3.2.48 \(\int \coth ^3(c+b x) \sinh (a+b x) \, dx\) [148]
Optimal. Leaf size=73 \[ -\frac {\cosh (a-c) \text {csch}(c+b x)}{b}-\frac {3 \tanh ^{-1}(\cosh (c+b x)) \sinh (a-c)}{2 b}-\frac {\coth (c+b x) \text {csch}(c+b x) \sinh (a-c)}{2 b}+\frac {\sinh (a+b x)}{b} \]
[Out]
-cosh(a-c)*csch(b*x+c)/b-3/2*arctanh(cosh(b*x+c))*sinh(a-c)/b-1/2*coth(b*x+c)*csch(b*x+c)*sinh(a-c)/b+sinh(b*x
+a)/b
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Rubi [A]
time = 0.08, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 = {5741, 5740,
2717, 3855, 2686, 8, 2691} \begin {gather*} -\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} \end {gather*}
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 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rule 2686
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 2691
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]
Rule 2717
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Rule 3855
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rule 5740
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 5741
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]
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)) \text {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*}
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Mathematica [A]
time = 0.24, size = 70, normalized size = 0.96 \begin {gather*} \frac {-12 \tanh ^{-1}\left (\cosh (c)+\sinh (c) \tanh \left (\frac {b x}{2}\right )\right ) \sinh (a-c)+\text {csch}^2(c+b x) (2 \sinh (a-2 c-b x)-5 \sinh (a+b x)+\sinh (a+2 c+3 b x))}{4 b} \end {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs.
\(2(69)=138\).
time = 1.59, size = 230, normalized size = 3.15
| | |
method |
result |
size |
| | |
risch |
\(\frac {{\mathrm e}^{b x +a}}{2 b}-\frac {{\mathrm e}^{-b x -a}}{2 b}+\frac {{\mathrm e}^{b x +a} \left (-3 \,{\mathrm e}^{2 b x +4 a +2 c}-{\mathrm e}^{2 b x +2 a +4 c}+{\mathrm e}^{4 a}+3 \,{\mathrm e}^{2 a +2 c}\right )}{2 b \left (-{\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}-\frac {3 \ln \left ({\mathrm e}^{b x +a}-{\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}\) |
\(230\) |
| | |
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(b*x+c)^3*sinh(b*x+a),x,method=_RETURNVERBOSE)
[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*ln(exp(b*x+a)+exp(a-c))/b*exp(-a-c)*exp(2*a)+3/4*ln(exp(b*x+a)+e
xp(a-c))/b*exp(-a-c)*exp(2*c)+3/4*ln(exp(b*x+a)-exp(a-c))/b*exp(-a-c)*exp(2*a)-3/4*ln(exp(b*x+a)-exp(a-c))/b*e
xp(-a-c)*exp(2*c)
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs.
\(2 (69) = 138\).
time = 0.27, size = 186, normalized size = 2.55 \begin {gather*} -\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 {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. 2372 vs.
\(2 (69) = 138\).
time = 0.45, size = 2372, normalized size = 32.49 \begin {gather*} \text {Too large to display} \end {gather*}
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))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sinh {\left (a + b x \right )} \coth ^{3}{\left (b x + c \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(b*x+c)**3*sinh(b*x+a),x)
[Out]
Integral(sinh(a + b*x)*coth(b*x + c)**3, x)
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 169 vs.
\(2 (69) = 138\).
time = 0.40, size = 169, normalized size = 2.32 \begin {gather*} -\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 + a + c\right )} + e^{a}\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 + a + c\right )} - e^{a} \right |}\right ) + \frac {2 \, {\left (3 \, e^{\left (3 \, b x + 5 \, a + 2 \, c\right )} + e^{\left (3 \, b x + 3 \, a + 4 \, c\right )} - e^{\left (b x + 5 \, a\right )} - 3 \, e^{\left (b x + 3 \, a + 2 \, c\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} - e^{\left (2 \, a\right )}\right )}^{2}} - 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )}}{4 \, b} \end {gather*}
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 + a + c) + e^a) - 3*(e^(2*a + c) - e^(3*c))*e^(-a - 2*
c)*log(abs(e^(b*x + a + c) - e^a)) + 2*(3*e^(3*b*x + 5*a + 2*c) + e^(3*b*x + 3*a + 4*c) - e^(b*x + 5*a) - 3*e^
(b*x + 3*a + 2*c))/(e^(2*b*x + 2*a + 2*c) - e^(2*a))^2 - 2*e^(b*x + a) + 2*e^(-b*x - a))/b
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (c+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(c + b*x)^3*sinh(a + b*x),x)
[Out]
int(coth(c + b*x)^3*sinh(a + b*x), x)
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