3.3.16 \(\int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx\) [216]

Optimal. Leaf size=311 \[ \frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {15 \text {ArcTan}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{4 b c \sqrt {\coth ^2(a c+b c x)}} \]

[Out]

exp(c*(b*x+a))*coth(b*c*x+a*c)/b/c/(coth(b*c*x+a*c)^2)^(1/2)-4*exp(c*(b*x+a))*coth(b*c*x+a*c)/b/c/(1+exp(2*c*(
b*x+a)))^4/(coth(b*c*x+a*c)^2)^(1/2)+26/3*exp(c*(b*x+a))*coth(b*c*x+a*c)/b/c/(1+exp(2*c*(b*x+a)))^3/(coth(b*c*
x+a*c)^2)^(1/2)-55/6*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)+25/4*
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)-15/4*arctan(exp(c*(b*x+a)))*
coth(b*c*x+a*c)/b/c/(coth(b*c*x+a*c)^2)^(1/2)

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Rubi [A]
time = 1.22, antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {6852, 2320, 398, 1828, 1171, 393, 209} \begin {gather*} -\frac {15 \text {ArcTan}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{4 b c \sqrt {\coth ^2(a c+b c x)}}+\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (e^{2 c (a+b x)}+1\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (e^{2 c (a+b x)}+1\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^4 \sqrt {\coth ^2(a c+b c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(c*(a + b*x))/(Coth[a*c + b*c*x]^2)^(5/2),x]

[Out]

(E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b*c*Sqrt[Coth[a*c + b*c*x]^2]) - (4*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(b
*c*(1 + E^(2*c*(a + b*x)))^4*Sqrt[Coth[a*c + b*c*x]^2]) + (26*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(3*b*c*(1 + E
^(2*c*(a + b*x)))^3*Sqrt[Coth[a*c + b*c*x]^2]) - (55*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(6*b*c*(1 + E^(2*c*(a
+ b*x)))^2*Sqrt[Coth[a*c + b*c*x]^2]) + (25*E^(c*(a + b*x))*Coth[a*c + b*c*x])/(4*b*c*(1 + E^(2*c*(a + b*x)))*
Sqrt[Coth[a*c + b*c*x]^2]) - (15*ArcTan[E^(c*(a + b*x))]*Coth[a*c + b*c*x])/(4*b*c*Sqrt[Coth[a*c + b*c*x]^2])

Rule 209

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

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

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 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + 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, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -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*} \int \frac {e^{c (a+b x)}}{\coth ^2(a c+b c x)^{5/2}} \, dx &=\frac {\coth (a c+b c x) \int e^{c (a+b x)} \tanh ^5(a c+b c x) \, dx}{\sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {\coth (a c+b c x) \text {Subst}\left (\int \frac {\left (-1+x^2\right )^5}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {\coth (a c+b c x) \text {Subst}\left (\int \left (1-\frac {2 \left (1+10 x^4+5 x^8\right )}{\left (1+x^2\right )^5}\right ) \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {(2 \coth (a c+b c x)) \text {Subst}\left (\int \frac {1+10 x^4+5 x^8}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {\coth (a c+b c x) \text {Subst}\left (\int \frac {8-120 x^2+40 x^4-40 x^6}{\left (1+x^2\right )^4} \, dx,x,e^{c (a+b x)}\right )}{4 b c \sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {\coth (a c+b c x) \text {Subst}\left (\int \frac {160-480 x^2+240 x^4}{\left (1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{24 b c \sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {\coth (a c+b c x) \text {Subst}\left (\int \frac {240-960 x^2}{\left (1+x^2\right )^2} \, dx,x,e^{c (a+b x)}\right )}{96 b c \sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {(15 \coth (a c+b c x)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{4 b c \sqrt {\coth ^2(a c+b c x)}}\\ &=\frac {e^{c (a+b x)} \coth (a c+b c x)}{b c \sqrt {\coth ^2(a c+b c x)}}-\frac {4 e^{c (a+b x)} \coth (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(a c+b c x)}}+\frac {26 e^{c (a+b x)} \coth (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\coth ^2(a c+b c x)}}-\frac {55 e^{c (a+b x)} \coth (a c+b c x)}{6 b c \left (1+e^{2 c (a+b x)}\right )^2 \sqrt {\coth ^2(a c+b c x)}}+\frac {25 e^{c (a+b x)} \coth (a c+b c x)}{4 b c \left (1+e^{2 c (a+b x)}\right ) \sqrt {\coth ^2(a c+b c x)}}-\frac {15 \tan ^{-1}\left (e^{c (a+b x)}\right ) \coth (a c+b c x)}{4 b c \sqrt {\coth ^2(a c+b c x)}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 133, normalized size = 0.43 \begin {gather*} \frac {\left (e^{c (a+b x)} \left (33+157 e^{2 c (a+b x)}+187 e^{4 c (a+b x)}+123 e^{6 c (a+b x)}+12 e^{8 c (a+b x)}\right )-45 \left (1+e^{2 c (a+b x)}\right )^4 \text {ArcTan}\left (e^{c (a+b x)}\right )\right ) \coth (c (a+b x))}{12 b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\coth ^2(c (a+b x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(c*(a + b*x))/(Coth[a*c + b*c*x]^2)^(5/2),x]

[Out]

((E^(c*(a + b*x))*(33 + 157*E^(2*c*(a + b*x)) + 187*E^(4*c*(a + b*x)) + 123*E^(6*c*(a + b*x)) + 12*E^(8*c*(a +
b*x))) - 45*(1 + E^(2*c*(a + b*x)))^4*ArcTan[E^(c*(a + b*x))])*Coth[c*(a + b*x)])/(12*b*c*(1 + E^(2*c*(a + b*
x)))^4*Sqrt[Coth[c*(a + b*x)]^2])

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Maple [C] Result contains complex when optimal does not.
time = 5.32, size = 324, normalized size = 1.04

method	result	size
risch	\(\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) {\mathrm e}^{c \left (b x +a \right )}}{\sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) b c}+\frac {{\mathrm e}^{c \left (b x +a \right )} \left (75 \,{\mathrm e}^{6 c \left (b x +a \right )}+115 \,{\mathrm e}^{4 c \left (b x +a \right )}+109 \,{\mathrm e}^{2 c \left (b x +a \right )}+21\right )}{12 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{3} \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, b c}+\frac {15 i \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \ln \left ({\mathrm e}^{c \left (b x +a \right )}-i\right )}{8 \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) c b}-\frac {15 i \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \ln \left ({\mathrm e}^{c \left (b x +a \right )}+i\right )}{8 \sqrt {\frac {\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) c b}\)	\(324\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(b*x+a))/(coth(b*c*x+a*c)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/((1+exp(2*c*(b*x+a)))^2/(exp(2*c*(b*x+a))-1)^2)^(1/2)/(exp(2*c*(b*x+a))-1)*(1+exp(2*c*(b*x+a)))*exp(c*(b*x+a
))/b/c+1/12/(1+exp(2*c*(b*x+a)))^3/(exp(2*c*(b*x+a))-1)/((1+exp(2*c*(b*x+a)))^2/(exp(2*c*(b*x+a))-1)^2)^(1/2)*
exp(c*(b*x+a))*(75*exp(6*c*(b*x+a))+115*exp(4*c*(b*x+a))+109*exp(2*c*(b*x+a))+21)/b/c+15/8*I*(1+exp(2*c*(b*x+a
)))/(exp(2*c*(b*x+a))-1)/((1+exp(2*c*(b*x+a)))^2/(exp(2*c*(b*x+a))-1)^2)^(1/2)/c/b*ln(exp(c*(b*x+a))-I)-15/8*I
*(1+exp(2*c*(b*x+a)))/(exp(2*c*(b*x+a))-1)/((1+exp(2*c*(b*x+a)))^2/(exp(2*c*(b*x+a))-1)^2)^(1/2)/c/b*ln(exp(c*
(b*x+a))+I)

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Maxima [A]
time = 0.48, size = 145, normalized size = 0.47 \begin {gather*} -\frac {15 \, \arctan \left (e^{\left (b c x + a c\right )}\right )}{4 \, b c} + \frac {12 \, e^{\left (9 \, b c x + 9 \, a c\right )} + 123 \, e^{\left (7 \, b c x + 7 \, a c\right )} + 187 \, e^{\left (5 \, b c x + 5 \, a c\right )} + 157 \, e^{\left (3 \, b c x + 3 \, a c\right )} + 33 \, e^{\left (b c x + a c\right )}}{12 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(coth(b*c*x+a*c)^2)^(5/2),x, algorithm="maxima")

[Out]

-15/4*arctan(e^(b*c*x + a*c))/(b*c) + 1/12*(12*e^(9*b*c*x + 9*a*c) + 123*e^(7*b*c*x + 7*a*c) + 187*e^(5*b*c*x
+ 5*a*c) + 157*e^(3*b*c*x + 3*a*c) + 33*e^(b*c*x + a*c))/(b*c*(e^(8*b*c*x + 8*a*c) + 4*e^(6*b*c*x + 6*a*c) + 6
*e^(4*b*c*x + 4*a*c) + 4*e^(2*b*c*x + 2*a*c) + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1226 vs. \(2 (281) = 562\).
time = 0.36, size = 1226, normalized size = 3.94 \begin {gather*} \frac {12 \, \cosh \left (b c x + a c\right )^{9} + 108 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{8} + 12 \, \sinh \left (b c x + a c\right )^{9} + 3 \, {\left (144 \, \cosh \left (b c x + a c\right )^{2} + 41\right )} \sinh \left (b c x + a c\right )^{7} + 123 \, \cosh \left (b c x + a c\right )^{7} + 21 \, {\left (48 \, \cosh \left (b c x + a c\right )^{3} + 41 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{6} + {\left (1512 \, \cosh \left (b c x + a c\right )^{4} + 2583 \, \cosh \left (b c x + a c\right )^{2} + 187\right )} \sinh \left (b c x + a c\right )^{5} + 187 \, \cosh \left (b c x + a c\right )^{5} + {\left (1512 \, \cosh \left (b c x + a c\right )^{5} + 4305 \, \cosh \left (b c x + a c\right )^{3} + 935 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{4} + {\left (1008 \, \cosh \left (b c x + a c\right )^{6} + 4305 \, \cosh \left (b c x + a c\right )^{4} + 1870 \, \cosh \left (b c x + a c\right )^{2} + 157\right )} \sinh \left (b c x + a c\right )^{3} + 157 \, \cosh \left (b c x + a c\right )^{3} + {\left (432 \, \cosh \left (b c x + a c\right )^{7} + 2583 \, \cosh \left (b c x + a c\right )^{5} + 1870 \, \cosh \left (b c x + a c\right )^{3} + 471 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 45 \, {\left (\cosh \left (b c x + a c\right )^{8} + 8 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{7} + \sinh \left (b c x + a c\right )^{8} + 4 \, {\left (7 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{6} + 4 \, \cosh \left (b c x + a c\right )^{6} + 8 \, {\left (7 \, \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 )^{5} + 2 \, {\left (35 \, \cosh \left (b c x + a c\right )^{4} + 30 \, \cosh \left (b c x + a c\right )^{2} + 3\right )} \sinh \left (b c x + a c\right )^{4} + 6 \, \cosh \left (b c x + a c\right )^{4} + 8 \, {\left (7 \, \cosh \left (b c x + a c\right )^{5} + 10 \, \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 )^{3} + 4 \, {\left (7 \, \cosh \left (b c x + a c\right )^{6} + 15 \, \cosh \left (b c x + a c\right )^{4} + 9 \, \cosh \left (b c x + a c\right )^{2} + 1\right )} \sinh \left (b c x + a c\right )^{2} + 4 \, \cosh \left (b c x + a c\right )^{2} + 8 \, {\left (\cosh \left (b c x + a c\right )^{7} + 3 \, \cosh \left (b c x + a c\right )^{5} + 3 \, \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 (108 \, \cosh \left (b c x + a c\right )^{8} + 861 \, \cosh \left (b c x + a c\right )^{6} + 935 \, \cosh \left (b c x + a c\right )^{4} + 471 \, \cosh \left (b c x + a c\right )^{2} + 33\right )} \sinh \left (b c x + a c\right ) + 33 \, \cosh \left (b c x + a c\right )}{12 \, {\left (b c \cosh \left (b c x + a c\right )^{8} + 8 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{7} + b c \sinh \left (b c x + a c\right )^{8} + 4 \, b c \cosh \left (b c x + a c\right )^{6} + 4 \, {\left (7 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\right )^{6} + 6 \, b c \cosh \left (b c x + a c\right )^{4} + 8 \, {\left (7 \, b c \cosh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{5} + 2 \, {\left (35 \, b c \cosh \left (b c x + a c\right )^{4} + 30 \, b c \cosh \left (b c x + a c\right )^{2} + 3 \, b c\right )} \sinh \left (b c x + a c\right )^{4} + 4 \, b c \cosh \left (b c x + a c\right )^{2} + 8 \, {\left (7 \, b c \cosh \left (b c x + a c\right )^{5} + 10 \, b c \cosh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{3} + 4 \, {\left (7 \, b c \cosh \left (b c x + a c\right )^{6} + 15 \, b c \cosh \left (b c x + a c\right )^{4} + 9 \, 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 + 8 \, {\left (b c \cosh \left (b c x + a c\right )^{7} + 3 \, b c \cosh \left (b c x + a c\right )^{5} + 3 \, 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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(coth(b*c*x+a*c)^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*cosh(b*c*x + a*c)^9 + 108*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^8 + 12*sinh(b*c*x + a*c)^9 + 3*(144*cos
h(b*c*x + a*c)^2 + 41)*sinh(b*c*x + a*c)^7 + 123*cosh(b*c*x + a*c)^7 + 21*(48*cosh(b*c*x + a*c)^3 + 41*cosh(b*
c*x + a*c))*sinh(b*c*x + a*c)^6 + (1512*cosh(b*c*x + a*c)^4 + 2583*cosh(b*c*x + a*c)^2 + 187)*sinh(b*c*x + a*c
)^5 + 187*cosh(b*c*x + a*c)^5 + (1512*cosh(b*c*x + a*c)^5 + 4305*cosh(b*c*x + a*c)^3 + 935*cosh(b*c*x + a*c))*
sinh(b*c*x + a*c)^4 + (1008*cosh(b*c*x + a*c)^6 + 4305*cosh(b*c*x + a*c)^4 + 1870*cosh(b*c*x + a*c)^2 + 157)*s
inh(b*c*x + a*c)^3 + 157*cosh(b*c*x + a*c)^3 + (432*cosh(b*c*x + a*c)^7 + 2583*cosh(b*c*x + a*c)^5 + 1870*cosh
(b*c*x + a*c)^3 + 471*cosh(b*c*x + a*c))*sinh(b*c*x + a*c)^2 - 45*(cosh(b*c*x + a*c)^8 + 8*cosh(b*c*x + a*c)*s
inh(b*c*x + a*c)^7 + sinh(b*c*x + a*c)^8 + 4*(7*cosh(b*c*x + a*c)^2 + 1)*sinh(b*c*x + a*c)^6 + 4*cosh(b*c*x +
a*c)^6 + 8*(7*cosh(b*c*x + a*c)^3 + 3*cosh(b*c*x + a*c))*sinh(b*c*x + a*c)^5 + 2*(35*cosh(b*c*x + a*c)^4 + 30*
cosh(b*c*x + a*c)^2 + 3)*sinh(b*c*x + a*c)^4 + 6*cosh(b*c*x + a*c)^4 + 8*(7*cosh(b*c*x + a*c)^5 + 10*cosh(b*c*
x + a*c)^3 + 3*cosh(b*c*x + a*c))*sinh(b*c*x + a*c)^3 + 4*(7*cosh(b*c*x + a*c)^6 + 15*cosh(b*c*x + a*c)^4 + 9*
cosh(b*c*x + a*c)^2 + 1)*sinh(b*c*x + a*c)^2 + 4*cosh(b*c*x + a*c)^2 + 8*(cosh(b*c*x + a*c)^7 + 3*cosh(b*c*x +
a*c)^5 + 3*cosh(b*c*x + a*c)^3 + cosh(b*c*x + a*c))*sinh(b*c*x + a*c) + 1)*arctan(cosh(b*c*x + a*c) + sinh(b*
c*x + a*c)) + (108*cosh(b*c*x + a*c)^8 + 861*cosh(b*c*x + a*c)^6 + 935*cosh(b*c*x + a*c)^4 + 471*cosh(b*c*x +
a*c)^2 + 33)*sinh(b*c*x + a*c) + 33*cosh(b*c*x + a*c))/(b*c*cosh(b*c*x + a*c)^8 + 8*b*c*cosh(b*c*x + a*c)*sinh
(b*c*x + a*c)^7 + b*c*sinh(b*c*x + a*c)^8 + 4*b*c*cosh(b*c*x + a*c)^6 + 4*(7*b*c*cosh(b*c*x + a*c)^2 + b*c)*si
nh(b*c*x + a*c)^6 + 6*b*c*cosh(b*c*x + a*c)^4 + 8*(7*b*c*cosh(b*c*x + a*c)^3 + 3*b*c*cosh(b*c*x + a*c))*sinh(b
*c*x + a*c)^5 + 2*(35*b*c*cosh(b*c*x + a*c)^4 + 30*b*c*cosh(b*c*x + a*c)^2 + 3*b*c)*sinh(b*c*x + a*c)^4 + 4*b*
c*cosh(b*c*x + a*c)^2 + 8*(7*b*c*cosh(b*c*x + a*c)^5 + 10*b*c*cosh(b*c*x + a*c)^3 + 3*b*c*cosh(b*c*x + a*c))*s
inh(b*c*x + a*c)^3 + 4*(7*b*c*cosh(b*c*x + a*c)^6 + 15*b*c*cosh(b*c*x + a*c)^4 + 9*b*c*cosh(b*c*x + a*c)^2 + b
*c)*sinh(b*c*x + a*c)^2 + b*c + 8*(b*c*cosh(b*c*x + a*c)^7 + 3*b*c*cosh(b*c*x + a*c)^5 + 3*b*c*cosh(b*c*x + a*
c)^3 + b*c*cosh(b*c*x + a*c))*sinh(b*c*x + a*c))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(coth(b*c*x+a*c)**2)**(5/2),x)

[Out]

Timed out

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Giac [A]
time = 0.41, size = 185, normalized size = 0.59 \begin {gather*} -\frac {{\left (45 \, \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 ) - 12 \, e^{\left (b c x\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) - \frac {75 \, e^{\left (7 \, b c x + 6 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 115 \, e^{\left (5 \, b c x + 4 \, a c\right )} \mathrm {sgn}\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right ) + 109 \, 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 ) + 21 \, 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 )}^{4}}\right )} e^{\left (a c\right )}}{12 \, b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(coth(b*c*x+a*c)^2)^(5/2),x, algorithm="giac")

[Out]

-1/12*(45*arctan(e^(b*c*x + a*c))*e^(-a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1) - 12*e^(b*c*x)*sgn(e^(2*b*c*x + 2*a*c)
- 1) - (75*e^(7*b*c*x + 6*a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1) + 115*e^(5*b*c*x + 4*a*c)*sgn(e^(2*b*c*x + 2*a*c)
- 1) + 109*e^(3*b*c*x + 2*a*c)*sgn(e^(2*b*c*x + 2*a*c) - 1) + 21*e^(b*c*x)*sgn(e^(2*b*c*x + 2*a*c) - 1))/(e^(
2*b*c*x + 2*a*c) + 1)^4)*e^(a*c)/(b*c)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left ({\mathrm {coth}\left (a\,c+b\,c\,x\right )}^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(a + b*x))/(coth(a*c + b*c*x)^2)^(5/2),x)

[Out]

int(exp(c*(a + b*x))/(coth(a*c + b*c*x)^2)^(5/2), x)

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