Integrand size = 25, antiderivative size = 197 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx=\frac {e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c}-\frac {2 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac {3 e^{c (a+b x)} \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )}-\frac {3 \text {arctanh}\left (e^{c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)} \tanh (a c+b c x)}{b c} \]
exp(c*(b*x+a))*(coth(b*c*x+a*c)^2)^(1/2)*tanh(b*c*x+a*c)/b/c-2*exp(c*(b*x+ a))*(coth(b*c*x+a*c)^2)^(1/2)*tanh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))^2+3 *exp(c*(b*x+a))*(coth(b*c*x+a*c)^2)^(1/2)*tanh(b*c*x+a*c)/b/c/(1-exp(2*c*( b*x+a)))-3*arctanh(exp(c*(b*x+a)))*(coth(b*c*x+a*c)^2)^(1/2)*tanh(b*c*x+a* c)/b/c
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 4.15 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.70 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx=-\frac {e^{-5 c (a+b x)} \coth ^2(c (a+b x))^{3/2} \left (-21 \left (252105+507305 e^{2 c (a+b x)}+173916 e^{4 c (a+b x)}-154296 e^{6 c (a+b x)}-73885 e^{8 c (a+b x)}+4887 e^{10 c (a+b x)}\right )-\frac {315 \left (-16807-28218 e^{2 c (a+b x)}+1173 e^{4 c (a+b x)}+17748 e^{6 c (a+b x)}+4299 e^{8 c (a+b x)}-1434 e^{10 c (a+b x)}+7 e^{12 c (a+b x)}\right ) \text {arctanh}\left (\sqrt {e^{2 c (a+b x)}}\right )}{\sqrt {e^{2 c (a+b x)}}}+384 e^{8 c (a+b x)} \left (1+e^{2 c (a+b x)}\right )^2 \left (7+5 e^{2 c (a+b x)}\right ) \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};e^{2 c (a+b x)}\right )+256 e^{8 c (a+b x)} \left (1+e^{2 c (a+b x)}\right )^3 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};e^{2 c (a+b x)}\right )\right ) \tanh ^3(c (a+b x))}{60480 b c} \]
-1/60480*((Coth[c*(a + b*x)]^2)^(3/2)*(-21*(252105 + 507305*E^(2*c*(a + b* x)) + 173916*E^(4*c*(a + b*x)) - 154296*E^(6*c*(a + b*x)) - 73885*E^(8*c*( a + b*x)) + 4887*E^(10*c*(a + b*x))) - (315*(-16807 - 28218*E^(2*c*(a + b* x)) + 1173*E^(4*c*(a + b*x)) + 17748*E^(6*c*(a + b*x)) + 4299*E^(8*c*(a + b*x)) - 1434*E^(10*c*(a + b*x)) + 7*E^(12*c*(a + b*x)))*ArcTanh[Sqrt[E^(2* c*(a + b*x))]])/Sqrt[E^(2*c*(a + b*x))] + 384*E^(8*c*(a + b*x))*(1 + E^(2* c*(a + b*x)))^2*(7 + 5*E^(2*c*(a + b*x)))*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, E^(2*c*(a + b*x))] + 256*E^(8*c*(a + b*x))*(1 + E^(2 *c*(a + b*x)))^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2 }, E^(2*c*(a + b*x))])*Tanh[c*(a + b*x)]^3)/(b*c*E^(5*c*(a + b*x)))
Time = 0.45 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.54, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7271, 2720, 25, 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.
| \(\displaystyle \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)} \int e^{c (a+b x)} \coth ^3(a c+b x c)dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)} \int -\frac {\left (1+e^{2 c (a+b x)}\right )^3}{\left (1-e^{2 c (a+b x)}\right )^3}de^{c (a+b x)}}{b c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)} \int \frac {\left (1+e^{2 c (a+b x)}\right )^3}{\left (1-e^{2 c (a+b x)}\right )^3}de^{c (a+b x)}}{b c}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle -\frac {\tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)} \int \left (\frac {2 \left (1+3 e^{4 c (a+b x)}\right )}{\left (1-e^{2 c (a+b x)}\right )^3}-1\right )de^{c (a+b x)}}{b c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (-3 \text {arctanh}\left (e^{c (a+b x)}\right )+e^{c (a+b x)}+\frac {3 e^{c (a+b x)}}{1-e^{2 c (a+b x)}}-\frac {2 e^{c (a+b x)}}{\left (1-e^{2 c (a+b x)}\right )^2}\right ) \tanh (a c+b c x) \sqrt {\coth ^2(a c+b c x)}}{b c}\) |
((E^(c*(a + b*x)) - (2*E^(c*(a + b*x)))/(1 - E^(2*c*(a + b*x)))^2 + (3*E^( c*(a + b*x)))/(1 - E^(2*c*(a + b*x))) - 3*ArcTanh[E^(c*(a + b*x))])*Sqrt[C oth[a*c + b*c*x]^2]*Tanh[a*c + b*c*x])/(b*c)
3.3.12.3.1 Defintions of rubi rules used
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]]]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.58 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(\frac {\operatorname {csgn}\left (\coth \left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{3}}{\sinh \left (b c x +a c \right )^{2}}-\frac {3 \cosh \left (b c x +a c \right )}{\sinh \left (b c x +a c \right )^{2}}+\frac {3 \,\operatorname {csch}\left (b c x +a c \right ) \coth \left (b c x +a c \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b c x +a c}\right )+\frac {\cosh \left (b c x +a c \right )^{2}}{\sinh \left (b c x +a c \right )}-\frac {2}{\sinh \left (b c x +a c \right )}\right )}{c b}\) | \(131\) |
| risch | \(\frac {\sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \left (2 \,{\mathrm e}^{5 c \left (b x +a \right )}+3 \,{\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )-3 \,{\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )-10 \,{\mathrm e}^{3 c \left (b x +a \right )}-6 \,{\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )+6 \,{\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )+4 \,{\mathrm e}^{c \left (b x +a \right )}+3 \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )-3 \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )\right )}{2 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) c b}\) | \(211\) |
csgn(coth(c*(b*x+a)))/c/b*(cosh(b*c*x+a*c)^3/sinh(b*c*x+a*c)^2-3/sinh(b*c* x+a*c)^2*cosh(b*c*x+a*c)+3/2*csch(b*c*x+a*c)*coth(b*c*x+a*c)-3*arctanh(exp (b*c*x+a*c))+1/sinh(b*c*x+a*c)*cosh(b*c*x+a*c)^2-2/sinh(b*c*x+a*c))
Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (179) = 358\).
Time = 0.26 (sec) , antiderivative size = 613, normalized size of antiderivative = 3.11 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx=\frac {2 \, \cosh \left (b c x + a c\right )^{5} + 10 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} + 2 \, \sinh \left (b c x + a c\right )^{5} + 10 \, {\left (2 \, \cosh \left (b c x + a c\right )^{2} - 1\right )} \sinh \left (b c x + a c\right )^{3} - 10 \, \cosh \left (b c x + a c\right )^{3} + 10 \, {\left (2 \, \cosh \left (b c x + a c\right )^{3} - 3 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 3 \, {\left (\cosh \left (b c x + a c\right )^{4} + 4 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + \sinh \left (b c x + a c\right )^{4} + 2 \, {\left (3 \, \cosh \left (b c x + a c\right )^{2} - 1\right )} \sinh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (\cosh \left (b c x + a c\right )^{3} - \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right ) + 1\right )} \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) + 1\right ) + 3 \, {\left (\cosh \left (b c x + a c\right )^{4} + 4 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + \sinh \left (b c x + a c\right )^{4} + 2 \, {\left (3 \, \cosh \left (b c x + a c\right )^{2} - 1\right )} \sinh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (\cosh \left (b c x + a c\right )^{3} - \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right ) + 1\right )} \log \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right ) - 1\right ) + 2 \, {\left (5 \, \cosh \left (b c x + a c\right )^{4} - 15 \, \cosh \left (b c x + a c\right )^{2} + 2\right )} \sinh \left (b c x + a c\right ) + 4 \, \cosh \left (b c x + a c\right )}{2 \, {\left (b c \cosh \left (b c x + a c\right )^{4} + 4 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + b c \sinh \left (b c x + a c\right )^{4} - 2 \, b c \cosh \left (b c x + a c\right )^{2} + 2 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{2} - b c\right )} \sinh \left (b c x + a c\right )^{2} + b c + 4 \, {\left (b c \cosh \left (b c x + a c\right )^{3} - b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )\right )}} \]
1/2*(2*cosh(b*c*x + a*c)^5 + 10*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^4 + 2* sinh(b*c*x + a*c)^5 + 10*(2*cosh(b*c*x + a*c)^2 - 1)*sinh(b*c*x + a*c)^3 - 10*cosh(b*c*x + a*c)^3 + 10*(2*cosh(b*c*x + a*c)^3 - 3*cosh(b*c*x + a*c)) *sinh(b*c*x + a*c)^2 - 3*(cosh(b*c*x + a*c)^4 + 4*cosh(b*c*x + a*c)*sinh(b *c*x + a*c)^3 + sinh(b*c*x + a*c)^4 + 2*(3*cosh(b*c*x + a*c)^2 - 1)*sinh(b *c*x + a*c)^2 - 2*cosh(b*c*x + a*c)^2 + 4*(cosh(b*c*x + a*c)^3 - cosh(b*c* x + a*c))*sinh(b*c*x + a*c) + 1)*log(cosh(b*c*x + a*c) + sinh(b*c*x + a*c) + 1) + 3*(cosh(b*c*x + a*c)^4 + 4*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^3 + sinh(b*c*x + a*c)^4 + 2*(3*cosh(b*c*x + a*c)^2 - 1)*sinh(b*c*x + a*c)^2 - 2*cosh(b*c*x + a*c)^2 + 4*(cosh(b*c*x + a*c)^3 - cosh(b*c*x + a*c))*sinh( b*c*x + a*c) + 1)*log(cosh(b*c*x + a*c) + sinh(b*c*x + a*c) - 1) + 2*(5*co sh(b*c*x + a*c)^4 - 15*cosh(b*c*x + a*c)^2 + 2)*sinh(b*c*x + a*c) + 4*cosh (b*c*x + a*c))/(b*c*cosh(b*c*x + a*c)^4 + 4*b*c*cosh(b*c*x + a*c)*sinh(b*c *x + a*c)^3 + b*c*sinh(b*c*x + a*c)^4 - 2*b*c*cosh(b*c*x + a*c)^2 + 2*(3*b *c*cosh(b*c*x + a*c)^2 - b*c)*sinh(b*c*x + a*c)^2 + b*c + 4*(b*c*cosh(b*c* x + a*c)^3 - b*c*cosh(b*c*x + a*c))*sinh(b*c*x + a*c))
Timed out. \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.57 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx=-\frac {3 \, \log \left (e^{\left (b c x + a c\right )} + 1\right )}{2 \, b c} + \frac {3 \, \log \left (e^{\left (b c x + a c\right )} - 1\right )}{2 \, b c} + \frac {e^{\left (5 \, b c x + 5 \, a c\right )} - 5 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 2 \, e^{\left (b c x + a c\right )}}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} - 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
-3/2*log(e^(b*c*x + a*c) + 1)/(b*c) + 3/2*log(e^(b*c*x + a*c) - 1)/(b*c) + (e^(5*b*c*x + 5*a*c) - 5*e^(3*b*c*x + 3*a*c) + 2*e^(b*c*x + a*c))/(b*c*(e ^(4*b*c*x + 4*a*c) - 2*e^(2*b*c*x + 2*a*c) + 1))
Time = 0.45 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.79 \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx=\frac {\frac {2 \, e^{\left (b c x + a c\right )}}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} - \frac {3 \, \log \left (e^{\left (b c x + a c\right )} + 1\right )}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} + \frac {3 \, \log \left ({\left | e^{\left (b c x + a c\right )} - 1 \right |}\right )}{\mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )} - \frac {2 \, {\left (3 \, e^{\left (3 \, b c x + 3 \, a c\right )} - e^{\left (b c x + a c\right )}\right )}}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{2} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}}{2 \, b c} \]
1/2*(2*e^(b*c*x + a*c)/sgn(e^(2*b*c*x + 2*a*c) - 1) - 3*log(e^(b*c*x + a*c ) + 1)/sgn(e^(2*b*c*x + 2*a*c) - 1) + 3*log(abs(e^(b*c*x + a*c) - 1))/sgn( e^(2*b*c*x + 2*a*c) - 1) - 2*(3*e^(3*b*c*x + 3*a*c) - e^(b*c*x + a*c))/((e ^(2*b*c*x + 2*a*c) - 1)^2*sgn(e^(2*b*c*x + 2*a*c) - 1)))/(b*c)
Timed out. \[ \int e^{c (a+b x)} \coth ^2(a c+b c x)^{3/2} \, dx=\int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\left ({\mathrm {coth}\left (a\,c+b\,c\,x\right )}^2\right )}^{3/2} \,d x \]