3.10 \(\int (a+b \text{csch}^2(c+d x))^{3/2} \, dx\)
Optimal. Leaf size=126 \[ \frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac{b \coth (c+d x) \sqrt{a+b \coth ^2(c+d x)-b}}{2 d}-\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{2 d} \]
[Out]
(a^(3/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d - ((3*a - b)*Sqrt[b]*ArcTanh[(Sqr
t[b]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/(2*d) - (b*Coth[c + d*x]*Sqrt[a - b + b*Coth[c + d*x]^2]
)/(2*d)
________________________________________________________________________________________
Rubi [A] time = 0.109502, antiderivative size = 126, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used =
{4128, 416, 523, 217, 206, 377} \[ \frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{d}-\frac{b \coth (c+d x) \sqrt{a+b \coth ^2(c+d x)-b}}{2 d}-\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a+b \coth ^2(c+d x)-b}}\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Csch[c + d*x]^2)^(3/2),x]
[Out]
(a^(3/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/d - ((3*a - b)*Sqrt[b]*ArcTanh[(Sqr
t[b]*Coth[c + d*x])/Sqrt[a - b + b*Coth[c + d*x]^2]])/(2*d) - (b*Coth[c + d*x]*Sqrt[a - b + b*Coth[c + d*x]^2]
)/(2*d)
Rule 4128
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]
Rule 416
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]
Rule 523
Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 377
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Rubi steps
\begin{align*} \int \left (a+b \text{csch}^2(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac{b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{2 d}-\frac{\operatorname{Subst}\left (\int \frac{-(a-b) (2 a-b)-(3 a-b) b x^2}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{2 d}\\ &=-\frac{b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{2 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{d}-\frac{((3 a-b) b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b+b x^2}} \, dx,x,\coth (c+d x)\right )}{2 d}\\ &=-\frac{b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{2 d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac{((3 a-b) b) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{2 d}\\ &=\frac{a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac{(3 a-b) \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \coth (c+d x)}{\sqrt{a-b+b \coth ^2(c+d x)}}\right )}{2 d}-\frac{b \coth (c+d x) \sqrt{a-b+b \coth ^2(c+d x)}}{2 d}\\ \end{align*}
Mathematica [A] time = 0.826508, size = 193, normalized size = 1.53 \[ \frac{\sinh ^3(c+d x) \left (a+b \text{csch}^2(c+d x)\right )^{3/2} \left (2 \sqrt{2} a^{3/2} \log \left (\sqrt{a \cosh (2 (c+d x))-a+2 b}+\sqrt{2} \sqrt{a} \cosh (c+d x)\right )+\sqrt{2} \sqrt{b} (b-3 a) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \cosh (c+d x)}{\sqrt{a \cosh (2 (c+d x))-a+2 b}}\right )-b \coth (c+d x) \text{csch}(c+d x) \sqrt{a \cosh (2 (c+d x))-a+2 b}\right )}{d (a \cosh (2 (c+d x))-a+2 b)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Csch[c + d*x]^2)^(3/2),x]
[Out]
((a + b*Csch[c + d*x]^2)^(3/2)*(Sqrt[2]*Sqrt[b]*(-3*a + b)*ArcTanh[(Sqrt[2]*Sqrt[b]*Cosh[c + d*x])/Sqrt[-a + 2
*b + a*Cosh[2*(c + d*x)]]] - b*Sqrt[-a + 2*b + a*Cosh[2*(c + d*x)]]*Coth[c + d*x]*Csch[c + d*x] + 2*Sqrt[2]*a^
(3/2)*Log[Sqrt[2]*Sqrt[a]*Cosh[c + d*x] + Sqrt[-a + 2*b + a*Cosh[2*(c + d*x)]]])*Sinh[c + d*x]^3)/(d*(-a + 2*b
+ a*Cosh[2*(c + d*x)])^(3/2))
________________________________________________________________________________________
Maple [F] time = 0.147, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*csch(d*x+c)^2)^(3/2),x)
[Out]
int((a+b*csch(d*x+c)^2)^(3/2),x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{csch}\left (d x + c\right )^{2} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*csch(d*x+c)^2)^(3/2),x, algorithm="maxima")
[Out]
integrate((b*csch(d*x + c)^2 + a)^(3/2), x)
________________________________________________________________________________________
Fricas [B] time = 4.68261, size = 17550, normalized size = 139.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*csch(d*x+c)^2)^(3/2),x, algorithm="fricas")
[Out]
[1/4*((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*
a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(a)*lo
g((a*b^2*cosh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(a*b^2 + b^3)*cos
h(d*x + c)^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + a*b^2 + b^3)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d*x + c)^3 + 3*(a
*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 + (70*a*b^2*cosh(d*x +
c)^4 + a^3 - 4*a^2*b + 9*a*b^2 + 30*(a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a*b^2*cosh(d*x + c)
^5 + 10*(a*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 - 2*(a^
3 - 3*a^2*b)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(a*b^2 + b^3)*cosh(d*x + c)^4 - a^3 + 3*a^2*b
+ 3*(a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x
+ c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x
+ c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c)^2 + (1
5*b^2*cosh(d*x + c)^4 + 18*b^2*cosh(d*x + c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^2 + a^2 + 2*(3*b^2*cosh(d*x + c)^5
+ 6*b^2*cosh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*sqrt((a*cosh(d*x + c)^2 + a*sin
h(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(2*a*b^2*cosh
(d*x + c)^7 + 3*(a*b^2 + b^3)*cosh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^3 - (a^3 - 3*a^2*b)*co
sh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^6 + 6*cosh(d*x + c)^5*sinh(d*x + c) + 15*cosh(d*x + c)^4*sinh(d*x +
c)^2 + 20*cosh(d*x + c)^3*sinh(d*x + c)^3 + 15*cosh(d*x + c)^2*sinh(d*x + c)^4 + 6*cosh(d*x + c)*sinh(d*x + c
)^5 + sinh(d*x + c)^6)) - ((3*a - b)*cosh(d*x + c)^4 + 4*(3*a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a - b)*s
inh(d*x + c)^4 - 2*(3*a - b)*cosh(d*x + c)^2 + 2*(3*(3*a - b)*cosh(d*x + c)^2 - 3*a + b)*sinh(d*x + c)^2 + 4*(
(3*a - b)*cosh(d*x + c)^3 - (3*a - b)*cosh(d*x + c))*sinh(d*x + c) + 3*a - b)*sqrt(b)*log(((a + b)*cosh(d*x +
c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 - 2*(a - 3*b)*cosh(d*x + c)^2 + 2*(3*
(a + b)*cosh(d*x + c)^2 - a + 3*b)*sinh(d*x + c)^2 + 2*sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c
) + sinh(d*x + c)^2 + 1)*sqrt(b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*c
osh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*((a + b)*cosh(d*x + c)^3 - (a - 3*b)*cosh(d*x + c))*sinh(d*
x + c) + a + b)/(cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 -
1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)) + (a*cosh(d*x
+ c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2
- a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(a)*log(-(a*cosh(d*x +
c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c
)^2 - a + b)*sinh(d*x + c)^2 + sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)
*sqrt(a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x +
c) + sinh(d*x + c)^2)) + 4*(a*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a)/(cosh(d*x + c)^2 + 2
*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x +
c) + b*sinh(d*x + c)^2 + b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(
d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 + d*sinh(d*
x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 - d*cosh(d
*x + c))*sinh(d*x + c) + d), 1/4*(2*((3*a - b)*cosh(d*x + c)^4 + 4*(3*a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (
3*a - b)*sinh(d*x + c)^4 - 2*(3*a - b)*cosh(d*x + c)^2 + 2*(3*(3*a - b)*cosh(d*x + c)^2 - 3*a + b)*sinh(d*x +
c)^2 + 4*((3*a - b)*cosh(d*x + c)^3 - (3*a - b)*cosh(d*x + c))*sinh(d*x + c) + 3*a - b)*sqrt(-b)*arctan(sqrt(2
)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqrt(-b)*sqrt((a*cosh(d*x + c)^2 + a
*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*cosh(d*x +
c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x
+ c)^2 - a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + (a*c
osh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x
+ c)^2 - a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(a)*log((a*b^2*c
osh(d*x + c)^8 + 8*a*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + a*b^2*sinh(d*x + c)^8 + 2*(a*b^2 + b^3)*cosh(d*x + c)
^6 + 2*(14*a*b^2*cosh(d*x + c)^2 + a*b^2 + b^3)*sinh(d*x + c)^6 + 4*(14*a*b^2*cosh(d*x + c)^3 + 3*(a*b^2 + b^3
)*cosh(d*x + c))*sinh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^4 + (70*a*b^2*cosh(d*x + c)^4 + a^3
- 4*a^2*b + 9*a*b^2 + 30*(a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(14*a*b^2*cosh(d*x + c)^5 + 10*(a
*b^2 + b^3)*cosh(d*x + c)^3 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + a^3 - 2*(a^3 - 3*a^2*
b)*cosh(d*x + c)^2 + 2*(14*a*b^2*cosh(d*x + c)^6 + 15*(a*b^2 + b^3)*cosh(d*x + c)^4 - a^3 + 3*a^2*b + 3*(a^3 -
4*a^2*b + 9*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + sqrt(2)*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh
(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*b^2*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + b^2)*sinh(d*x + c)^4 +
4*(5*b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c)^2 + (15*b^2*cosh
(d*x + c)^4 + 18*b^2*cosh(d*x + c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^2 + a^2 + 2*(3*b^2*cosh(d*x + c)^5 + 6*b^2*c
osh(d*x + c)^3 - (a^2 - 4*a*b)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)
^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(2*a*b^2*cosh(d*x + c)^
7 + 3*(a*b^2 + b^3)*cosh(d*x + c)^5 + (a^3 - 4*a^2*b + 9*a*b^2)*cosh(d*x + c)^3 - (a^3 - 3*a^2*b)*cosh(d*x + c
))*sinh(d*x + c))/(cosh(d*x + c)^6 + 6*cosh(d*x + c)^5*sinh(d*x + c) + 15*cosh(d*x + c)^4*sinh(d*x + c)^2 + 20
*cosh(d*x + c)^3*sinh(d*x + c)^3 + 15*cosh(d*x + c)^2*sinh(d*x + c)^4 + 6*cosh(d*x + c)*sinh(d*x + c)^5 + sinh
(d*x + c)^6)) + (a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)
^2 + 2*(3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*
sqrt(a)*log(-(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - b)*cosh(d*x +
c)^2 + 2*(3*a*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 + sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x
+ c) + sinh(d*x + c)^2 - 1)*sqrt(a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 -
2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) + 4*(a*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c
) + a)/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*sqrt(2)*(b*cosh(d*x + c)^2 + 2
*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)
/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*
sinh(d*x + c)^3 + d*sinh(d*x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(d
*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d), -1/4*(2*(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*
x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 4*(a*cosh(d
*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(-a)*arctan(sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)
*sinh(d*x + c) + b*sinh(d*x + c)^2 + a)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(
d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*b*cosh(d*x + c)^4 + 4*a*b*cosh(d*x + c)*sinh
(d*x + c)^3 + a*b*sinh(d*x + c)^4 - (a^2 - 3*a*b)*cosh(d*x + c)^2 + (6*a*b*cosh(d*x + c)^2 - a^2 + 3*a*b)*sinh
(d*x + c)^2 + a^2 + 2*(2*a*b*cosh(d*x + c)^3 - (a^2 - 3*a*b)*cosh(d*x + c))*sinh(d*x + c))) + 2*(a*cosh(d*x +
c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 -
a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(-a)*arctan(sqrt(2)*(cosh(
d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*
x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*cosh(d*x + c)^4 +
4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 -
a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + ((3*a - b)*c
osh(d*x + c)^4 + 4*(3*a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a - b)*sinh(d*x + c)^4 - 2*(3*a - b)*cosh(d*x
+ c)^2 + 2*(3*(3*a - b)*cosh(d*x + c)^2 - 3*a + b)*sinh(d*x + c)^2 + 4*((3*a - b)*cosh(d*x + c)^3 - (3*a - b)*
cosh(d*x + c))*sinh(d*x + c) + 3*a - b)*sqrt(b)*log(((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*
x + c)^3 + (a + b)*sinh(d*x + c)^4 - 2*(a - 3*b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 - a + 3*b)*sin
h(d*x + c)^2 + 2*sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqrt(b)*sqrt(
(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x
+ c)^2)) + 4*((a + b)*cosh(d*x + c)^3 - (a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + a + b)/(cosh(d*x + c)^4 + 4*c
osh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2*cosh(d*x + c)^2
+ 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)) + 2*sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*
sinh(d*x + c) + b*sinh(d*x + c)^2 + b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2
- 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 +
d*sinh(d*x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3
- d*cosh(d*x + c))*sinh(d*x + c) + d), -1/2*((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d
*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - a*cosh(
d*x + c))*sinh(d*x + c) + a)*sqrt(-a)*arctan(sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*
sinh(d*x + c)^2 + a)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh
(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*b*cosh(d*x + c)^4 + 4*a*b*cosh(d*x + c)*sinh(d*x + c)^3 + a*b*s
inh(d*x + c)^4 - (a^2 - 3*a*b)*cosh(d*x + c)^2 + (6*a*b*cosh(d*x + c)^2 - a^2 + 3*a*b)*sinh(d*x + c)^2 + a^2 +
2*(2*a*b*cosh(d*x + c)^3 - (a^2 - 3*a*b)*cosh(d*x + c))*sinh(d*x + c))) + (a*cosh(d*x + c)^4 + 4*a*cosh(d*x +
c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*a*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a)*sinh(d*x + c)^2 +
4*(a*cosh(d*x + c)^3 - a*cosh(d*x + c))*sinh(d*x + c) + a)*sqrt(-a)*arctan(sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d
*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-a)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(
cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sin
h(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a + 2*b)*sinh(d*x +
c)^2 + 4*(a*cosh(d*x + c)^3 - (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) - ((3*a - b)*cosh(d*x + c)^4 + 4*(3
*a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (3*a - b)*sinh(d*x + c)^4 - 2*(3*a - b)*cosh(d*x + c)^2 + 2*(3*(3*a -
b)*cosh(d*x + c)^2 - 3*a + b)*sinh(d*x + c)^2 + 4*((3*a - b)*cosh(d*x + c)^3 - (3*a - b)*cosh(d*x + c))*sinh(d
*x + c) + 3*a - b)*sqrt(-b)*arctan(sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2
+ 1)*sqrt(-b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d
*x + c) + sinh(d*x + c)^2))/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(a
- 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (a - 2*b)*
cosh(d*x + c))*sinh(d*x + c) + a)) + sqrt(2)*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x
+ c)^2 + b)*sqrt((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 - a + 2*b)/(cosh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*
x + c) + sinh(d*x + c)^2)))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 + d*sinh(d*x + c)^4 - 2*d*c
osh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*
x + c) + d)]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{csch}^{2}{\left (c + d x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*csch(d*x+c)**2)**(3/2),x)
[Out]
Integral((a + b*csch(c + d*x)**2)**(3/2), x)
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*csch(d*x+c)^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: RuntimeError