Integrand size = 15, antiderivative size = 44 \[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx=-\frac {3 \text {arctanh}(\cosh (a+b x))}{2 b}+\frac {\cosh (a+b x)}{b}-\frac {\coth (a+b x) \text {csch}(a+b x)}{2 b} \] Output:
-3/2*arctanh(cosh(b*x+a))/b+cosh(b*x+a)/b-1/2*coth(b*x+a)*csch(b*x+a)/b
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.93 \[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx=\frac {\cosh (a+b x)}{b}-\frac {\text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {3 \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {3 \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {\text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b} \] Input:
Integrate[Cosh[a + b*x]*Coth[a + b*x]^3,x]
Output:
Cosh[a + b*x]/b - Csch[(a + b*x)/2]^2/(8*b) - (3*Log[Cosh[(a + b*x)/2]])/( 2*b) + (3*Log[Sinh[(a + b*x)/2]])/(2*b) - Sech[(a + b*x)/2]^2/(8*b)
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 26, 3072, 252, 262, 219}
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.
\(\displaystyle \int \cosh (a+b x) \coth ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin \left (i a+i b x+\frac {\pi }{2}\right ) \tan \left (i a+i b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin \left (\frac {1}{2} (2 i a+\pi )+i b x\right ) \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^3dx\) |
\(\Big \downarrow \) 3072 |
\(\displaystyle \frac {\int \frac {\cosh ^4(a+b x)}{\left (1-\cosh ^2(a+b x)\right )^2}d\cosh (a+b x)}{b}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {\frac {\cosh ^3(a+b x)}{2 \left (1-\cosh ^2(a+b x)\right )}-\frac {3}{2} \int \frac {\cosh ^2(a+b x)}{1-\cosh ^2(a+b x)}d\cosh (a+b x)}{b}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\frac {\cosh ^3(a+b x)}{2 \left (1-\cosh ^2(a+b x)\right )}-\frac {3}{2} \left (\int \frac {1}{1-\cosh ^2(a+b x)}d\cosh (a+b x)-\cosh (a+b x)\right )}{b}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\cosh ^3(a+b x)}{2 \left (1-\cosh ^2(a+b x)\right )}-\frac {3}{2} (\text {arctanh}(\cosh (a+b x))-\cosh (a+b x))}{b}\) |
Input:
Int[Cosh[a + b*x]*Coth[a + b*x]^3,x]
Output:
((-3*(ArcTanh[Cosh[a + b*x]] - Cosh[a + b*x]))/2 + Cosh[a + b*x]^3/(2*(1 - Cosh[a + b*x]^2)))/b
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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\frac {\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{2}}-\frac {3 \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {3 \coth \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(62\) |
default | \(\frac {\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{2}}-\frac {3 \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {3 \coth \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(62\) |
risch | \(\frac {{\mathrm e}^{b x +a}}{2 b}+\frac {{\mathrm e}^{-b x -a}}{2 b}-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}\) | \(90\) |
Input:
int(cosh(b*x+a)*coth(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/b*(1/sinh(b*x+a)^2*cosh(b*x+a)^3-3*cosh(b*x+a)/sinh(b*x+a)^2+3/2*coth(b* x+a)*csch(b*x+a)-3*arctanh(exp(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (40) = 80\).
Time = 0.09 (sec) , antiderivative size = 612, normalized size of antiderivative = 13.91 \[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate(cosh(b*x+a)*coth(b*x+a)^3,x, algorithm="fricas")
Output:
1/2*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - 3*cosh(b*x + a)^4 + 4*(5*cosh (b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 - 6* cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 3*cosh(b*x + a)^2 - 3*(cosh(b*x + a )^5 + 5*cosh(b*x + a)*sinh(b*x + a)^4 + sinh(b*x + a)^5 + 2*(5*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^3 - 2*cosh(b*x + a)^3 + 2*(5*cosh(b*x + a)^3 - 3*c osh(b*x + a))*sinh(b*x + a)^2 + (5*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 + 1 )*sinh(b*x + a) + cosh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 3*(cosh(b*x + a)^5 + 5*cosh(b*x + a)*sinh(b*x + a)^4 + sinh(b*x + a)^5 + 2 *(5*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^3 - 2*cosh(b*x + a)^3 + 2*(5*cosh(b *x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^2 + (5*cosh(b*x + a)^4 - 6*cosh (b*x + a)^2 + 1)*sinh(b*x + a) + cosh(b*x + a))*log(cosh(b*x + a) + sinh(b *x + a) - 1) + 6*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 - cosh(b*x + a))*sin h(b*x + a) + 1)/(b*cosh(b*x + a)^5 + 5*b*cosh(b*x + a)*sinh(b*x + a)^4 + b *sinh(b*x + a)^5 - 2*b*cosh(b*x + a)^3 + 2*(5*b*cosh(b*x + a)^2 - b)*sinh( b*x + a)^3 + 2*(5*b*cosh(b*x + a)^3 - 3*b*cosh(b*x + a))*sinh(b*x + a)^2 + b*cosh(b*x + a) + (5*b*cosh(b*x + a)^4 - 6*b*cosh(b*x + a)^2 + b)*sinh(b* x + a))
\[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx=\int \cosh {\left (a + b x \right )} \coth ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(cosh(b*x+a)*coth(b*x+a)**3,x)
Output:
Integral(cosh(a + b*x)*coth(a + b*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (40) = 80\).
Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.45 \[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx=\frac {e^{\left (-b x - a\right )}}{2 \, b} - \frac {3 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac {3 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} - \frac {4 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} - 1}{2 \, b {\left (e^{\left (-b x - a\right )} - 2 \, e^{\left (-3 \, b x - 3 \, a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}\right )}} \] Input:
integrate(cosh(b*x+a)*coth(b*x+a)^3,x, algorithm="maxima")
Output:
1/2*e^(-b*x - a)/b - 3/2*log(e^(-b*x - a) + 1)/b + 3/2*log(e^(-b*x - a) - 1)/b - 1/2*(4*e^(-2*b*x - 2*a) + e^(-4*b*x - 4*a) - 1)/(b*(e^(-b*x - a) - 2*e^(-3*b*x - 3*a) + e^(-5*b*x - 5*a)))
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (40) = 80\).
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.39 \[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx=-\frac {\frac {4 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4} - 2 \, e^{\left (b x + a\right )} - 2 \, e^{\left (-b x - a\right )} + 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) - 3 \, \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{4 \, b} \] Input:
integrate(cosh(b*x+a)*coth(b*x+a)^3,x, algorithm="giac")
Output:
-1/4*(4*(e^(b*x + a) + e^(-b*x - a))/((e^(b*x + a) + e^(-b*x - a))^2 - 4) - 2*e^(b*x + a) - 2*e^(-b*x - a) + 3*log(e^(b*x + a) + e^(-b*x - a) + 2) - 3*log(e^(b*x + a) + e^(-b*x - a) - 2))/b
Time = 0.94 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.55 \[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx=\frac {{\mathrm {e}}^{a+b\,x}}{2\,b}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {{\mathrm {e}}^{-a-b\,x}}{2\,b}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \] Input:
int(cosh(a + b*x)*coth(a + b*x)^3,x)
Output:
exp(a + b*x)/(2*b) - (3*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(-b^2)^(1/ 2) + exp(- a - b*x)/(2*b) - (2*exp(a + b*x))/(b*(exp(4*a + 4*b*x) - 2*exp( 2*a + 2*b*x) + 1)) - exp(a + b*x)/(b*(exp(2*a + 2*b*x) - 1))
Time = 0.22 (sec) , antiderivative size = 202, normalized size of antiderivative = 4.59 \[ \int \cosh (a+b x) \coth ^3(a+b x) \, dx=\frac {e^{6 b x +6 a}+3 e^{5 b x +5 a} \mathrm {log}\left (e^{b x +a}-1\right )-3 e^{5 b x +5 a} \mathrm {log}\left (e^{b x +a}+1\right )-3 e^{4 b x +4 a}-6 e^{3 b x +3 a} \mathrm {log}\left (e^{b x +a}-1\right )+6 e^{3 b x +3 a} \mathrm {log}\left (e^{b x +a}+1\right )-3 e^{2 b x +2 a}+3 e^{b x +a} \mathrm {log}\left (e^{b x +a}-1\right )-3 e^{b x +a} \mathrm {log}\left (e^{b x +a}+1\right )+1}{2 e^{b x +a} b \left (e^{4 b x +4 a}-2 e^{2 b x +2 a}+1\right )} \] Input:
int(cosh(b*x+a)*coth(b*x+a)^3,x)
Output:
(e**(6*a + 6*b*x) + 3*e**(5*a + 5*b*x)*log(e**(a + b*x) - 1) - 3*e**(5*a + 5*b*x)*log(e**(a + b*x) + 1) - 3*e**(4*a + 4*b*x) - 6*e**(3*a + 3*b*x)*lo g(e**(a + b*x) - 1) + 6*e**(3*a + 3*b*x)*log(e**(a + b*x) + 1) - 3*e**(2*a + 2*b*x) + 3*e**(a + b*x)*log(e**(a + b*x) - 1) - 3*e**(a + b*x)*log(e**( a + b*x) + 1) + 1)/(2*e**(a + b*x)*b*(e**(4*a + 4*b*x) - 2*e**(2*a + 2*b*x ) + 1))