Integrand size = 25, antiderivative size = 193 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {3 \arctan \left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}} \]
exp(c*(b*x+a))*coth(b*c*x+a*c)/b/c/(coth(b*c*x+a*c)^2)^(1/2)-2*exp(c*(b*x+ a))*coth(b*c*x+a*c)/b/c/(1+exp(2*c*(b*x+a)))^2/(coth(b*c*x+a*c)^2)^(1/2)+3 *exp(c*(b*x+a))*coth(b*c*x+a*c)/b/c/(1+exp(2*c*(b*x+a)))/(coth(b*c*x+a*c)^ 2)^(1/2)-3*arctan(exp(c*(b*x+a)))*coth(b*c*x+a*c)/b/c/(coth(b*c*x+a*c)^2)^ (1/2)
Time = 0.35 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.54 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\frac {\left (e^{c (a+b x)} \left (2+5 e^{2 c (a+b x)}+e^{4 c (a+b x)}\right )-3 \left (1+e^{2 c (a+b x)}\right )^2 \arctan \left (e^{c (a+b x)}\right )\right ) \coth (c (a+b x))}{b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(c (a+b x))}} \]
((E^(c*(a + b*x))*(2 + 5*E^(2*c*(a + b*x)) + E^(4*c*(a + b*x))) - 3*(1 + E ^(2*c*(a + b*x)))^2*ArcTan[E^(c*(a + b*x))])*Coth[c*(a + b*x)])/(b*c*(1 + E^(2*c*(a + b*x)))^2*Sqrt[Coth[c*(a + b*x)]^2])
Time = 0.89 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.53, 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.
\(\displaystyle \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 7271 |
\(\displaystyle \frac {\coth (a c+b c x) \int e^{c (a+b x)} \tanh ^3(a c+b x c)dx}{\sqrt {\coth ^2(a c+b c x)}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\coth (a c+b c x) \int -\frac {\left (1-e^{2 c (a+b x)}\right )^3}{\left (1+e^{2 c (a+b x)}\right )^3}de^{c (a+b x)}}{b c \sqrt {\coth ^2(a c+b c x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\coth (a c+b c x) \int \frac {\left (1-e^{2 c (a+b x)}\right )^3}{\left (1+e^{2 c (a+b x)}\right )^3}de^{c (a+b x)}}{b c \sqrt {\coth ^2(a c+b c x)}}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle -\frac {\coth (a c+b c x) \int \left (\frac {2 \left (1+3 e^{4 c (a+b x)}\right )}{\left (1+e^{2 c (a+b x)}\right )^3}-1\right )de^{c (a+b x)}}{b c \sqrt {\coth ^2(a c+b c x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (-3 \arctan \left (e^{c (a+b x)}\right )+e^{c (a+b x)}+\frac {3 e^{c (a+b x)}}{e^{2 c (a+b x)}+1}-\frac {2 e^{c (a+b x)}}{\left (e^{2 c (a+b x)}+1\right )^2}\right ) \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}\) |
((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*ArcTan[E^(c*(a + b*x))])*Coth[a* c + b*c*x])/(b*c*Sqrt[Coth[a*c + b*c*x]^2])
3.3.15.3.1 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 = 0.61 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\coth \left (c \left (b x +a \right )\right )\right ) \left (\frac {\sinh \left (b c x +a c \right )^{2}}{\cosh \left (b c x +a c \right )}+\frac {2}{\cosh \left (b c x +a c \right )}+\frac {\sinh \left (b c x +a c \right )^{3}}{\cosh \left (b c x +a c \right )^{2}}+\frac {3 \sinh \left (b c x +a c \right )}{\cosh \left (b c x +a c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (b c x +a c \right ) \tanh \left (b c x +a c \right )}{2}-3 \arctan \left ({\mathrm e}^{b c x +a c}\right )\right )}{c b}\) | \(131\) |
risch | \(\frac {3 i {\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )-3 i {\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )+2 \,{\mathrm e}^{5 c \left (b x +a \right )}+6 i {\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )-6 i {\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )+10 \,{\mathrm e}^{3 c \left (b x +a \right )}+3 i \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )-3 i \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )+4 \,{\mathrm e}^{c \left (b x +a \right )}}{2 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, c b}\) | \(223\) |
csgn(coth(c*(b*x+a)))/c/b*(sinh(b*c*x+a*c)^2/cosh(b*c*x+a*c)+2/cosh(b*c*x+ a*c)+sinh(b*c*x+a*c)^3/cosh(b*c*x+a*c)^2+3*sinh(b*c*x+a*c)/cosh(b*c*x+a*c) ^2-3/2*sech(b*c*x+a*c)*tanh(b*c*x+a*c)-3*arctan(exp(b*c*x+a*c)))
Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (179) = 358\).
Time = 0.27 (sec) , antiderivative size = 458, normalized size of antiderivative = 2.37 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\frac {\cosh \left (b c x + a c\right )^{5} + 5 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} + \sinh \left (b c x + a c\right )^{5} + 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{3} + 5 \, \cosh \left (b c x + a c\right )^{3} + 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{3} + 3 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 3 \, {\left (\cosh \left (b c x + a c\right )^{4} + 4 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + \sinh \left (b c x + a c\right )^{4} + 2 \, {\left (3 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{2} + 2 \, \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (\cosh \left (b c x + a c\right )^{3} + \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right ) + 1\right )} \arctan \left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right ) + {\left (5 \, \cosh \left (b c x + a c\right )^{4} + 15 \, \cosh \left (b c x + a c\right )^{2} + 2\right )} \sinh \left (b c x + a c\right ) + 2 \, \cosh \left (b c x + a c\right )}{b c \cosh \left (b c x + a c\right )^{4} + 4 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{3} + b c \sinh \left (b c x + a c\right )^{4} + 2 \, b c \cosh \left (b c x + a c\right )^{2} + 2 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\right )^{2} + b c + 4 \, {\left (b c \cosh \left (b c x + a c\right )^{3} + b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )} \]
(cosh(b*c*x + a*c)^5 + 5*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^4 + sinh(b*c* x + a*c)^5 + 5*(2*cosh(b*c*x + a*c)^2 + 1)*sinh(b*c*x + a*c)^3 + 5*cosh(b* c*x + a*c)^3 + 5*(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))*si nh(b*c*x + a*c) + 1)*arctan(cosh(b*c*x + a*c) + sinh(b*c*x + a*c)) + (5*co sh(b*c*x + a*c)^4 + 15*cosh(b*c*x + a*c)^2 + 2)*sinh(b*c*x + a*c) + 2*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))
\[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=e^{a c} \int \frac {e^{b c x}}{\left (\coth ^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.47 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=-\frac {3 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{b c} + \frac {e^{\left (5 \, b c x + 5 \, a c\right )} + 5 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 2 \, e^{\left (b c x + a c\right )}}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
-3*arctan(e^(b*c*x + a*c))/(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 = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.67 \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=-\frac {{\left (3 \, \arctan \left (e^{\left (b c x + a c\right )}\right ) e^{\left (-a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - e^{\left (b c x\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \frac {3 \, e^{\left (3 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + e^{\left (b c x\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{2}}\right )} e^{\left (a c\right )}}{b c} \]
-(3*arctan(e^(b*c*x + a*c))*e^(-a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1) - e^(b*c *x)*sgn(e^(2*b*c*x + 2*a*c) - 1) - (3*e^(3*b*c*x + 2*a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1) + e^(b*c*x)*sgn(e^(2*b*c*x + 2*a*c) - 1))/(e^(2*b*c*x + 2*a*c ) + 1)^2)*e^(a*c)/(b*c)
Timed out. \[ \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left ({\mathrm {coth}\left (a\,c+b\,c\,x\right )}^2\right )}^{3/2}} \,d x \]