3.1.10 \(\int (a+b \text {csch}^2(c+d x))^{3/2} \, dx\) [10]
Optimal. Leaf size=126 \[ \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{d}-\frac {(3 a-b) \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth (c+d x)}{\sqrt {a-b+b \coth ^2(c+d x)}}\right )}{2 d}-\frac {b \coth (c+d x) \sqrt {a-b+b \coth ^2(c+d x)}}{2 d} \]
[Out]
a^(3/2)*arctanh(coth(d*x+c)*a^(1/2)/(a-b+b*coth(d*x+c)^2)^(1/2))/d-1/2*(3*a-b)*arctanh(coth(d*x+c)*b^(1/2)/(a-
b+b*coth(d*x+c)^2)^(1/2))*b^(1/2)/d-1/2*b*coth(d*x+c)*(a-b+b*coth(d*x+c)^2)^(1/2)/d
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Rubi [A]
time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, 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 = {4213, 427, 537,
223, 212, 385} \begin {gather*} \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} \end {gather*}
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 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 223
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 385
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]
Rule 427
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 537
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 4213
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]
Rubi steps
\begin {align*} \int \left (a+b \text {csch}^2(c+d x)\right )^{3/2} \, dx &=\frac {\text {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 {\text {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 \text {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) \text {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 \text {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) \text {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*}
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Mathematica [A]
time = 0.55, size = 193, normalized size = 1.53 \begin {gather*} \frac {\left (a+b \text {csch}^2(c+d x)\right )^{3/2} \left (\sqrt {2} \sqrt {b} (-3 a+b) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \cosh (c+d x)}{\sqrt {-a+2 b+a \cosh (2 (c+d x))}}\right )-b \sqrt {-a+2 b+a \cosh (2 (c+d x))} \coth (c+d x) \text {csch}(c+d x)+2 \sqrt {2} a^{3/2} \log \left (\sqrt {2} \sqrt {a} \cosh (c+d x)+\sqrt {-a+2 b+a \cosh (2 (c+d x))}\right )\right ) \sinh ^3(c+d x)}{d (-a+2 b+a \cosh (2 (c+d x)))^{3/2}} \end {gather*}
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))
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Maple [F]
time = 1.70, size = 0, normalized size = 0.00 \[\int \left (a +b \mathrm {csch}\left (d x +c \right )^{2}\right )^{\frac {3}{2}}\, dx\]
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)
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1272 vs.
\(2 (108) = 216\).
time = 0.62, size = 6645, normalized size = 52.74 \begin {gather*} \text {Too large to display} \end {gather*}
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...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
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)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(ex
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b/sinh(c + d*x)^2)^(3/2),x)
[Out]
int((a + b/sinh(c + d*x)^2)^(3/2), x)
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