\(\int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\) [137]

Optimal result

Integrand size = 23, antiderivative size = 262 \[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))} \]

[Out]

1/4*b*arctanh((a+b*sech(d*x+c))^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/d-1/4*b*arctanh((a+b*sech(d*x+c))^(1/2)/(a+b)^(
1/2))/(a+b)^(3/2)/d+2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)-arctanh((a+b*sech(d*x+c))^(1/2)/(a-b)
^(1/2))/d/(a-b)^(1/2)-arctanh((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2))/d/(a+b)^(1/2)-1/4*(a+b*sech(d*x+c))^(1/2)/(
a+b)/d/(1-sech(d*x+c))-1/4*(a+b*sech(d*x+c))^(1/2)/(a-b)/d/(1+sech(d*x+c))

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3970, 912, 1252, 212, 205, 213} \[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{3/2}}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a+b) (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 d (a-b) (\text {sech}(c+d x)+1)} \]

[In]

Int[Coth[c + d*x]^3/Sqrt[a + b*Sech[c + d*x]],x]

[Out]

(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/(S
qrt[a - b]*d) + (b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) - (b*ArcTanh[Sqrt[a + b
*Sech[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) - ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b
]*d) - Sqrt[a + b*Sech[c + d*x]]/(4*(a + b)*d*(1 - Sech[c + d*x])) - Sqrt[a + b*Sech[c + d*x]]/(4*(a - b)*d*(1
+ Sech[c + d*x]))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 1252

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

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*} \text {integral}& = -\frac {b^4 \text {Subst}\left (\int \frac {1}{x \sqrt {a+x} \left (b^2-x^2\right )^2} \, dx,x,b \text {sech}(c+d x)\right )}{d} \\ & = -\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^2\right ) \left (-a^2+b^2+2 a x^2-x^4\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d} \\ & = -\frac {\left (2 b^4\right ) \text {Subst}\left (\int \left (-\frac {1}{b^4 \left (a-x^2\right )}+\frac {1}{4 b^3 \left (a+b-x^2\right )^2}+\frac {1}{2 b^4 \left (a+b-x^2\right )}-\frac {1}{4 b^3 \left (-a+b+x^2\right )^2}-\frac {1}{2 b^4 \left (-a+b+x^2\right )}\right ) \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {\text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {2 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {b \text {Subst}\left (\int \frac {1}{\left (a+b-x^2\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d}+\frac {b \text {Subst}\left (\int \frac {1}{\left (-a+b+x^2\right )^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))}-\frac {b \text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 (a-b) d}-\frac {b \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 (a+b) d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a+b) d (1-\text {sech}(c+d x))}-\frac {\sqrt {a+b \text {sech}(c+d x)}}{4 (a-b) d (1+\text {sech}(c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.13 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.07 \[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=-\frac {-\frac {b^2 \arctan \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {-a+b}}\right )}{(-a+b)^{3/2}}-\frac {8 b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {4 b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {4 a \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}+4 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )-\frac {b \sqrt {a+b \text {sech}(c+d x)}}{(a+b) (-1+\text {sech}(c+d x))}+\frac {b \sqrt {a+b \text {sech}(c+d x)}}{(a-b) (1+\text {sech}(c+d x))}}{4 b d} \]

[In]

Integrate[Coth[c + d*x]^3/Sqrt[a + b*Sech[c + d*x]],x]

[Out]

-1/4*(-((b^2*ArcTan[Sqrt[a + b*Sech[c + d*x]]/Sqrt[-a + b]])/(-a + b)^(3/2)) - (8*b*ArcTanh[Sqrt[a + b*Sech[c
+ d*x]]/Sqrt[a]])/Sqrt[a] + (4*b*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]])/Sqrt[a - b] + (b^2*ArcTanh[Sq
rt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/(a + b)^(3/2) - (4*a*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/Sqr
t[a + b] + 4*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]] - (b*Sqrt[a + b*Sech[c + d*x]])/((a +
b)*(-1 + Sech[c + d*x])) + (b*Sqrt[a + b*Sech[c + d*x]])/((a - b)*(1 + Sech[c + d*x])))/(b*d)

Maple [F]

\[\int \frac {\coth \left (d x +c \right )^{3}}{\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}d x\]

[In]

int(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x)

[Out]

int(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1870 vs. \(2 (218) = 436\).

Time = 4.30 (sec) , antiderivative size = 20300, normalized size of antiderivative = 77.48 \[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {\coth ^{3}{\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]

[In]

integrate(coth(d*x+c)**3/(a+b*sech(d*x+c))**(1/2),x)

[Out]

Integral(coth(c + d*x)**3/sqrt(a + b*sech(c + d*x)), x)

Maxima [F]

\[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\coth \left (d x + c\right )^{3}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(coth(d*x + c)^3/sqrt(b*sech(d*x + c) + a), x)

Giac [F]

\[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\coth \left (d x + c\right )^{3}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(coth(d*x+c)^3/(a+b*sech(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(coth(d*x + c)^3/sqrt(b*sech(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^3(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]

[In]

int(coth(c + d*x)^3/(a + b/cosh(c + d*x))^(1/2),x)

[Out]

int(coth(c + d*x)^3/(a + b/cosh(c + d*x))^(1/2), x)