Integrand size = 22, antiderivative size = 47 \[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=2 a^2 \sqrt {x}-\frac {4 a b \text {arctanh}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d}-\frac {2 b^2 \coth \left (c+d \sqrt {x}\right )}{d} \] Output:
2*a^2*x^(1/2)-4*a*b*arctanh(cosh(c+d*x^(1/2)))/d-2*b^2*coth(c+d*x^(1/2))/d
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=-\frac {b^2 \coth \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )-2 a \left (a c+a d \sqrt {x}-2 b \log \left (\cosh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )+2 b \log \left (\sinh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )\right )\right )+b^2 \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{d} \] Input:
Integrate[(a + b*Csch[c + d*Sqrt[x]])^2/Sqrt[x],x]
Output:
-((b^2*Coth[(c + d*Sqrt[x])/2] - 2*a*(a*c + a*d*Sqrt[x] - 2*b*Log[Cosh[(c + d*Sqrt[x])/2]] + 2*b*Log[Sinh[(c + d*Sqrt[x])/2]]) + b^2*Tanh[(c + d*Sqr t[x])/2])/d)
Time = 0.38 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5960, 3042, 4260, 25, 26, 3042, 25, 26, 4254, 24, 4257}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 5960 |
\(\displaystyle 2 \int \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2d\sqrt {x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int \left (a+i b \csc \left (i c+i d \sqrt {x}\right )\right )^2d\sqrt {x}\) |
\(\Big \downarrow \) 4260 |
\(\displaystyle 2 \left (2 i a b \int -i \text {csch}\left (c+d \sqrt {x}\right )d\sqrt {x}-b^2 \int -\text {csch}^2\left (c+d \sqrt {x}\right )d\sqrt {x}+a^2 \sqrt {x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (2 i a b \int -i \text {csch}\left (c+d \sqrt {x}\right )d\sqrt {x}+b^2 \int \text {csch}^2\left (c+d \sqrt {x}\right )d\sqrt {x}+a^2 \sqrt {x}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 \left (2 a b \int \text {csch}\left (c+d \sqrt {x}\right )d\sqrt {x}+b^2 \int \text {csch}^2\left (c+d \sqrt {x}\right )d\sqrt {x}+a^2 \sqrt {x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \left (2 a b \int i \csc \left (i c+i d \sqrt {x}\right )d\sqrt {x}+b^2 \int -\csc \left (i c+i d \sqrt {x}\right )^2d\sqrt {x}+a^2 \sqrt {x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 \left (2 a b \int i \csc \left (i c+i d \sqrt {x}\right )d\sqrt {x}-b^2 \int \csc \left (i c+i d \sqrt {x}\right )^2d\sqrt {x}+a^2 \sqrt {x}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 \left (2 i a b \int \csc \left (i c+i d \sqrt {x}\right )d\sqrt {x}-b^2 \int \csc \left (i c+i d \sqrt {x}\right )^2d\sqrt {x}+a^2 \sqrt {x}\right )\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle 2 \left (2 i a b \int \csc \left (i c+i d \sqrt {x}\right )d\sqrt {x}-\frac {i b^2 \int 1d\left (-i \coth \left (c+d \sqrt {x}\right )\right )}{d}+a^2 \sqrt {x}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle 2 \left (2 i a b \int \csc \left (i c+i d \sqrt {x}\right )d\sqrt {x}+a^2 \sqrt {x}-\frac {b^2 \coth \left (c+d \sqrt {x}\right )}{d}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle 2 \left (a^2 \sqrt {x}-\frac {2 a b \text {arctanh}\left (\cosh \left (c+d \sqrt {x}\right )\right )}{d}-\frac {b^2 \coth \left (c+d \sqrt {x}\right )}{d}\right )\) |
Input:
Int[(a + b*Csch[c + d*Sqrt[x]])^2/Sqrt[x],x]
Output:
2*(a^2*Sqrt[x] - (2*a*b*ArcTanh[Cosh[c + d*Sqrt[x]]])/d - (b^2*Coth[c + d* Sqrt[x]])/d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Simp[2*a*b Int[Csc[c + d*x], x], x] + Simp[b^2 Int[Csc[c + d*x]^2, x] , x]) /; FreeQ[{a, b, c, d}, x]
Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
Time = 0.49 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (c +d \sqrt {x}\right )-8 a b \,\operatorname {arctanh}\left ({\mathrm e}^{c +d \sqrt {x}}\right )-2 b^{2} \coth \left (c +d \sqrt {x}\right )}{d}\) | \(44\) |
default | \(\frac {2 a^{2} \left (c +d \sqrt {x}\right )-8 a b \,\operatorname {arctanh}\left ({\mathrm e}^{c +d \sqrt {x}}\right )-2 b^{2} \coth \left (c +d \sqrt {x}\right )}{d}\) | \(44\) |
parts | \(2 a^{2} \sqrt {x}-\frac {2 b^{2} \coth \left (c +d \sqrt {x}\right )}{d}+\frac {4 a b \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )}{d}\) | \(45\) |
Input:
int((a+b*csch(c+d*x^(1/2)))^2/x^(1/2),x,method=_RETURNVERBOSE)
Output:
2/d*(a^2*(c+d*x^(1/2))-4*a*b*arctanh(exp(c+d*x^(1/2)))-b^2*coth(c+d*x^(1/2 )))
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (41) = 82\).
Time = 0.12 (sec) , antiderivative size = 271, normalized size of antiderivative = 5.77 \[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=\frac {2 \, {\left (a^{2} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, a^{2} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + a^{2} d \sqrt {x} \sinh \left (d \sqrt {x} + c\right )^{2} - a^{2} d \sqrt {x} - 2 \, b^{2} - 2 \, {\left (a b \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + a b \sinh \left (d \sqrt {x} + c\right )^{2} - a b\right )} \log \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right ) + 1\right ) + 2 \, {\left (a b \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + a b \sinh \left (d \sqrt {x} + c\right )^{2} - a b\right )} \log \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right ) - 1\right )\right )}}{d \cosh \left (d \sqrt {x} + c\right )^{2} + 2 \, d \cosh \left (d \sqrt {x} + c\right ) \sinh \left (d \sqrt {x} + c\right ) + d \sinh \left (d \sqrt {x} + c\right )^{2} - d} \] Input:
integrate((a+b*csch(c+d*x^(1/2)))^2/x^(1/2),x, algorithm="fricas")
Output:
2*(a^2*d*sqrt(x)*cosh(d*sqrt(x) + c)^2 + 2*a^2*d*sqrt(x)*cosh(d*sqrt(x) + c)*sinh(d*sqrt(x) + c) + a^2*d*sqrt(x)*sinh(d*sqrt(x) + c)^2 - a^2*d*sqrt( x) - 2*b^2 - 2*(a*b*cosh(d*sqrt(x) + c)^2 + 2*a*b*cosh(d*sqrt(x) + c)*sinh (d*sqrt(x) + c) + a*b*sinh(d*sqrt(x) + c)^2 - a*b)*log(cosh(d*sqrt(x) + c) + sinh(d*sqrt(x) + c) + 1) + 2*(a*b*cosh(d*sqrt(x) + c)^2 + 2*a*b*cosh(d* sqrt(x) + c)*sinh(d*sqrt(x) + c) + a*b*sinh(d*sqrt(x) + c)^2 - a*b)*log(co sh(d*sqrt(x) + c) + sinh(d*sqrt(x) + c) - 1))/(d*cosh(d*sqrt(x) + c)^2 + 2 *d*cosh(d*sqrt(x) + c)*sinh(d*sqrt(x) + c) + d*sinh(d*sqrt(x) + c)^2 - d)
\[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=\int \frac {\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}{\sqrt {x}}\, dx \] Input:
integrate((a+b*csch(c+d*x**(1/2)))**2/x**(1/2),x)
Output:
Integral((a + b*csch(c + d*sqrt(x)))**2/sqrt(x), x)
Time = 0.14 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=2 \, a^{2} \sqrt {x} + \frac {4 \, a b \log \left (\tanh \left (\frac {1}{2} \, d \sqrt {x} + \frac {1}{2} \, c\right )\right )}{d} + \frac {4 \, b^{2}}{d {\left (e^{\left (-2 \, d \sqrt {x} - 2 \, c\right )} - 1\right )}} \] Input:
integrate((a+b*csch(c+d*x^(1/2)))^2/x^(1/2),x, algorithm="maxima")
Output:
2*a^2*sqrt(x) + 4*a*b*log(tanh(1/2*d*sqrt(x) + 1/2*c))/d + 4*b^2/(d*(e^(-2 *d*sqrt(x) - 2*c) - 1))
Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=\frac {2 \, {\left (d \sqrt {x} + c\right )} a^{2}}{d} - \frac {4 \, a b \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right )}{d} + \frac {4 \, a b \log \left ({\left | e^{\left (d \sqrt {x} + c\right )} - 1 \right |}\right )}{d} - \frac {4 \, b^{2}}{d {\left (e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - 1\right )}} \] Input:
integrate((a+b*csch(c+d*x^(1/2)))^2/x^(1/2),x, algorithm="giac")
Output:
2*(d*sqrt(x) + c)*a^2/d - 4*a*b*log(e^(d*sqrt(x) + c) + 1)/d + 4*a*b*log(a bs(e^(d*sqrt(x) + c) - 1))/d - 4*b^2/(d*(e^(2*d*sqrt(x) + 2*c) - 1))
Time = 2.58 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=2\,a^2\,\sqrt {x}-\frac {4\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,\sqrt {x}}-1\right )}-\frac {8\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}} \] Input:
int((a + b/sinh(c + d*x^(1/2)))^2/x^(1/2),x)
Output:
2*a^2*x^(1/2) - (4*b^2)/(d*(exp(2*c + 2*d*x^(1/2)) - 1)) - (8*atan((a*b*ex p(d*x^(1/2))*exp(c)*(-d^2)^(1/2))/(d*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(- d^2)^(1/2)
Time = 0.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.09 \[ \int \frac {\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2}{\sqrt {x}} \, dx=\frac {2 \sqrt {x}\, e^{2 \sqrt {x}\, d +2 c} a^{2} d +4 e^{2 \sqrt {x}\, d +2 c} \mathrm {log}\left (e^{\sqrt {x}\, d +c}-1\right ) a b -4 e^{2 \sqrt {x}\, d +2 c} \mathrm {log}\left (e^{\sqrt {x}\, d +c}+1\right ) a b -4 e^{2 \sqrt {x}\, d +2 c} b^{2}-2 \sqrt {x}\, a^{2} d -4 \,\mathrm {log}\left (e^{\sqrt {x}\, d +c}-1\right ) a b +4 \,\mathrm {log}\left (e^{\sqrt {x}\, d +c}+1\right ) a b}{d \left (e^{2 \sqrt {x}\, d +2 c}-1\right )} \] Input:
int((a+b*csch(c+d*x^(1/2)))^2/x^(1/2),x)
Output:
(2*(sqrt(x)*e**(2*sqrt(x)*d + 2*c)*a**2*d + 2*e**(2*sqrt(x)*d + 2*c)*log(e **(sqrt(x)*d + c) - 1)*a*b - 2*e**(2*sqrt(x)*d + 2*c)*log(e**(sqrt(x)*d + c) + 1)*a*b - 2*e**(2*sqrt(x)*d + 2*c)*b**2 - sqrt(x)*a**2*d - 2*log(e**(s qrt(x)*d + c) - 1)*a*b + 2*log(e**(sqrt(x)*d + c) + 1)*a*b))/(d*(e**(2*sqr t(x)*d + 2*c) - 1))