b-coth(b*x+c)*csc h(b*x+c)/b)*sinh(a-c)
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07 \[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=\frac {12 \text {arctanh}\left (\cosh (c)+\sinh (c) \tanh \left (\frac {b x}{2}\right )\right ) \sinh (a-c)-\text {csch}^3(c+b x) (4 \cosh (a-c)+3 \sinh (a-c) \sinh (2 (c+b x)))}{12 b} \] Input:
Integrate[Cosh[a + b*x]*Csch[c + b*x]^4,x]
Output:
(12*ArcTanh[Cosh[c] + Sinh[c]*Tanh[(b*x)/2]]*Sinh[a - c] - Csch[c + b*x]^3 *(4*Cosh[a - c] + 3*Sinh[a - c]*Sinh[2*(c + b*x)]))/(12*b)
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {6159, 3042, 25, 26, 3086, 15, 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.
\(\displaystyle \int \cosh (a+b x) \text {csch}^4(b x+c) \, dx\) |
\(\Big \downarrow \) 6159 |
\(\displaystyle \sinh (a-c) \int \text {csch}^3(c+b x)dx+\cosh (a-c) \int \coth (c+b x) \text {csch}^3(c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh (a-c) \int -i \csc (i c+i b x)^3dx+\cosh (a-c) \int -\sec \left (i c+i b x-\frac {\pi }{2}\right )^3 \tan \left (i c+i b x-\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \sinh (a-c) \int -i \csc (i c+i b x)^3dx-\cosh (a-c) \int \sec \left (\frac {1}{2} (2 i c-\pi )+i b x\right )^3 \tan \left (\frac {1}{2} (2 i c-\pi )+i b x\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \sinh (a-c) \int \csc (i c+i b x)^3dx-\cosh (a-c) \int \sec \left (\frac {1}{2} (2 i c-\pi )+i b x\right )^3 \tan \left (\frac {1}{2} (2 i c-\pi )+i b x\right )dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {i \cosh (a-c) \int -\text {csch}^2(c+b x)d(-i \text {csch}(c+b x))}{b}-i \sinh (a-c) \int \csc (i c+i b x)^3dx\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\cosh (a-c) \text {csch}^3(b x+c)}{3 b}-i \sinh (a-c) \int \csc (i c+i b x)^3dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle -\frac {\cosh (a-c) \text {csch}^3(b x+c)}{3 b}-i \sinh (a-c) \left (\frac {1}{2} \int -i \text {csch}(c+b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\cosh (a-c) \text {csch}^3(b x+c)}{3 b}-i \sinh (a-c) \left (-\frac {1}{2} i \int \text {csch}(c+b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cosh (a-c) \text {csch}^3(b x+c)}{3 b}-i \sinh (a-c) \left (-\frac {1}{2} i \int i \csc (i c+i b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\cosh (a-c) \text {csch}^3(b x+c)}{3 b}-i \sinh (a-c) \left (\frac {1}{2} \int \csc (i c+i b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\cosh (a-c) \text {csch}^3(b x+c)}{3 b}-i \sinh (a-c) \left (\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
Input:
Int[Cosh[a + b*x]*Csch[c + b*x]^4,x]
Output:
-1/3*(Cosh[a - c]*Csch[c + b*x]^3)/b - I*(((I/2)*ArcTanh[Cosh[c + b*x]])/b - ((I/2)*Coth[c + b*x]*Csch[c + b*x])/b)*Sinh[a - c]
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 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[Cosh[v_]*Csch[w_]^(n_.), x_Symbol] :> Simp[Cosh[v - w] Int[Coth[w]*Cs ch[w]^(n - 1), x], x] + Simp[Sinh[v - w] Int[Csch[w]^(n - 1), x], x] /; G tQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Leaf count of result is larger than twice the leaf count of optimal. \(234\) vs. \(2(57)=114\).
Time = 2.04 (sec) , antiderivative size = 235, normalized size of antiderivative = 3.85
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (-3 \,{\mathrm e}^{4 b x +6 a +4 c}+3 \,{\mathrm e}^{4 b x +4 a +6 c}-8 \,{\mathrm e}^{2 b x +6 a +2 c}-8 \,{\mathrm e}^{2 b x +4 a +4 c}+3 \,{\mathrm e}^{6 a}-3 \,{\mathrm e}^{4 a +2 c}\right )}{6 b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{3}}+\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}-\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}+\frac {\ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}\) | \(235\) |
Input:
int(cosh(b*x+a)*csch(b*x+c)^4,x,method=_RETURNVERBOSE)
Output:
-1/6*exp(b*x+a)*(-3*exp(4*b*x+6*a+4*c)+3*exp(4*b*x+4*a+6*c)-8*exp(2*b*x+6* a+2*c)-8*exp(2*b*x+4*a+4*c)+3*exp(6*a)-3*exp(4*a+2*c))/b/(-exp(2*b*x+2*a+2 *c)+exp(2*a))^3+1/4*ln(exp(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2*a)-1/4*ln(ex p(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2*c)-1/4*ln(exp(b*x+a)-exp(a-c))/b*exp( -a-c)*exp(2*a)+1/4*ln(exp(b*x+a)-exp(a-c))/b*exp(-a-c)*exp(2*c)
Leaf count of result is larger than twice the leaf count of optimal. 2696 vs. \(2 (57) = 114\).
Time = 0.10 (sec) , antiderivative size = 2696, normalized size of antiderivative = 44.20 \[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=\text {Too large to display} \] Input:
integrate(cosh(b*x+a)*csch(b*x+c)^4,x, algorithm="fricas")
Output:
-1/12*(6*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^5 + 6*(cosh(-a + c)^2 - 2*cosh (-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^5 - 30*(2*cosh(b *x + c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(- a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^4 + 16*(cosh(-a + c)^2 + 1)*cos h(b*x + c)^3 + 4*(15*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (15*cosh(b*x + c)^2 + 4)*sinh(-a + c)^2 + 4*cosh(-a + c)^2 - 2*(15*cosh(b*x + c)^2*cosh( -a + c) + 4*cosh(-a + c))*sinh(-a + c) + 4)*sinh(b*x + c)^3 + 12*(5*(cosh( -a + c)^2 - 1)*cosh(b*x + c)^3 + (5*cosh(b*x + c)^3 + 4*cosh(b*x + c))*sin h(-a + c)^2 + 4*(cosh(-a + c)^2 + 1)*cosh(b*x + c) - 2*(5*cosh(b*x + c)^3* cosh(-a + c) + 4*cosh(b*x + c)*cosh(-a + c))*sinh(-a + c))*sinh(b*x + c)^2 + 2*(3*cosh(b*x + c)^5 + 8*cosh(b*x + c)^3 - 3*cosh(b*x + c))*sinh(-a + c )^2 - 6*(cosh(-a + c)^2 - 1)*cosh(b*x + c) - 3*((cosh(-a + c)^2 - 1)*cosh( b*x + c)^6 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^ 2 - 1)*sinh(b*x + c)^6 - 6*(2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - co sh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^5 - 3*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^4 + 3*(5*(cosh(-a + c)^2 - 1 )*cosh(b*x + c)^2 + (5*cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^ 2 - 2*(5*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c))*sinh(-a + c) + 1)*si nh(b*x + c)^4 + 4*(5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^3 + (5*cosh(b*x + c)^3 - 3*cosh(b*x + c))*sinh(-a + c)^2 - 3*(cosh(-a + c)^2 - 1)*cosh(b*...
\[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=\int \cosh {\left (a + b x \right )} \operatorname {csch}^{4}{\left (b x + c \right )}\, dx \] Input:
integrate(cosh(b*x+a)*csch(b*x+c)**4,x)
Output:
Integral(cosh(a + b*x)*csch(b*x + c)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (57) = 114\).
Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 3.13 \[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{4 \, b} - \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{4 \, b} + \frac {3 \, {\left (e^{\left (2 \, a + 4 \, c\right )} - e^{\left (6 \, c\right )}\right )} e^{\left (-b x - a\right )} + 8 \, {\left (e^{\left (4 \, a + 2 \, c\right )} + e^{\left (2 \, a + 4 \, c\right )}\right )} e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, {\left (e^{\left (6 \, a\right )} - e^{\left (4 \, a + 2 \, c\right )}\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{6 \, b {\left (3 \, e^{\left (-2 \, b x + 4 \, c\right )} - 3 \, e^{\left (-4 \, b x + 2 \, c\right )} + e^{\left (-6 \, b x\right )} - e^{\left (6 \, c\right )}\right )}} \] Input:
integrate(cosh(b*x+a)*csch(b*x+c)^4,x, algorithm="maxima")
Output:
1/4*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) + e^c)/b - 1/4*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) - e^c)/b + 1/6*(3*(e^(2*a + 4*c) - e^(6*c ))*e^(-b*x - a) + 8*(e^(4*a + 2*c) + e^(2*a + 4*c))*e^(-3*b*x - 3*a) - 3*( e^(6*a) - e^(4*a + 2*c))*e^(-5*b*x - 5*a))/(b*(3*e^(-2*b*x + 4*c) - 3*e^(- 4*b*x + 2*c) + e^(-6*b*x) - e^(6*c)))
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (57) = 114\).
Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.70 \[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=\frac {{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right )}{4 \, b} - \frac {{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right )}{4 \, b} - \frac {{\left (3 \, e^{\left (5 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (5 \, b x + 6 \, c\right )} + 8 \, e^{\left (3 \, b x + 2 \, a + 2 \, c\right )} + 8 \, e^{\left (3 \, b x + 4 \, c\right )} - 3 \, e^{\left (b x + 2 \, a\right )} + 3 \, e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{6 \, b {\left (e^{\left (2 \, b x + 2 \, c\right )} - 1\right )}^{3}} \] Input:
integrate(cosh(b*x+a)*csch(b*x+c)^4,x, algorithm="giac")
Output:
1/4*(e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(e^(b*x + c) + 1)/b - 1/4*(e^( 2*a + c) - e^(3*c))*e^(-a - 2*c)*log(abs(e^(b*x + c) - 1))/b - 1/6*(3*e^(5 *b*x + 2*a + 4*c) - 3*e^(5*b*x + 6*c) + 8*e^(3*b*x + 2*a + 2*c) + 8*e^(3*b *x + 4*c) - 3*e^(b*x + 2*a) + 3*e^(b*x + 2*c))*e^(-a)/(b*(e^(2*b*x + 2*c) - 1)^3)
Timed out. \[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{{\mathrm {sinh}\left (c+b\,x\right )}^4} \,d x \] Input:
int(cosh(a + b*x)/sinh(c + b*x)^4,x)
Output:
int(cosh(a + b*x)/sinh(c + b*x)^4, x)
Time = 0.23 (sec) , antiderivative size = 478, normalized size of antiderivative = 7.84 \[ \int \cosh (a+b x) \text {csch}^4(c+b x) \, dx=\frac {-3 e^{6 b x +2 a +6 c} \mathrm {log}\left (e^{b x +c}-1\right )+3 e^{6 b x +2 a +6 c} \mathrm {log}\left (e^{b x +c}+1\right )+3 e^{6 b x +8 c} \mathrm {log}\left (e^{b x +c}-1\right )-3 e^{6 b x +8 c} \mathrm {log}\left (e^{b x +c}+1\right )-6 e^{5 b x +2 a +5 c}+6 e^{5 b x +7 c}+9 e^{4 b x +2 a +4 c} \mathrm {log}\left (e^{b x +c}-1\right )-9 e^{4 b x +2 a +4 c} \mathrm {log}\left (e^{b x +c}+1\right )-9 e^{4 b x +6 c} \mathrm {log}\left (e^{b x +c}-1\right )+9 e^{4 b x +6 c} \mathrm {log}\left (e^{b x +c}+1\right )-16 e^{3 b x +2 a +3 c}-16 e^{3 b x +5 c}-9 e^{2 b x +2 a +2 c} \mathrm {log}\left (e^{b x +c}-1\right )+9 e^{2 b x +2 a +2 c} \mathrm {log}\left (e^{b x +c}+1\right )+9 e^{2 b x +4 c} \mathrm {log}\left (e^{b x +c}-1\right )-9 e^{2 b x +4 c} \mathrm {log}\left (e^{b x +c}+1\right )+6 e^{b x +2 a +c}-6 e^{b x +3 c}+3 e^{2 a} \mathrm {log}\left (e^{b x +c}-1\right )-3 e^{2 a} \mathrm {log}\left (e^{b x +c}+1\right )-3 e^{2 c} \mathrm {log}\left (e^{b x +c}-1\right )+3 e^{2 c} \mathrm {log}\left (e^{b x +c}+1\right )}{12 e^{a +c} b \left (e^{6 b x +6 c}-3 e^{4 b x +4 c}+3 e^{2 b x +2 c}-1\right )} \] Input:
int(cosh(b*x+a)*csch(b*x+c)^4,x)
Output:
( - 3*e**(2*a + 6*b*x + 6*c)*log(e**(b*x + c) - 1) + 3*e**(2*a + 6*b*x + 6 *c)*log(e**(b*x + c) + 1) + 3*e**(6*b*x + 8*c)*log(e**(b*x + c) - 1) - 3*e **(6*b*x + 8*c)*log(e**(b*x + c) + 1) - 6*e**(2*a + 5*b*x + 5*c) + 6*e**(5 *b*x + 7*c) + 9*e**(2*a + 4*b*x + 4*c)*log(e**(b*x + c) - 1) - 9*e**(2*a + 4*b*x + 4*c)*log(e**(b*x + c) + 1) - 9*e**(4*b*x + 6*c)*log(e**(b*x + c) - 1) + 9*e**(4*b*x + 6*c)*log(e**(b*x + c) + 1) - 16*e**(2*a + 3*b*x + 3*c ) - 16*e**(3*b*x + 5*c) - 9*e**(2*a + 2*b*x + 2*c)*log(e**(b*x + c) - 1) + 9*e**(2*a + 2*b*x + 2*c)*log(e**(b*x + c) + 1) + 9*e**(2*b*x + 4*c)*log(e **(b*x + c) - 1) - 9*e**(2*b*x + 4*c)*log(e**(b*x + c) + 1) + 6*e**(2*a + b*x + c) - 6*e**(b*x + 3*c) + 3*e**(2*a)*log(e**(b*x + c) - 1) - 3*e**(2*a )*log(e**(b*x + c) + 1) - 3*e**(2*c)*log(e**(b*x + c) - 1) + 3*e**(2*c)*lo g(e**(b*x + c) + 1))/(12*e**(a + c)*b*(e**(6*b*x + 6*c) - 3*e**(4*b*x + 4* c) + 3*e**(2*b*x + 2*c) - 1))