Integrand size = 16, antiderivative size = 113 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {e^{a+b x}}{b}+\frac {8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac {5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {3 \text {arctanh}\left (e^{a+b x}\right )}{b} \] Output:
exp(b*x+a)/b+8/3*exp(b*x+a)/b/(1-exp(2*b*x+2*a))^3-14/3*exp(b*x+a)/b/(1-ex p(2*b*x+2*a))^2+5*exp(b*x+a)/b/(1-exp(2*b*x+2*a))-3*arctanh(exp(b*x+a))/b
Time = 10.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {-24 e^{a+b x}+50 e^{3 (a+b x)}-48 e^{5 (a+b x)}+6 e^{7 (a+b x)}+9 \left (-1+e^{2 (a+b x)}\right )^3 \log \left (1-e^{a+b x}\right )-9 \left (-1+e^{2 (a+b x)}\right )^3 \log \left (1+e^{a+b x}\right )}{6 b \left (-1+e^{2 (a+b x)}\right )^3} \] Input:
Integrate[E^(a + b*x)*Coth[a + b*x]^4,x]
Output:
(-24*E^(a + b*x) + 50*E^(3*(a + b*x)) - 48*E^(5*(a + b*x)) + 6*E^(7*(a + b *x)) + 9*(-1 + E^(2*(a + b*x)))^3*Log[1 - E^(a + b*x)] - 9*(-1 + E^(2*(a + b*x)))^3*Log[1 + E^(a + b*x)])/(6*b*(-1 + E^(2*(a + b*x)))^3)
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2720, 300, 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.
\(\displaystyle \int e^{a+b x} \coth ^4(a+b x) \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\int \frac {\left (1+e^{2 a+2 b x}\right )^4}{\left (1-e^{2 a+2 b x}\right )^4}de^{a+b x}}{b}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \frac {\int \left (\frac {8 e^{2 a+2 b x} \left (1+e^{4 a+4 b x}\right )}{\left (1-e^{2 a+2 b x}\right )^4}+1\right )de^{a+b x}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-3 \text {arctanh}\left (e^{a+b x}\right )+e^{a+b x}+\frac {5 e^{a+b x}}{1-e^{2 a+2 b x}}-\frac {14 e^{a+b x}}{3 \left (1-e^{2 a+2 b x}\right )^2}+\frac {8 e^{a+b x}}{3 \left (1-e^{2 a+2 b x}\right )^3}}{b}\) |
Input:
Int[E^(a + b*x)*Coth[a + b*x]^4,x]
Output:
(E^(a + b*x) + (8*E^(a + b*x))/(3*(1 - E^(2*a + 2*b*x))^3) - (14*E^(a + b* x))/(3*(1 - E^(2*a + 2*b*x))^2) + (5*E^(a + b*x))/(1 - E^(2*a + 2*b*x)) - 3*ArcTanh[E^(a + b*x)])/b
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Time = 0.75 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{b}-\frac {{\mathrm e}^{b x +a} \left (15 \,{\mathrm e}^{4 b x +4 a}-16 \,{\mathrm e}^{2 b x +2 a}+9\right )}{3 b \left (-1+{\mathrm e}^{2 b x +2 a}\right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}\) | \(88\) |
derivativedivides | \(\frac {\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{2}}-\frac {3 \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {3 \,\operatorname {csch}\left (b x +a \right ) \coth \left (b x +a \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )+\frac {\cosh \left (b x +a \right )^{4}}{\sinh \left (b x +a \right )^{3}}-\frac {4 \cosh \left (b x +a \right )^{2}}{\sinh \left (b x +a \right )^{3}}+\frac {8}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(107\) |
default | \(\frac {\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{2}}-\frac {3 \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {3 \,\operatorname {csch}\left (b x +a \right ) \coth \left (b x +a \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )+\frac {\cosh \left (b x +a \right )^{4}}{\sinh \left (b x +a \right )^{3}}-\frac {4 \cosh \left (b x +a \right )^{2}}{\sinh \left (b x +a \right )^{3}}+\frac {8}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(107\) |
Input:
int(exp(b*x+a)*coth(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
exp(b*x+a)/b-1/3*exp(b*x+a)*(15*exp(4*b*x+4*a)-16*exp(2*b*x+2*a)+9)/b/(-1+ exp(2*b*x+2*a))^3+3/2/b*ln(exp(b*x+a)-1)-3/2/b*ln(exp(b*x+a)+1)
Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (95) = 190\).
Time = 0.11 (sec) , antiderivative size = 796, normalized size of antiderivative = 7.04 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(b*x+a)*coth(b*x+a)^4,x, algorithm="fricas")
Output:
1/6*(6*cosh(b*x + a)^7 + 42*cosh(b*x + a)*sinh(b*x + a)^6 + 6*sinh(b*x + a )^7 + 6*(21*cosh(b*x + a)^2 - 8)*sinh(b*x + a)^5 - 48*cosh(b*x + a)^5 + 30 *(7*cosh(b*x + a)^3 - 8*cosh(b*x + a))*sinh(b*x + a)^4 + 10*(21*cosh(b*x + a)^4 - 48*cosh(b*x + a)^2 + 5)*sinh(b*x + a)^3 + 50*cosh(b*x + a)^3 + 6*( 21*cosh(b*x + a)^5 - 80*cosh(b*x + a)^3 + 25*cosh(b*x + a))*sinh(b*x + a)^ 2 - 9*(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*co sh(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 + 6*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) - 1)*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 9*(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 + 6*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^3 + cosh(b*x + a))*sinh(b *x + a) - 1)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 6*(7*cosh(b*x + a)^6 - 40*cosh(b*x + a)^4 + 25*cosh(b*x + a)^2 - 4)*sinh(b*x + a) - 24*cosh(b* x + a))/(b*cosh(b*x + a)^6 + 6*b*cosh(b*x + a)*sinh(b*x + a)^5 + b*sinh(b* x + a)^6 - 3*b*cosh(b*x + a)^4 + 3*(5*b*cosh(b*x + a)^2 - b)*sinh(b*x + a) ^4 + 4*(5*b*cosh(b*x + a)^3 - 3*b*cosh(b*x + a))*sinh(b*x + a)^3 + 3*b*...
\[ \int e^{a+b x} \coth ^4(a+b x) \, dx=e^{a} \int e^{b x} \coth ^{4}{\left (a + b x \right )}\, dx \] Input:
integrate(exp(b*x+a)*coth(b*x+a)**4,x)
Output:
exp(a)*Integral(exp(b*x)*coth(a + b*x)**4, x)
Time = 0.04 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.97 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {e^{\left (b x + a\right )}}{b} - \frac {3 \, \log \left (e^{\left (b x + a\right )} + 1\right )}{2 \, b} + \frac {3 \, \log \left (e^{\left (b x + a\right )} - 1\right )}{2 \, b} - \frac {15 \, e^{\left (5 \, b x + 5 \, a\right )} - 16 \, e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}}{3 \, b {\left (e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \] Input:
integrate(exp(b*x+a)*coth(b*x+a)^4,x, algorithm="maxima")
Output:
e^(b*x + a)/b - 3/2*log(e^(b*x + a) + 1)/b + 3/2*log(e^(b*x + a) - 1)/b - 1/3*(15*e^(5*b*x + 5*a) - 16*e^(3*b*x + 3*a) + 9*e^(b*x + a))/(b*(e^(6*b*x + 6*a) - 3*e^(4*b*x + 4*a) + 3*e^(2*b*x + 2*a) - 1))
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=-\frac {\frac {2 \, {\left (15 \, e^{\left (5 \, b x + 5 \, a\right )} - 16 \, e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} - 6 \, e^{\left (b x + a\right )} + 9 \, \log \left (e^{\left (b x + a\right )} + 1\right ) - 9 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \] Input:
integrate(exp(b*x+a)*coth(b*x+a)^4,x, algorithm="giac")
Output:
-1/6*(2*(15*e^(5*b*x + 5*a) - 16*e^(3*b*x + 3*a) + 9*e^(b*x + a))/(e^(2*b* x + 2*a) - 1)^3 - 6*e^(b*x + a) + 9*log(e^(b*x + a) + 1) - 9*log(abs(e^(b* x + a) - 1)))/b
Time = 0.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.42 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {\frac {4\,{\mathrm {e}}^{a+b\,x}}{3\,b}+\frac {4\,{\mathrm {e}}^{5\,a+5\,b\,x}}{3\,b}}{3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {11\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \] Input:
int(coth(a + b*x)^4*exp(a + b*x),x)
Output:
exp(a + b*x)/b - (3*atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b))/(-b^2)^(1/2) - ((4*exp(a + b*x))/(3*b) + (4*exp(5*a + 5*b*x))/(3*b))/(3*exp(2*a + 2*b*x) - 3*exp(4*a + 4*b*x) + exp(6*a + 6*b*x) - 1) - (2*exp(a + b*x))/(b*(exp(4 *a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - (11*exp(a + b*x))/(3*b*(exp(2*a + 2*b*x) - 1))
Time = 0.23 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.17 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {6 e^{7 b x +7 a}+9 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}-1\right )-9 e^{6 b x +6 a} \mathrm {log}\left (e^{b x +a}+1\right )-48 e^{5 b x +5 a}-27 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )+27 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )+50 e^{3 b x +3 a}+27 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )-27 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )-24 e^{b x +a}-9 \,\mathrm {log}\left (e^{b x +a}-1\right )+9 \,\mathrm {log}\left (e^{b x +a}+1\right )}{6 b \left (e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1\right )} \] Input:
int(exp(b*x+a)*coth(b*x+a)^4,x)
Output:
(6*e**(7*a + 7*b*x) + 9*e**(6*a + 6*b*x)*log(e**(a + b*x) - 1) - 9*e**(6*a + 6*b*x)*log(e**(a + b*x) + 1) - 48*e**(5*a + 5*b*x) - 27*e**(4*a + 4*b*x )*log(e**(a + b*x) - 1) + 27*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1) + 50*e **(3*a + 3*b*x) + 27*e**(2*a + 2*b*x)*log(e**(a + b*x) - 1) - 27*e**(2*a + 2*b*x)*log(e**(a + b*x) + 1) - 24*e**(a + b*x) - 9*log(e**(a + b*x) - 1) + 9*log(e**(a + b*x) + 1))/(6*b*(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3 *e**(2*a + 2*b*x) - 1))