\(\int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx\) [238]

Optimal result

Integrand size = 25, antiderivative size = 197 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx=\frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\tanh ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right ) \sqrt {\tanh ^2(a c+b c x)}}-\frac {3 \text {arctanh}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}} \]

[Out]

exp(c*(b*x+a))*tanh(b*c*x+a*c)/b/c/(tanh(b*c*x+a*c)^2)^(1/2)-2*exp(c*(b*x+a))*tanh(b*c*x+a*c)/b/c/(1-exp(2*c*(
b*x+a)))^2/(tanh(b*c*x+a*c)^2)^(1/2)+3*exp(c*(b*x+a))*tanh(b*c*x+a*c)/b/c/(1-exp(2*c*(b*x+a)))/(tanh(b*c*x+a*c
)^2)^(1/2)-3*arctanh(exp(c*(b*x+a)))*tanh(b*c*x+a*c)/b/c/(tanh(b*c*x+a*c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6852, 2320, 398, 1172, 12, 294, 213} \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx=-\frac {3 \text {arctanh}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}+\frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right ) \sqrt {\tanh ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\tanh ^2(a c+b c x)}} \]

[In]

Int[E^(c*(a + b*x))/(Tanh[a*c + b*c*x]^2)^(3/2),x]

[Out]

(E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c + b*c*x]^2]) - (2*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b
*c*(1 - E^(2*c*(a + b*x)))^2*Sqrt[Tanh[a*c + b*c*x]^2]) + (3*E^(c*(a + b*x))*Tanh[a*c + b*c*x])/(b*c*(1 - E^(2
*c*(a + b*x)))*Sqrt[Tanh[a*c + b*c*x]^2]) - (3*ArcTanh[E^(c*(a + b*x))]*Tanh[a*c + b*c*x])/(b*c*Sqrt[Tanh[a*c
+ b*c*x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 1172

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + c*
x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, Simp[(-R)*x*((d + e
*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx +
R*(2*q + 3), x], x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6852

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[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]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\tanh (a c+b c x) \int e^{c (a+b x)} \coth ^3(a c+b c x) \, dx}{\sqrt {\tanh ^2(a c+b c x)}} \\ & = \frac {\tanh (a c+b c x) \text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{\left (-1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\tanh ^2(a c+b c x)}} \\ & = \frac {\tanh (a c+b c x) \text {Subst}\left (\int \left (1+\frac {2 \left (1+3 x^4\right )}{\left (-1+x^2\right )^3}\right ) \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\tanh ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}+\frac {(2 \tanh (a c+b c x)) \text {Subst}\left (\int \frac {1+3 x^4}{\left (-1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\tanh ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\tanh ^2(a c+b c x)}}+\frac {\tanh (a c+b c x) \text {Subst}\left (\int \frac {12 x^2}{\left (-1+x^2\right )^2} \, dx,x,e^{c (a+b x)}\right )}{2 b c \sqrt {\tanh ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\tanh ^2(a c+b c x)}}+\frac {(6 \tanh (a c+b c x)) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^2\right )^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\tanh ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\tanh ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right ) \sqrt {\tanh ^2(a c+b c x)}}+\frac {(3 \tanh (a c+b c x)) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\tanh ^2(a c+b c x)}} \\ & = \frac {e^{c (a+b x)} \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}}-\frac {2 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\tanh ^2(a c+b c x)}}+\frac {3 e^{c (a+b x)} \tanh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right ) \sqrt {\tanh ^2(a c+b c x)}}-\frac {3 \text {arctanh}\left (e^{c (a+b x)}\right ) \tanh (a c+b c x)}{b c \sqrt {\tanh ^2(a c+b c x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 7.78 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.70 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx=-\frac {e^{-5 c (a+b x)} \left (-21 \left (252105+507305 e^{2 c (a+b x)}+173916 e^{4 c (a+b x)}-154296 e^{6 c (a+b x)}-73885 e^{8 c (a+b x)}+4887 e^{10 c (a+b x)}\right )-\frac {315 \left (-16807-28218 e^{2 c (a+b x)}+1173 e^{4 c (a+b x)}+17748 e^{6 c (a+b x)}+4299 e^{8 c (a+b x)}-1434 e^{10 c (a+b x)}+7 e^{12 c (a+b x)}\right ) \text {arctanh}\left (\sqrt {e^{2 c (a+b x)}}\right )}{\sqrt {e^{2 c (a+b x)}}}+384 e^{8 c (a+b x)} \left (1+e^{2 c (a+b x)}\right )^2 \left (7+5 e^{2 c (a+b x)}\right ) \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};e^{2 c (a+b x)}\right )+256 e^{8 c (a+b x)} \left (1+e^{2 c (a+b x)}\right )^3 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};e^{2 c (a+b x)}\right )\right ) \tanh ^3(c (a+b x))}{60480 b c \tanh ^2(c (a+b x))^{3/2}} \]

[In]

Integrate[E^(c*(a + b*x))/(Tanh[a*c + b*c*x]^2)^(3/2),x]

[Out]

-1/60480*((-21*(252105 + 507305*E^(2*c*(a + b*x)) + 173916*E^(4*c*(a + b*x)) - 154296*E^(6*c*(a + b*x)) - 7388
5*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)))*ArcTan
h[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))*(Tanh[c*(a + b*x)]^2)^(3/2))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.47 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\operatorname {csgn}\left (\tanh \left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{3}}{\sinh \left (b c x +a c \right )^{2}}-\frac {3 \cosh \left (b c x +a c \right )}{\sinh \left (b c x +a c \right )^{2}}+\frac {3 \,\operatorname {csch}\left (b c x +a c \right ) \coth \left (b c x +a c \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b c x +a c}\right )+\frac {\cosh \left (b c x +a c \right )^{2}}{\sinh \left (b c x +a c \right )}-\frac {2}{\sinh \left (b c x +a c \right )}\right )}{c b}\)	\(131\)
risch	\(\frac {2 \,{\mathrm e}^{5 c \left (b x +a \right )}+3 \,{\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )-3 \,{\mathrm e}^{4 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )-10 \,{\mathrm e}^{3 c \left (b x +a \right )}-6 \,{\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )+6 \,{\mathrm e}^{2 c \left (b x +a \right )} \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )+4 \,{\mathrm e}^{c \left (b x +a \right )}+3 \ln \left ({\mathrm e}^{c \left (b x +a \right )}-1\right )-3 \ln \left ({\mathrm e}^{c \left (b x +a \right )}+1\right )}{2 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, c b}\)	\(211\)

[In]

int(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

csgn(tanh(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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (179) = 358\).

Time = 0.27 (sec) , antiderivative size = 613, normalized size of antiderivative = 3.11 \[ \int \frac {e^{c (a+b x)}}{\tanh ^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 )}} \]

[In]

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(3/2),x, algorithm="fricas")

[Out]

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*cosh(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))

Sympy [F]

\[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx=e^{a c} \int \frac {e^{b c x}}{\left (\tanh ^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)**2)**(3/2),x)

[Out]

exp(a*c)*Integral(exp(b*c*x)/(tanh(a*c + b*c*x)**2)**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.57 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx=-\frac {3 \, \log \left (e^{\left (b c x + a c\right )} + 1\right )}{2 \, b c} + \frac {3 \, \log \left (e^{\left (b c x + a c\right )} - 1\right )}{2 \, b c} + \frac {e^{\left (5 \, b c x + 5 \, a c\right )} - 5 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 2 \, e^{\left (b c x + a c\right )}}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} - 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]

[In]

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(3/2),x, algorithm="maxima")

[Out]

-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))

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.82 \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx=\frac {2 \, e^{\left (b c x + a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - 3 \, \log \left (e^{\left (b c x + a c\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 3 \, \log \left ({\left | e^{\left (b c x + a c\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \frac {2 \, {\left (3 \, e^{\left (3 \, b c x + 3 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - e^{\left (b c x + a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )\right )}}{{\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{2}}}{2 \, b c} \]

[In]

integrate(exp(c*(b*x+a))/(tanh(b*c*x+a*c)^2)^(3/2),x, algorithm="giac")

[Out]

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)*sgn(e^(2*b*c*x + 2*a
*c) - 1) - e^(b*c*x + a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1))/(e^(2*b*c*x + 2*a*c) - 1)^2)/(b*c)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{c (a+b x)}}{\tanh ^2(a c+b c x)^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left ({\mathrm {tanh}\left (a\,c+b\,c\,x\right )}^2\right )}^{3/2}} \,d x \]

[In]

int(exp(c*(a + b*x))/(tanh(a*c + b*c*x)^2)^(3/2),x)

[Out]

int(exp(c*(a + b*x))/(tanh(a*c + b*c*x)^2)^(3/2), x)