Integrand size = 15, antiderivative size = 73 \[ \int \cosh (a+b x) \coth ^3(c+b x) \, dx=-\frac {3 \text {arctanh}(\cosh (c+b x)) \cosh (a-c)}{2 b}+\frac {\cosh (a+b x)}{b}-\frac {\cosh (a-c) \coth (c+b x) \text {csch}(c+b x)}{2 b}-\frac {\text {csch}(c+b x) \sinh (a-c)}{b} \]
-3/2*arctanh(cosh(b*x+c))*cosh(a-c)/b+cosh(b*x+a)/b-1/2*cosh(a-c)*coth(b*x +c)*csch(b*x+c)/b-csch(b*x+c)*sinh(a-c)/b
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int \cosh (a+b x) \coth ^3(c+b x) \, dx=\frac {-12 \text {arctanh}\left (\cosh (c)+\sinh (c) \tanh \left (\frac {b x}{2}\right )\right ) \cosh (a-c)+(2 \cosh (a-2 c-b x)-5 \cosh (a+b x)+\cosh (a+2 c+3 b x)) \text {csch}^2(c+b x)}{4 b} \]
(-12*ArcTanh[Cosh[c] + Sinh[c]*Tanh[(b*x)/2]]*Cosh[a - c] + (2*Cosh[a - 2* c - b*x] - 5*Cosh[a + b*x] + Cosh[a + 2*c + 3*b*x])*Csch[c + b*x]^2)/(4*b)
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.133, Rules used = {6155, 3042, 26, 3091, 26, 3042, 26, 4257, 6156, 3042, 3086, 24, 6155, 3042, 26, 3118, 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.
\(\displaystyle \int \cosh (a+b x) \coth ^3(b x+c) \, dx\) |
\(\Big \downarrow \) 6155 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx+\cosh (a-c) \int \coth ^2(c+b x) \text {csch}(c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx+\cosh (a-c) \int -i \sec \left (i c+i b x-\frac {\pi }{2}\right ) \tan \left (i c+i b x-\frac {\pi }{2}\right )^2dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx-i \cosh (a-c) \int \sec \left (\frac {1}{2} (2 i c-\pi )+i b x\right ) \tan \left (\frac {1}{2} (2 i c-\pi )+i b x\right )^2dx\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx-i \cosh (a-c) \left (-\frac {1}{2} \int -i \text {csch}(c+b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx-i \cosh (a-c) \left (\frac {1}{2} i \int \text {csch}(c+b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx-i \cosh (a-c) \left (\frac {1}{2} i \int i \csc (i c+i b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx-i \cosh (a-c) \left (-\frac {1}{2} \int \csc (i c+i b x)dx-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \int \coth ^2(c+b x) \sinh (a+b x)dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 6156 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx+\sinh (a-c) \int \coth (c+b x) \text {csch}(c+b x)dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx+\sinh (a-c) \int \sec \left (i c+i b x-\frac {\pi }{2}\right ) \tan \left (i c+i b x-\frac {\pi }{2}\right )dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx-\frac {i \sinh (a-c) \int 1d(-i \text {csch}(c+b x))}{b}-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \int \cosh (a+b x) \coth (c+b x)dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 6155 |
\(\displaystyle \cosh (a-c) \int \text {csch}(c+b x)dx+\int \sinh (a+b x)dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \cosh (a-c) \int i \csc (i c+i b x)dx+\int -i \sin (i a+i b x)dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \cosh (a-c) \int \csc (i c+i b x)dx-i \int \sin (i a+i b x)dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle i \cosh (a-c) \int \csc (i c+i b x)dx-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}+\frac {\cosh (a+b x)}{b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\cosh (a-c) \text {arctanh}(\cosh (b x+c))}{b}-i \cosh (a-c) \left (-\frac {i \text {arctanh}(\cosh (b x+c))}{2 b}-\frac {i \coth (b x+c) \text {csch}(b x+c)}{2 b}\right )-\frac {\sinh (a-c) \text {csch}(b x+c)}{b}+\frac {\cosh (a+b x)}{b}\) |
-((ArcTanh[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b - I*Cosh[a - c ]*(((-1/2*I)*ArcTanh[Cosh[c + b*x]])/b - ((I/2)*Coth[c + b*x]*Csch[c + b*x ])/b) - (Csch[c + b*x]*Sinh[a - c])/b
3.2.60.3.1 Defintions of rubi rules used
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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cosh[v_]*Coth[w_]^(n_.), x_Symbol] :> Int[Sinh[v]*Coth[w]^(n - 1), x] + Simp[Cosh[v - w] Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ [w, v] && FreeQ[v - w, x]
Int[Coth[w_]^(n_.)*Sinh[v_], x_Symbol] :> Int[Cosh[v]*Coth[w]^(n - 1), x] + Simp[Sinh[v - w] Int[Csch[w]*Coth[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ [w, v] && FreeQ[v - w, x]
Leaf count of result is larger than twice the leaf count of optimal. \(227\) vs. \(2(69)=138\).
Time = 0.43 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.12
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left (-3 \,{\mathrm e}^{2 b x +4 a +2 c}+{\mathrm e}^{2 b x +2 a +4 c}+{\mathrm e}^{4 a}-3 \,{\mathrm e}^{2 a +2 c}\right )}{2 b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}\) | \(228\) |
1/2*exp(b*x+a)/b+1/2*exp(-b*x-a)/b+1/2*exp(b*x+a)*(-3*exp(2*b*x+4*a+2*c)+e xp(2*b*x+2*a+4*c)+exp(4*a)-3*exp(2*a+2*c))/b/(-exp(2*b*x+2*a+2*c)+exp(2*a) )^2+3/4*ln(exp(b*x+a)-exp(a-c))/b*exp(-a-c)*exp(2*a)+3/4*ln(exp(b*x+a)-exp (a-c))/b*exp(-a-c)*exp(2*c)-3/4*ln(exp(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2* a)-3/4*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. 2372 vs. \(2 (69) = 138\).
Time = 0.28 (sec) , antiderivative size = 2372, normalized size of antiderivative = 32.49 \[ \int \cosh (a+b x) \coth ^3(c+b x) \, dx=\text {Too large to display} \]
1/4*(2*cosh(b*x + c)^6*cosh(-a + c)^2 + 2*(cosh(-a + c)^2 - 2*cosh(-a + c) *sinh(-a + c) + sinh(-a + c)^2)*sinh(b*x + c)^6 + 12*(cosh(b*x + c)*cosh(- a + c)^2 - 2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) + cosh(b*x + c)*sinh( -a + c)^2)*sinh(b*x + c)^5 - 2*(5*cosh(-a + c)^2 - 2)*cosh(b*x + c)^4 + 2* (15*cosh(b*x + c)^2*cosh(-a + c)^2 + 5*(3*cosh(b*x + c)^2 - 1)*sinh(-a + c )^2 - 5*cosh(-a + c)^2 - 10*(3*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c) )*sinh(-a + c) + 2)*sinh(b*x + c)^4 + 8*(5*cosh(b*x + c)^3*cosh(-a + c)^2 + 5*(cosh(b*x + c)^3 - cosh(b*x + c))*sinh(-a + c)^2 - (5*cosh(-a + c)^2 - 2)*cosh(b*x + c) - 10*(cosh(b*x + c)^3*cosh(-a + c) - cosh(b*x + c)*cosh( -a + c))*sinh(-a + c))*sinh(b*x + c)^3 + 2*(2*cosh(-a + c)^2 - 5)*cosh(b*x + c)^2 + 2*(15*cosh(b*x + c)^4*cosh(-a + c)^2 - 6*(5*cosh(-a + c)^2 - 2)* cosh(b*x + c)^2 + (15*cosh(b*x + c)^4 - 30*cosh(b*x + c)^2 + 2)*sinh(-a + c)^2 + 2*cosh(-a + c)^2 - 2*(15*cosh(b*x + c)^4*cosh(-a + c) - 30*cosh(b*x + c)^2*cosh(-a + c) + 2*cosh(-a + c))*sinh(-a + c) - 5)*sinh(b*x + c)^2 + 2*(cosh(b*x + c)^6 - 5*cosh(b*x + c)^4 + 2*cosh(b*x + c)^2)*sinh(-a + c)^ 2 - 3*((cosh(-a + c)^2 + 1)*cosh(b*x + c)^5 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 + 1)*sinh(b*x + c)^5 - 5*(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 - 2*(cosh(-a + c)^2 + 1)*cosh(b*x + c)^3 + 2*(5*(cosh(-a + c)^2 + 1)*cosh(b*x + c)^2 + (5*cosh(b*x + c)^2...
\[ \int \cosh (a+b x) \coth ^3(c+b x) \, dx=\int \cosh {\left (a + b x \right )} \coth ^{3}{\left (b x + c \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (69) = 138\).
Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.52 \[ \int \cosh (a+b x) \coth ^3(c+b x) \, dx=-\frac {3 \, {\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{4 \, b} + \frac {3 \, {\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{4 \, b} + \frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {{\left (5 \, e^{\left (2 \, a + 2 \, c\right )} - e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} - {\left (2 \, e^{\left (4 \, a\right )} - 3 \, e^{\left (2 \, a + 2 \, c\right )}\right )} e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, c\right )}}{2 \, b {\left (e^{\left (-b x - a + 4 \, c\right )} - 2 \, e^{\left (-3 \, b x - a + 2 \, c\right )} + e^{\left (-5 \, b x - a\right )}\right )}} \]
-3/4*(e^(2*a) + e^(2*c))*e^(-a - c)*log(e^(-b*x) + e^c)/b + 3/4*(e^(2*a) + e^(2*c))*e^(-a - c)*log(e^(-b*x) - e^c)/b + 1/2*e^(-b*x - a)/b - 1/2*((5* e^(2*a + 2*c) - e^(4*c))*e^(-2*b*x - 2*a) - (2*e^(4*a) - 3*e^(2*a + 2*c))* e^(-4*b*x - 4*a) - e^(4*c))/(b*(e^(-b*x - a + 4*c) - 2*e^(-3*b*x - a + 2*c ) + e^(-5*b*x - a)))
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (69) = 138\).
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.29 \[ \int \cosh (a+b x) \coth ^3(c+b x) \, dx=-\frac {3 \, {\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + a + c\right )} + e^{a}\right ) - 3 \, {\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + a + c\right )} - e^{a} \right |}\right ) + \frac {2 \, {\left (3 \, e^{\left (3 \, b x + 5 \, a + 2 \, c\right )} - e^{\left (3 \, b x + 3 \, a + 4 \, c\right )} - e^{\left (b x + 5 \, a\right )} + 3 \, e^{\left (b x + 3 \, a + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} - e^{\left (2 \, a\right )}\right )}^{2}} - 2 \, e^{\left (b x + a\right )} - 2 \, e^{\left (-b x - a\right )}}{4 \, b} \]
-1/4*(3*(e^(2*a + c) + e^(3*c))*e^(-a - 2*c)*log(e^(b*x + a + c) + e^a) - 3*(e^(2*a + c) + e^(3*c))*e^(-a - 2*c)*log(abs(e^(b*x + a + c) - e^a)) + 2 *(3*e^(3*b*x + 5*a + 2*c) - e^(3*b*x + 3*a + 4*c) - e^(b*x + 5*a) + 3*e^(b *x + 3*a + 2*c))/(e^(2*b*x + 2*a + 2*c) - e^(2*a))^2 - 2*e^(b*x + a) - 2*e ^(-b*x - a))/b
Timed out. \[ \int \cosh (a+b x) \coth ^3(c+b x) \, dx=\int \mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {coth}\left (c+b\,x\right )}^3 \,d x \]