3.148 \(\int \coth ^3(c+b x) \sinh (a+b x) \, dx\)
Optimal. Leaf size=73 \[ -\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{3 \sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{2 b}-\frac{\sinh (a-c) \coth (b x+c) \text{csch}(b x+c)}{2 b}+\frac{\sinh (a+b x)}{b} \]
[Out]
-((Cosh[a - c]*Csch[c + b*x])/b) - (3*ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/(2*b) - (Coth[c + b*x]*Csch[c + b*x]
*Sinh[a - c])/(2*b) + Sinh[a + b*x]/b
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Rubi [A] time = 0.0874259, antiderivative size = 73, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used =
{5622, 5621, 2637, 3770, 2606, 8, 2611} \[ -\frac{\cosh (a-c) \text{csch}(b x+c)}{b}-\frac{3 \sinh (a-c) \tanh ^{-1}(\cosh (b x+c))}{2 b}-\frac{\sinh (a-c) \coth (b x+c) \text{csch}(b x+c)}{2 b}+\frac{\sinh (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
Int[Coth[c + b*x]^3*Sinh[a + b*x],x]
[Out]
-((Cosh[a - c]*Csch[c + b*x])/b) - (3*ArcTanh[Cosh[c + b*x]]*Sinh[a - c])/(2*b) - (Coth[c + b*x]*Csch[c + b*x]
*Sinh[a - c])/(2*b) + Sinh[a + b*x]/b
Rule 5622
Int[Coth[w_]^(n_.)*Sinh[v_], x_Symbol] :> Int[Cosh[v]*Coth[w]^(n - 1), x] + Dist[Sinh[v - w], Int[Csch[w]*Coth
[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Rule 5621
Int[Cosh[v_]*Coth[w_]^(n_.), x_Symbol] :> Int[Sinh[v]*Coth[w]^(n - 1), x] + Dist[Cosh[v - w], Int[Csch[w]*Coth
[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Rule 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rule 2606
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 2611
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]
Rubi steps
\begin{align*} \int \coth ^3(c+b x) \sinh (a+b x) \, dx &=\sinh (a-c) \int \coth ^2(c+b x) \text{csch}(c+b x) \, dx+\int \cosh (a+b x) \coth ^2(c+b x) \, dx\\ &=-\frac{\coth (c+b x) \text{csch}(c+b x) \sinh (a-c)}{2 b}+\cosh (a-c) \int \coth (c+b x) \text{csch}(c+b x) \, dx+\frac{1}{2} \sinh (a-c) \int \text{csch}(c+b x) \, dx+\int \coth (c+b x) \sinh (a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cosh (c+b x)) \sinh (a-c)}{2 b}-\frac{\coth (c+b x) \text{csch}(c+b x) \sinh (a-c)}{2 b}-\frac{(i \cosh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(c+b x))}{b}+\sinh (a-c) \int \text{csch}(c+b x) \, dx+\int \cosh (a+b x) \, dx\\ &=-\frac{\cosh (a-c) \text{csch}(c+b x)}{b}-\frac{3 \tanh ^{-1}(\cosh (c+b x)) \sinh (a-c)}{2 b}-\frac{\coth (c+b x) \text{csch}(c+b x) \sinh (a-c)}{2 b}+\frac{\sinh (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.327516, size = 70, normalized size = 0.96 \[ \frac{\text{csch}^2(b x+c) (2 \sinh (a-b x-2 c)+\sinh (a+3 b x+2 c)-5 \sinh (a+b x))-12 \sinh (a-c) \tanh ^{-1}\left (\sinh (c) \tanh \left (\frac{b x}{2}\right )+\cosh (c)\right )}{4 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + b*x]^3*Sinh[a + b*x],x]
[Out]
(-12*ArcTanh[Cosh[c] + Sinh[c]*Tanh[(b*x)/2]]*Sinh[a - c] + Csch[c + b*x]^2*(2*Sinh[a - 2*c - b*x] - 5*Sinh[a
+ b*x] + Sinh[a + 2*c + 3*b*x]))/(4*b)
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Maple [B] time = 0.053, size = 230, normalized size = 3.2 \begin{align*}{\frac{{{\rm e}^{bx+a}}}{2\,b}}-{\frac{{{\rm e}^{-bx-a}}}{2\,b}}+{\frac{{{\rm e}^{bx+a}} \left ( -3\,{{\rm e}^{2\,bx+4\,a+2\,c}}-{{\rm e}^{2\,bx+2\,a+4\,c}}+{{\rm e}^{4\,a}}+3\,{{\rm e}^{2\,a+2\,c}} \right ) }{2\,b \left ( -{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) ^{2}}}-{\frac{3\,\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{4\,b}}+{\frac{3\,\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{4\,b}}+{\frac{3\,\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{4\,b}}-{\frac{3\,\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{4\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(b*x+c)^3*sinh(b*x+a),x)
[Out]
1/2*exp(b*x+a)/b-1/2*exp(-b*x-a)/b+1/2*exp(b*x+a)*(-3*exp(2*b*x+4*a+2*c)-exp(2*b*x+2*a+4*c)+exp(4*a)+3*exp(2*a
+2*c))/b/(-exp(2*b*x+2*a+2*c)+exp(2*a))^2-3/4/b*ln(exp(b*x+a)+exp(a-c))*exp(-a-c)*exp(2*a)+3/4/b*ln(exp(b*x+a)
+exp(a-c))*exp(-a-c)*exp(2*c)+3/4/b*ln(exp(b*x+a)-exp(a-c))*exp(-a-c)*exp(2*a)-3/4/b*ln(exp(b*x+a)-exp(a-c))*e
xp(-a-c)*exp(2*c)
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Maxima [B] time = 1.22753, size = 251, normalized size = 3.44 \begin{align*} -\frac{3 \,{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{4 \, b} + \frac{3 \,{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{4 \, b} - \frac{e^{\left (-b x - a\right )}}{2 \, b} - \frac{{\left (5 \, e^{\left (2 \, a + 2 \, c\right )} + e^{\left (4 \, c\right )}\right )} e^{\left (-2 \, b x - 2 \, a\right )} -{\left (2 \, e^{\left (4 \, a\right )} + 3 \, e^{\left (2 \, a + 2 \, c\right )}\right )} e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, c\right )}}{2 \, b{\left (e^{\left (-b x - a + 4 \, c\right )} - 2 \, e^{\left (-3 \, b x - a + 2 \, c\right )} + e^{\left (-5 \, b x - a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(b*x+c)^3*sinh(b*x+a),x, algorithm="maxima")
[Out]
-3/4*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) + e^c)/b + 3/4*(e^(2*a) - e^(2*c))*e^(-a - c)*log(e^(-b*x) -
e^c)/b - 1/2*e^(-b*x - a)/b - 1/2*((5*e^(2*a + 2*c) + e^(4*c))*e^(-2*b*x - 2*a) - (2*e^(4*a) + 3*e^(2*a + 2*c)
)*e^(-4*b*x - 4*a) - e^(4*c))/(b*(e^(-b*x - a + 4*c) - 2*e^(-3*b*x - a + 2*c) + e^(-5*b*x - a)))
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Fricas [B] time = 2.35647, size = 6496, normalized size = 88.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(b*x+c)^3*sinh(b*x+a),x, algorithm="fricas")
[Out]
1/4*(2*cosh(b*x + c)^6*cosh(-a + c)^2 + 2*(cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2)*sinh
(b*x + c)^6 + 12*(cosh(b*x + c)*cosh(-a + c)^2 - 2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) + cosh(b*x + c)*sin
h(-a + c)^2)*sinh(b*x + c)^5 - 2*(5*cosh(-a + c)^2 + 2)*cosh(b*x + c)^4 + 2*(15*cosh(b*x + c)^2*cosh(-a + c)^2
+ 5*(3*cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - 5*cosh(-a + c)^2 - 10*(3*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a
+ c))*sinh(-a + c) - 2)*sinh(b*x + c)^4 + 8*(5*cosh(b*x + c)^3*cosh(-a + c)^2 + 5*(cosh(b*x + c)^3 - cosh(b*x
+ c))*sinh(-a + c)^2 - (5*cosh(-a + c)^2 + 2)*cosh(b*x + c) - 10*(cosh(b*x + c)^3*cosh(-a + c) - cosh(b*x + c)
*cosh(-a + c))*sinh(-a + c))*sinh(b*x + c)^3 + 2*(2*cosh(-a + c)^2 + 5)*cosh(b*x + c)^2 + 2*(15*cosh(b*x + c)^
4*cosh(-a + c)^2 - 6*(5*cosh(-a + c)^2 + 2)*cosh(b*x + c)^2 + (15*cosh(b*x + c)^4 - 30*cosh(b*x + c)^2 + 2)*si
nh(-a + c)^2 + 2*cosh(-a + c)^2 - 2*(15*cosh(b*x + c)^4*cosh(-a + c) - 30*cosh(b*x + c)^2*cosh(-a + c) + 2*cos
h(-a + c))*sinh(-a + c) + 5)*sinh(b*x + c)^2 + 2*(cosh(b*x + c)^6 - 5*cosh(b*x + c)^4 + 2*cosh(b*x + c)^2)*sin
h(-a + c)^2 - 3*((cosh(-a + c)^2 - 1)*cosh(b*x + c)^5 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-
a + c)^2 - 1)*sinh(b*x + c)^5 - 5*(2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 -
(cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^4 - 2*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^3 + 2*(5*(cosh(-a +
c)^2 - 1)*cosh(b*x + c)^2 + (5*cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(5*cosh(b*x + c)^2*co
sh(-a + c) - cosh(-a + c))*sinh(-a + c) + 1)*sinh(b*x + c)^3 + 2*(5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^3 + (5*
cosh(b*x + c)^3 - 3*cosh(b*x + c))*sinh(-a + c)^2 - 3*(cosh(-a + c)^2 - 1)*cosh(b*x + c) - 2*(5*cosh(b*x + c)^
3*cosh(-a + c) - 3*cosh(b*x + c)*cosh(-a + c))*sinh(-a + c))*sinh(b*x + c)^2 + (cosh(b*x + c)^5 - 2*cosh(b*x +
c)^3 + cosh(b*x + c))*sinh(-a + c)^2 + (cosh(-a + c)^2 - 1)*cosh(b*x + c) + (5*(cosh(-a + c)^2 - 1)*cosh(b*x
+ c)^4 - 6*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (5*cosh(b*x + c)^4 - 6*cosh(b*x + c)^2 + 1)*sinh(-a + c)^2 +
cosh(-a + c)^2 - 2*(5*cosh(b*x + c)^4*cosh(-a + c) - 6*cosh(b*x + c)^2*cosh(-a + c) + cosh(-a + c))*sinh(-a +
c) - 1)*sinh(b*x + c) - 2*(cosh(b*x + c)^5*cosh(-a + c) - 2*cosh(b*x + c)^3*cosh(-a + c) + cosh(b*x + c)*cosh
(-a + c))*sinh(-a + c))*log(cosh(b*x + c) + sinh(b*x + c) + 1) + 3*((cosh(-a + c)^2 - 1)*cosh(b*x + c)^5 + (co
sh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^5 - 5*(2*cosh(b*x + c)*cosh(-a
+ c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^4 - 2*(co
sh(-a + c)^2 - 1)*cosh(b*x + c)^3 + 2*(5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (5*cosh(b*x + c)^2 - 1)*sinh(-
a + c)^2 - cosh(-a + c)^2 - 2*(5*cosh(b*x + c)^2*cosh(-a + c) - cosh(-a + c))*sinh(-a + c) + 1)*sinh(b*x + c)^
3 + 2*(5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^3 + (5*cosh(b*x + c)^3 - 3*cosh(b*x + c))*sinh(-a + c)^2 - 3*(cosh
(-a + c)^2 - 1)*cosh(b*x + c) - 2*(5*cosh(b*x + c)^3*cosh(-a + c) - 3*cosh(b*x + c)*cosh(-a + c))*sinh(-a + c)
)*sinh(b*x + c)^2 + (cosh(b*x + c)^5 - 2*cosh(b*x + c)^3 + cosh(b*x + c))*sinh(-a + c)^2 + (cosh(-a + c)^2 - 1
)*cosh(b*x + c) + (5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^4 - 6*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (5*cosh(b
*x + c)^4 - 6*cosh(b*x + c)^2 + 1)*sinh(-a + c)^2 + cosh(-a + c)^2 - 2*(5*cosh(b*x + c)^4*cosh(-a + c) - 6*cos
h(b*x + c)^2*cosh(-a + c) + cosh(-a + c))*sinh(-a + c) - 1)*sinh(b*x + c) - 2*(cosh(b*x + c)^5*cosh(-a + c) -
2*cosh(b*x + c)^3*cosh(-a + c) + cosh(b*x + c)*cosh(-a + c))*sinh(-a + c))*log(cosh(b*x + c) + sinh(b*x + c) -
1) + 4*(3*cosh(b*x + c)^5*cosh(-a + c)^2 - 2*(5*cosh(-a + c)^2 + 2)*cosh(b*x + c)^3 + (3*cosh(b*x + c)^5 - 10
*cosh(b*x + c)^3 + 2*cosh(b*x + c))*sinh(-a + c)^2 + (2*cosh(-a + c)^2 + 5)*cosh(b*x + c) - 2*(3*cosh(b*x + c)
^5*cosh(-a + c) - 10*cosh(b*x + c)^3*cosh(-a + c) + 2*cosh(b*x + c)*cosh(-a + c))*sinh(-a + c))*sinh(b*x + c)
- 4*(cosh(b*x + c)^6*cosh(-a + c) - 5*cosh(b*x + c)^4*cosh(-a + c) + 2*cosh(b*x + c)^2*cosh(-a + c))*sinh(-a +
c) - 2)/(b*cosh(b*x + c)^5*cosh(-a + c) + (b*cosh(-a + c) - b*sinh(-a + c))*sinh(b*x + c)^5 - 2*b*cosh(b*x +
c)^3*cosh(-a + c) + 5*(b*cosh(b*x + c)*cosh(-a + c) - b*cosh(b*x + c)*sinh(-a + c))*sinh(b*x + c)^4 + 2*(5*b*c
osh(b*x + c)^2*cosh(-a + c) - b*cosh(-a + c) - (5*b*cosh(b*x + c)^2 - b)*sinh(-a + c))*sinh(b*x + c)^3 + b*cos
h(b*x + c)*cosh(-a + c) + 2*(5*b*cosh(b*x + c)^3*cosh(-a + c) - 3*b*cosh(b*x + c)*cosh(-a + c) - (5*b*cosh(b*x
+ c)^3 - 3*b*cosh(b*x + c))*sinh(-a + c))*sinh(b*x + c)^2 + (5*b*cosh(b*x + c)^4*cosh(-a + c) - 6*b*cosh(b*x
+ c)^2*cosh(-a + c) + b*cosh(-a + c) - (5*b*cosh(b*x + c)^4 - 6*b*cosh(b*x + c)^2 + b)*sinh(-a + c))*sinh(b*x
+ c) - (b*cosh(b*x + c)^5 - 2*b*cosh(b*x + c)^3 + b*cosh(b*x + c))*sinh(-a + c))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(b*x+c)**3*sinh(b*x+a),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 1.21814, size = 207, normalized size = 2.84 \begin{align*} -\frac{3 \,{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\right ) - 3 \,{\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (b x + c\right )} - 1 \right |}\right ) + \frac{2 \,{\left (3 \, e^{\left (3 \, b x + 2 \, a + 2 \, c\right )} + e^{\left (3 \, b x + 4 \, c\right )} - e^{\left (b x + 2 \, a\right )} - 3 \, e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{{\left (e^{\left (2 \, b x + 2 \, c\right )} - 1\right )}^{2}} - 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(b*x+c)^3*sinh(b*x+a),x, algorithm="giac")
[Out]
-1/4*(3*(e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(e^(b*x + c) + 1) - 3*(e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log
(abs(e^(b*x + c) - 1)) + 2*(3*e^(3*b*x + 2*a + 2*c) + e^(3*b*x + 4*c) - e^(b*x + 2*a) - 3*e^(b*x + 2*c))*e^(-a
)/(e^(2*b*x + 2*c) - 1)^2 - 2*e^(b*x + a) + 2*e^(-b*x - a))/b