3.2.28 \(\int \coth (c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx\) [128]
Optimal. Leaf size=106 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d}-\frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d} \]
[Out]
2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))*a^(1/2)/d-arctanh((a+b*sech(d*x+c))^(1/2)/(a-b)^(1/2))*(a-b)^(1/2)/
d-arctanh((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2))*(a+b)^(1/2)/d
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Rubi [A]
time = 0.12, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3970, 912,
1301, 212, 213} \begin {gather*} \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d}-\frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Coth[c + d*x]*Sqrt[a + b*Sech[c + d*x]],x]
[Out]
(2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt
[a - b]])/d - (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]])/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 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 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 \coth (c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx &=-\frac {b^2 \text {Subst}\left (\int \frac {\sqrt {a+x}}{x \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 {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 {a}{b^2 \left (a-x^2\right )}+\frac {a+b}{2 b^2 \left (a+b-x^2\right )}+\frac {-a+b}{2 b^2 \left (-a+b+x^2\right )}\right ) \, 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 {(a-b) \text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}-\frac {(a+b) \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \text {sech}(c+d x)}\right )}{d}\\ &=\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{d}-\frac {\sqrt {a-b} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{d}-\frac {\sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 1.29, size = 211, normalized size = 1.99 \begin {gather*} \frac {\left (-\sqrt {-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 )+2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b+a \cosh (c+d x)}}{\sqrt {a \cosh (c+d x)}}\right )-\sqrt {a+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 )\right ) \sqrt {a \cosh (c+d x)} \sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a} d \sqrt {b+a \cosh (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Coth[c + d*x]*Sqrt[a + b*Sech[c + d*x]],x]
[Out]
((-(Sqrt[-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])]) + 2*Sqrt[a]*ArcTanh[Sqrt[b + a*Cosh[c + d*x]]/Sqrt[a*Cosh[c + d*x]]] - Sqrt[a + b]*ArcTanh[(a - a*C
osh[c + d*x] + Sqrt[a*Cosh[c + d*x]]*Sqrt[b + a*Cosh[c + d*x]])/(Sqrt[a]*Sqrt[a + b])])*Sqrt[a*Cosh[c + d*x]]*
Sqrt[a + b*Sech[c + d*x]])/(Sqrt[a]*d*Sqrt[b + a*Cosh[c + d*x]])
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Maple [F]
time = 2.89, size = 0, normalized size = 0.00 \[\int \coth \left (d x +c \right ) \sqrt {a +b \,\mathrm {sech}\left (d x +c \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(d*x+c)*(a+b*sech(d*x+c))^(1/2),x)
[Out]
int(coth(d*x+c)*(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)*(a+b*sech(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*sech(d*x + c) + a)*coth(d*x + c), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 410 vs.
\(2 (88) = 176\).
time = 0.73, size = 8620, normalized size = 81.32 \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))^(1/2),x, algorithm="fricas")
[Out]
[1/4*(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)*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)) + 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)*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*c
osh(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*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)/c
osh(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)))/d, -1/4*(4*sqrt(-a)*arctan((cos
h(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) + b)/cosh(d
*x + c))/(a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + b*cosh(d*x + c) + (2*a*cosh(d*x + c) + b)*sinh(d*x + c) + a)
) - 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)*s
inh(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)*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)) - sqrt(a + b)*log(-((8*a^2 + 8*a
*b + b^2)*cosh(d*x + c)^4 + (8*a^2 + 8*a*b + b^...
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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 {\left (c + d x \right )}\, 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))**(1/2),x)
[Out]
Integral(sqrt(a + b*sech(c + d*x))*coth(c + d*x), 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))^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(b*sech(d*x + c) + a)*coth(d*x + c), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {coth}\left (c+d\,x\right )\,\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)*(a + b/cosh(c + d*x))^(1/2),x)
[Out]
int(coth(c + d*x)*(a + b/cosh(c + d*x))^(1/2), x)
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