3.2.29 \(\int \coth ^3(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx\) [129]

Optimal. Leaf size=217 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 \sqrt {a-b} d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 \sqrt {a+b} d}-\frac {\coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)}}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.23, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3970, 912, 1329, 1192, 12, 1107, 212, 1184, 213} \begin {gather*} -\frac {\coth ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)}}{2 d}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 d \sqrt {a-b}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 d \sqrt {a+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 1107

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/(b/
2 - q/2 + c*x^2), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 1184

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

Rule 1192

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*(a*b*e - d*(b^2 - 2*a
*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1329

Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[
f^2/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 2)*(a*e + c*d*x^2)*(a + b*x^2 + c*x^4)^p, x], x] - Dist[d*e*(f^2/(
c*d^2 - b*d*e + a*e^2)), Int[(f*x)^(m - 2)*((a + b*x^2 + c*x^4)^(p + 1)/(d + e*x^2)), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 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*} \int \coth ^3(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx &=-\frac {b^4 \text {Subst}\left (\int \frac {\sqrt {a+x}}{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 {x^2}{\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^2\right ) \text {Subst}\left (\int \frac {-a^2+b^2+a x^2}{\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 a b^2\right ) \text {Subst}\left (\int \frac {1}{\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 {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}-\frac {\left (2 a b^2\right ) \text {Subst}\left (\int \left (-\frac {1}{b^2 \left (a-x^2\right )}+\frac {1}{2 b^2 \left (a+b-x^2\right )}-\frac {1}{2 b^2 \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 {6 b^2 \left (a^2-b^2\right )}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 \left (a^2-b^2\right ) d}\\ &=\frac {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}-\frac {a \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {a \text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{2 d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}+\frac {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 d}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{4 d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{4 \sqrt {a-b} d}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{4 \sqrt {a+b} d}+\frac {b^2 \sqrt {a+b \text {sech}(c+d x)}}{2 d \left (a^2-b^2-2 a (a+b \text {sech}(c+d x))+(a+b \text {sech}(c+d x))^2\right )}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 18.14, size = 456, normalized size = 2.10 \begin {gather*} \frac {\left (\frac {-\sqrt {a} \sqrt {-a+b} (2 a+3 b) \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 ) \cosh (c+d x)+2 i \sqrt {a} \sqrt {-a+b} \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)}+\sqrt {a+b} \left (\sqrt {a} (2 a-3 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)+2 \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 )}{\sqrt {-a+b} \sqrt {a+b} \sqrt {a \cosh (c+d x)} \sqrt {b+a \cosh (c+d x)}}-2 \coth ^2(c+d x)\right ) \sqrt {a+b \text {sech}(c+d x)}}{4 d} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]
time = 3.09, size = 0, normalized size = 0.00 \[\int \left (\coth ^{3}\left (d x +c \right )\right ) \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1399 vs. \(2 (179) = 358\).
time = 1.13, size = 16532, normalized size = 76.18 \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)^3*(a+b*sech(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[-1/16*(((4*a^2 + a*b - 3*b^2)*cosh(d*x + c)^4 + 4*(4*a^2 + a*b - 3*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a^
2 + a*b - 3*b^2)*sinh(d*x + c)^4 - 2*(4*a^2 + a*b - 3*b^2)*cosh(d*x + c)^2 + 2*(3*(4*a^2 + a*b - 3*b^2)*cosh(d
*x + c)^2 - 4*a^2 - a*b + 3*b^2)*sinh(d*x + c)^2 + 4*a^2 + a*b - 3*b^2 + 4*((4*a^2 + a*b - 3*b^2)*cosh(d*x + c
)^3 - (4*a^2 + a*b - 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)*co
sh(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)*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)) - ((4*a^2 - a*b - 3*b^2)*cosh(d*x + c)^4 + 4*(4*a^2 - a*b - 3*b^2)*cosh(d*x + c)*sinh(d*
x + c)^3 + (4*a^2 - a*b - 3*b^2)*sinh(d*x + c)^4 - 2*(4*a^2 - a*b - 3*b^2)*cosh(d*x + c)^2 + 2*(3*(4*a^2 - a*b
- 3*b^2)*cosh(d*x + c)^2 - 4*a^2 + a*b + 3*b^2)*sinh(d*x + c)^2 + 4*a^2 - a*b - 3*b^2 + 4*((4*a^2 - a*b - 3*b
^2)*cosh(d*x + c)^3 - (4*a^2 - a*b - 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*co
sh(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)) - 8*((a^2 - b^2)*cosh(d*x + c)^4 + 4*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x
+ c)^3 + (a^2 - b^2)*sinh(d*x + c)^4 - 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - b^2)*cosh(d*x + c)^2 - a^2
+ b^2)*sinh(d*x + c)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(d*x + c)^3 - (a^2 - b^2)*cosh(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*cosh(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^2 - b^2)*cosh(d*x + c)^4 + 4*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 +
(a^2 - b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - b^2)*cosh(d*x + c)^2 + a^2 - b^2)*s
inh(d*x + c)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((
a*cosh(d*x + c) + b)/cosh(d*x + c)))/((a^2 - b^2)*d*cosh(d*x + c)^4 + 4*(a^2 - b^2)*d*cosh(d*x + c)*sinh(d*x +
c)^3 + (a^2 - b^2)*d*sinh(d*x + c)^4 - 2*(a^2 - b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^2 - b^2)*d*cosh(d*x + c)^2 -
(a^2 - b^2)*d)*sinh(d*x + c)^2 + (a^2 - b^2)*d...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}} \coth ^{3}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {coth}\left (c+d\,x\right )}^3\,\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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