3.157 \(\int \cosh (a+b x) \tanh ^3(c+b x) \, dx\)
Optimal. Leaf size=72 \[ -\frac{3 \sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{2 b}+\frac{\cosh (a-c) \text{sech}(b x+c)}{b}+\frac{\sinh (a-c) \tanh (b x+c) \text{sech}(b x+c)}{2 b}+\frac{\cosh (a+b x)}{b} \]
[Out]
Cosh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x])/b - (3*ArcTan[Sinh[c + b*x]]*Sinh[a - c])/(2*b) + (Sech[c + b*x]
*Sinh[a - c]*Tanh[c + b*x])/(2*b)
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Rubi [A] time = 0.0787169, antiderivative size = 72, 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 =
{5623, 5620, 2638, 3770, 2606, 8, 2611} \[ -\frac{3 \sinh (a-c) \tan ^{-1}(\sinh (b x+c))}{2 b}+\frac{\cosh (a-c) \text{sech}(b x+c)}{b}+\frac{\sinh (a-c) \tanh (b x+c) \text{sech}(b x+c)}{2 b}+\frac{\cosh (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
Int[Cosh[a + b*x]*Tanh[c + b*x]^3,x]
[Out]
Cosh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x])/b - (3*ArcTan[Sinh[c + b*x]]*Sinh[a - c])/(2*b) + (Sech[c + b*x]
*Sinh[a - c]*Tanh[c + b*x])/(2*b)
Rule 5623
Int[Cosh[v_]*Tanh[w_]^(n_.), x_Symbol] :> Int[Sinh[v]*Tanh[w]^(n - 1), x] - Dist[Sinh[v - w], Int[Sech[w]*Tanh
[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Rule 5620
Int[Sinh[v_]*Tanh[w_]^(n_.), x_Symbol] :> Int[Cosh[v]*Tanh[w]^(n - 1), x] - Dist[Cosh[v - w], Int[Sech[w]*Tanh
[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[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 \cosh (a+b x) \tanh ^3(c+b x) \, dx &=-\left (\sinh (a-c) \int \text{sech}(c+b x) \tanh ^2(c+b x) \, dx\right )+\int \sinh (a+b x) \tanh ^2(c+b x) \, dx\\ &=\frac{\text{sech}(c+b x) \sinh (a-c) \tanh (c+b x)}{2 b}-\cosh (a-c) \int \text{sech}(c+b x) \tanh (c+b x) \, dx-\frac{1}{2} \sinh (a-c) \int \text{sech}(c+b x) \, dx+\int \cosh (a+b x) \tanh (c+b x) \, dx\\ &=-\frac{\tan ^{-1}(\sinh (c+b x)) \sinh (a-c)}{2 b}+\frac{\text{sech}(c+b x) \sinh (a-c) \tanh (c+b x)}{2 b}+\frac{\cosh (a-c) \operatorname{Subst}(\int 1 \, dx,x,\text{sech}(c+b x))}{b}-\sinh (a-c) \int \text{sech}(c+b x) \, dx+\int \sinh (a+b x) \, dx\\ &=\frac{\cosh (a+b x)}{b}+\frac{\cosh (a-c) \text{sech}(c+b x)}{b}-\frac{3 \tan ^{-1}(\sinh (c+b x)) \sinh (a-c)}{2 b}+\frac{\text{sech}(c+b x) \sinh (a-c) \tanh (c+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.308795, size = 115, normalized size = 1.6 \[ \frac{\text{sech}(c) \text{sech}^2(b x+c) (-\cosh (a-b x-c))+\text{sech}(c) \text{sech}^2(b x+c) \cosh (a+b x-c)+\text{sech}(c) \cosh (a-2 c) \text{sech}(b x+c)+\cosh (a) (3 \text{sech}(c) \text{sech}(b x+c)+4 \cosh (b x))-12 \sinh (a-c) \tan ^{-1}\left (\cosh (c) \tanh \left (\frac{b x}{2}\right )+\sinh (c)\right )+4 \sinh (a) \sinh (b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Cosh[a + b*x]*Tanh[c + b*x]^3,x]
[Out]
(Cosh[a - 2*c]*Sech[c]*Sech[c + b*x] - Cosh[a - c - b*x]*Sech[c]*Sech[c + b*x]^2 + Cosh[a - c + b*x]*Sech[c]*S
ech[c + b*x]^2 + Cosh[a]*(4*Cosh[b*x] + 3*Sech[c]*Sech[c + b*x]) - 12*ArcTan[Sinh[c] + Cosh[c]*Tanh[(b*x)/2]]*
Sinh[a - c] + 4*Sinh[a]*Sinh[b*x])/(4*b)
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Maple [C] time = 0.091, size = 238, normalized size = 3.3 \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{{\frac{3\,i}{4}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{a}} \right ) ^{2}}{b}}-{\frac{{\frac{3\,i}{4}}\ln \left ({{\rm e}^{bx+a}}-i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{c}} \right ) ^{2}}{b}}-{\frac{{\frac{3\,i}{4}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{a}} \right ) ^{2}}{b}}+{\frac{{\frac{3\,i}{4}}\ln \left ({{\rm e}^{bx+a}}+i{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}} \left ({{\rm e}^{c}} \right ) ^{2}}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cosh(b*x+a)*tanh(b*x+c)^3,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*I/b*ln(exp(b*x+a)-I*exp(a-c))*exp(-a-c)*exp(a)^2-3/4*I/b*ln(exp(b*
x+a)-I*exp(a-c))*exp(-a-c)*exp(c)^2-3/4*I/b*ln(exp(b*x+a)+I*exp(a-c))*exp(-a-c)*exp(a)^2+3/4*I/b*ln(exp(b*x+a)
+I*exp(a-c))*exp(-a-c)*exp(c)^2
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Maxima [B] time = 1.7538, size = 201, normalized size = 2.79 \begin{align*} \frac{3 \,{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{2 \, 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(cosh(b*x+a)*tanh(b*x+c)^3,x, algorithm="maxima")
[Out]
3/2*(e^(2*a) - e^(2*c))*arctan(e^(-b*x - c))*e^(-a - 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 = 2.00538, size = 4772, normalized size = 66.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(b*x+a)*tanh(b*x+c)^3,x, algorithm="fricas")
[Out]
1/2*(cosh(b*x + c)^6*cosh(-a + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2)*sinh(b*x
+ c)^6 + 6*(cosh(b*x + c)*cosh(-a + c)^2 - 2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) + cosh(b*x + c)*sinh(-a
+ c)^2)*sinh(b*x + c)^5 + (5*cosh(-a + c)^2 + 2)*cosh(b*x + c)^4 + (15*cosh(b*x + c)^2*cosh(-a + c)^2 + 5*(3*c
osh(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))*sin
h(-a + c) + 2)*sinh(b*x + c)^4 + 4*(5*cosh(b*x + c)^3*cosh(-a + c)^2 + 5*(cosh(b*x + c)^3 + cosh(b*x + c))*sin
h(-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*cosh(-a + c)^2 + 5)*cosh(b*x + c)^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)*sinh(-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*cosh(-a + c))*si
nh(-a + c) + 5)*sinh(b*x + c)^2 + (cosh(b*x + c)^6 + 5*cosh(b*x + c)^4 + 2*cosh(b*x + c)^2)*sinh(-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)*s
inh(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*cosh(-a + c) + co
sh(-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*(cos
h(-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))*arctan(cosh(b*x + c) + sinh(b*x + c)) + 2*(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) - 2*(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) + 1)/(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*cosh(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*cosh(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*c
osh(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*c
osh(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., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (a + b x \right )} \tanh ^{3}{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(b*x+a)*tanh(b*x+c)**3,x)
[Out]
Integral(cosh(a + b*x)*tanh(b*x + c)**3, x)
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Giac [A] time = 1.25087, size = 154, normalized size = 2.14 \begin{align*} -\frac{3 \,{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )} - \frac{{\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}} - e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(b*x+a)*tanh(b*x+c)^3,x, algorithm="giac")
[Out]
-1/2*(3*(e^(2*a) - e^(2*c))*arctan(e^(b*x + c))*e^(-a - c) - (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 - e^(b*x + a) - e^(-b*x - a))/b