3.2.45 \(\int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\) [145]
Optimal. Leaf size=142 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {2 b^2}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}} \]
[Out]
2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-arctanh((a+b*sech(d*x+c))^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/
d-arctanh((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2))/(a+b)^(3/2)/d+2*b^2/a/(a^2-b^2)/d/(a+b*sech(d*x+c))^(1/2)
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Rubi [A]
time = 0.16, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3970, 912,
1301, 212} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}+\frac {2 b^2}{a d \left (a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d (a-b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d (a+b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]
[Out]
(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(a^(3/2)*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/((
a - b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/((a + b)^(3/2)*d) + (2*b^2)/(a*(a^2 - b^2)*d*
Sqrt[a + b*Sech[c + d*x]])
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 912
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 + a*e^2)/e^2 - 2*c*
d*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]
Rule 1301
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]
Rule 3970
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Rubi steps
\begin {align*} \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx &=-\frac {b^2 \text {Subst}\left (\int \frac {1}{x (a+x)^{3/2} \left (b^2-x^2\right )} \, dx,x,b \text {sech}(c+d x)\right )}{d}\\ &=-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (-a+x^2\right ) \left (-a^2+b^2+2 a x^2-x^4\right )} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \left (\frac {1}{a \left (a^2-b^2\right ) x^2}-\frac {1}{a b^2 \left (a-x^2\right )}+\frac {1}{2 (a-b) b^2 \left (a-b-x^2\right )}+\frac {1}{2 b^2 (a+b) \left (a+b-x^2\right )}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=\frac {2 b^2}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{a d}-\frac {\text {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{(a-b) d}-\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{(a+b) d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {2 b^2}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 4.42, size = 483, normalized size = 3.40 \begin {gather*} \frac {\text {sech}(c+d x) \left (\frac {4 a b^2 (b+a \cosh (c+d x))}{a^2-b^2}+\frac {(b+a \cosh (c+d x))^{3/2} \left (a^{3/2} (-a+b)^{5/2} \tanh ^{-1}\left (\frac {a-a \cosh (c+d x)+\sqrt {a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {a+b}}\right )-\frac {1}{2} (a+b) \left (-2 i \sqrt {a} (-a+b)^{3/2} \tanh ^{-1}\left (\frac {a-a \cosh (c+d x)-i \sqrt {-a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {a+b}}\right ) \sqrt {-a^2 \cosh ^2(c+d x)}+2 \sqrt {a+b} \left (a^{3/2} (a+b) \text {ArcTan}\left (\frac {a+a \cosh (c+d x)-\sqrt {a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {-a+b}}\right ) \cosh (c+d x)+(a-b) \left (4 \sqrt {-a+b} \text {ArcTan}\left (\frac {\sqrt {b+a \cosh (c+d x)}}{\sqrt {-a \cosh (c+d x)}}\right )-i \sqrt {a} \text {ArcTan}\left (\frac {a+a \cosh (c+d x)+i \sqrt {-a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}{\sqrt {a} \sqrt {-a+b}}\right )\right ) \sqrt {-a^2 \cosh ^2(c+d x)}\right )\right ) \text {sech}(c+d x)\right )}{(-a+b)^{3/2} (a+b)^{3/2} \sqrt {a \cosh (c+d x)}}\right )}{2 a^2 d (a+b \text {sech}(c+d x))^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[Coth[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]
[Out]
(Sech[c + d*x]*((4*a*b^2*(b + a*Cosh[c + d*x]))/(a^2 - b^2) + ((b + a*Cosh[c + d*x])^(3/2)*(a^(3/2)*(-a + b)^(
5/2)*ArcTanh[(a - a*Cosh[c + d*x] + Sqrt[a*Cosh[c + d*x]]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a]*Sqrt[a + b])] -
((a + b)*((-2*I)*Sqrt[a]*(-a + b)^(3/2)*ArcTanh[(a - a*Cosh[c + d*x] - I*Sqrt[-(a*Cosh[c + d*x])]*Sqrt[b + a*C
osh[c + d*x]])/(Sqrt[a]*Sqrt[a + b])]*Sqrt[-(a^2*Cosh[c + d*x]^2)] + 2*Sqrt[a + b]*(a^(3/2)*(a + b)*ArcTan[(a
+ a*Cosh[c + d*x] - Sqrt[a*Cosh[c + d*x]]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a]*Sqrt[-a + b])]*Cosh[c + d*x] + (
a - b)*(4*Sqrt[-a + b]*ArcTan[Sqrt[b + a*Cosh[c + d*x]]/Sqrt[-(a*Cosh[c + d*x])]] - I*Sqrt[a]*ArcTan[(a + a*Co
sh[c + d*x] + I*Sqrt[-(a*Cosh[c + d*x])]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a]*Sqrt[-a + b])])*Sqrt[-(a^2*Cosh[c
+ d*x]^2)]))*Sech[c + d*x])/2))/((-a + b)^(3/2)*(a + b)^(3/2)*Sqrt[a*Cosh[c + d*x]])))/(2*a^2*d*(a + b*Sech[c
+ d*x])^(3/2))
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Maple [F]
time = 2.56, size = 0, normalized size = 0.00 \[\int \frac {\coth \left (d x +c \right )}{\left (a +b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x)
[Out]
int(coth(d*x+c)/(a+b*sech(d*x+c))^(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(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
integrate(coth(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1134 vs.
\(2 (122) = 244\).
time = 3.85, size = 14412, normalized size = 101.49 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
[-1/4*((a^5 + 2*a^4*b + a^3*b^2 + (a^5 + 2*a^4*b + a^3*b^2)*cosh(d*x + c)^2 + (a^5 + 2*a^4*b + a^3*b^2)*sinh(d
*x + c)^2 + 2*(a^4*b + 2*a^3*b^2 + a^2*b^3)*cosh(d*x + c) + 2*(a^4*b + 2*a^3*b^2 + a^2*b^3 + (a^5 + 2*a^4*b +
a^3*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cosh(d*x + c)^4 + (8*a^2 - 8*a*
b + b^2)*sinh(d*x + c)^4 + 4*(4*a*b - 3*b^2)*cosh(d*x + c)^3 + 4*(4*a*b - 3*b^2 + (8*a^2 - 8*a*b + b^2)*cosh(d
*x + c))*sinh(d*x + c)^3 + 2*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*(3*(8*a^2 - 8*a*b + b^2)*cosh(d*x + c
)^2 + 8*a^2 - 8*a*b + 3*b^2 + 6*(4*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*((2*a
- b)*cosh(d*x + c)^4 + (2*a - b)*sinh(d*x + c)^4 + 2*b*cosh(d*x + c)^3 + 2*(2*(2*a - b)*cosh(d*x + c) + b)*si
nh(d*x + c)^3 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*(2*a - b)*cosh(d*x + c)^2 + 3*b*cosh(d*x + c) + 2*a - b)*si
nh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(2*(2*a - b)*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 2*(2*a - b)*cosh(d*
x + c) + b)*sinh(d*x + c) + 2*a - b)*sqrt(a - b)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 4*(4*a*b - 3*b^2)
*cosh(d*x + c) + 4*((8*a^2 - 8*a*b + b^2)*cosh(d*x + c)^3 + 3*(4*a*b - 3*b^2)*cosh(d*x + c)^2 + 4*a*b - 3*b^2
+ (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^4 + 4*(cosh(d*x + c) + 1)*sinh(d*x + c)
^3 + sinh(d*x + c)^4 + 4*cosh(d*x + c)^3 + 6*(cosh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sinh(d*x + c)^2 + 6*cosh(
d*x + c)^2 + 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)^2 + 3*cosh(d*x + c) + 1)*sinh(d*x + c) + 4*cosh(d*x + c) + 1
)) - (a^5 - 2*a^4*b + a^3*b^2 + (a^5 - 2*a^4*b + a^3*b^2)*cosh(d*x + c)^2 + (a^5 - 2*a^4*b + a^3*b^2)*sinh(d*x
+ c)^2 + 2*(a^4*b - 2*a^3*b^2 + a^2*b^3)*cosh(d*x + c) + 2*(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^
3*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cosh(d*x + c)^4 + (8*a^2 + 8*a*b
+ b^2)*sinh(d*x + c)^4 + 4*(4*a*b + 3*b^2)*cosh(d*x + c)^3 + 4*(4*a*b + 3*b^2 + (8*a^2 + 8*a*b + b^2)*cosh(d*x
+ c))*sinh(d*x + c)^3 + 2*(8*a^2 + 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*(3*(8*a^2 + 8*a*b + b^2)*cosh(d*x + c)^
2 + 8*a^2 + 8*a*b + 3*b^2 + 6*(4*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 8*a^2 + 8*a*b + b^2 - 4*((2*a +
b)*cosh(d*x + c)^4 + (2*a + b)*sinh(d*x + c)^4 + 2*b*cosh(d*x + c)^3 + 2*(2*(2*a + b)*cosh(d*x + c) + b)*sinh
(d*x + c)^3 + 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*(2*a + b)*cosh(d*x + c)^2 + 3*b*cosh(d*x + c) + 2*a + b)*sinh
(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(2*(2*a + b)*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 2*(2*a + b)*cosh(d*x
+ c) + b)*sinh(d*x + c) + 2*a + b)*sqrt(a + b)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 4*(4*a*b + 3*b^2)*c
osh(d*x + c) + 4*((8*a^2 + 8*a*b + b^2)*cosh(d*x + c)^3 + 3*(4*a*b + 3*b^2)*cosh(d*x + c)^2 + 4*a*b + 3*b^2 +
(8*a^2 + 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^4 + 4*(cosh(d*x + c) - 1)*sinh(d*x + c)^3
+ sinh(d*x + c)^4 - 4*cosh(d*x + c)^3 + 6*(cosh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c)^2 + 6*cosh(d*
x + c)^2 + 4*(cosh(d*x + c)^3 - 3*cosh(d*x + c)^2 + 3*cosh(d*x + c) - 1)*sinh(d*x + c) - 4*cosh(d*x + c) + 1))
- 2*(a^5 - 2*a^3*b^2 + a*b^4 + (a^5 - 2*a^3*b^2 + a*b^4)*cosh(d*x + c)^2 + (a^5 - 2*a^3*b^2 + a*b^4)*sinh(d*x
+ c)^2 + 2*(a^4*b - 2*a^2*b^3 + b^5)*cosh(d*x + c) + 2*(a^4*b - 2*a^2*b^3 + b^5 + (a^5 - 2*a^3*b^2 + a*b^4)*c
osh(d*x + c))*sinh(d*x + c))*sqrt(a)*log(-(2*a^2*cosh(d*x + c)^4 + 2*a^2*sinh(d*x + c)^4 + 4*a*b*cosh(d*x + c)
^3 + 4*(2*a^2*cosh(d*x + c) + a*b)*sinh(d*x + c)^3 + 4*a*b*cosh(d*x + c) + (4*a^2 + b^2)*cosh(d*x + c)^2 + (12
*a^2*cosh(d*x + c)^2 + 12*a*b*cosh(d*x + c) + 4*a^2 + b^2)*sinh(d*x + c)^2 + 2*a^2 + 2*(a*cosh(d*x + c)^4 + a*
sinh(d*x + c)^4 + b*cosh(d*x + c)^3 + (4*a*cosh(d*x + c) + b)*sinh(d*x + c)^3 + 2*a*cosh(d*x + c)^2 + (6*a*cos
h(d*x + c)^2 + 3*b*cosh(d*x + c) + 2*a)*sinh(d*x + c)^2 + b*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 + 3*b*cosh(d*
x + c)^2 + 4*a*cosh(d*x + c) + b)*sinh(d*x + c) + a)*sqrt(a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 2*(4*
a^2*cosh(d*x + c)^3 + 6*a*b*cosh(d*x + c)^2 + 2*a*b + (4*a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x +
c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 8*(a^3*b^2 - a*b^4 + (a^3*b^2 - a*b^4)*cosh(d*x + c
)^2 + 2*(a^3*b^2 - a*b^4)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b^2 - a*b^4)*sinh(d*x + c)^2)*sqrt((a*cosh(d*x +
c) + b)/cosh(d*x + c)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*sinh(d*
x + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*cosh(d*x + c) + (a^7 - 2*a^5*b^2 + a^3*b^4)*d + 2*((a^7 - 2*a^5*b
^2 + a^3*b^4)*d*cosh(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d)*sinh(d*x + c)), 1/4*(2*(a^5 - 2*a^4*b + a^3*b
^2 + (a^5 - 2*a^4*b + a^3*b^2)*cosh(d*x + c)^2 + (a^5 - 2*a^4*b + a^3*b^2)*sinh(d*x + c)^2 + 2*(a^4*b - 2*a^3*
b^2 + a^2*b^3)*cosh(d*x + c) + 2*(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*cosh(d*x + c))*sinh(
d*x + c))*sqrt(-a - b)*arctan(2*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)*sqrt(-
a - b)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)/(a+b*sech(d*x+c))**(3/2),x)
[Out]
Integral(coth(c + d*x)/(a + b*sech(c + d*x))**(3/2), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="giac")
[Out]
integrate(coth(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {coth}\left (c+d\,x\right )}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(c + d*x)/(a + b/cosh(c + d*x))^(3/2),x)
[Out]
int(coth(c + d*x)/(a + b/cosh(c + d*x))^(3/2), x)
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