3.1.69 \(\int \frac {1}{\sqrt {x} (a+b \text {sech}(c+d \sqrt {x}))^2} \, dx\) [69]

Optimal. Leaf size=127 \[ \frac {2 \sqrt {x}}{a^2}-\frac {4 b \left (2 a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {2 b^2 \tanh \left (c+d \sqrt {x}\right )}{a \left (a^2-b^2\right ) d \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5544, 3870, 4004, 3916, 2738, 214} \begin {gather*} -\frac {4 b \left (2 a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {2 b^2 \tanh \left (c+d \sqrt {x}\right )}{a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )}+\frac {2 \sqrt {x}}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]])^2),x]

[Out]

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

Rule 214

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3870

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[c + d*x]*((a + b*Csc[c + d*x])^(n +
1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 5544

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.37, size = 232, normalized size = 1.83 \begin {gather*} \frac {2 \left (a \left (\left (a^2-b^2\right )^{3/2} \left (c+d \sqrt {x}\right )+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \cosh \left (c+d \sqrt {x}\right )+b \left (\left (a^2-b^2\right )^{3/2} \left (c+d \sqrt {x}\right )+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right )}{\sqrt {a^2-b^2}}\right )+a b \sqrt {a^2-b^2} \sinh \left (c+d \sqrt {x}\right )\right )\right )}{a^2 (a-b) (a+b) \sqrt {a^2-b^2} d \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(a + b*Sech[c + d*Sqrt[x]])^2),x]

[Out]

(2*(a*((a^2 - b^2)^(3/2)*(c + d*Sqrt[x]) + (4*a^2*b - 2*b^3)*ArcTan[((-a + b)*Tanh[(c + d*Sqrt[x])/2])/Sqrt[a^
2 - b^2]])*Cosh[c + d*Sqrt[x]] + b*((a^2 - b^2)^(3/2)*(c + d*Sqrt[x]) + (4*a^2*b - 2*b^3)*ArcTan[((-a + b)*Tan
h[(c + d*Sqrt[x])/2])/Sqrt[a^2 - b^2]] + a*b*Sqrt[a^2 - b^2]*Sinh[c + d*Sqrt[x]])))/(a^2*(a - b)*(a + b)*Sqrt[
a^2 - b^2]*d*(b + a*Cosh[c + d*Sqrt[x]]))

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Maple [A]
time = 3.89, size = 177, normalized size = 1.39

method	result	size
derivativedivides	\(\frac {-\frac {4 b \left (-\frac {a b \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )+a +b \right )}+\frac {\left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}-\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )-1\right )}{a^{2}}+\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+1\right )}{a^{2}}}{d}\)	\(177\)
default	\(\frac {-\frac {4 b \left (-\frac {a b \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )\right )+a +b \right )}+\frac {\left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}-\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )-1\right )}{a^{2}}+\frac {2 \ln \left (\tanh \left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right )+1\right )}{a^{2}}}{d}\)	\(177\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sech(c+d*x^(1/2)))^2/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/d*(-2*b/a^2*(-1/(a^2-b^2)*a*b*tanh(1/2*c+1/2*d*x^(1/2))/(a*tanh(1/2*c+1/2*d*x^(1/2))^2-b*tanh(1/2*c+1/2*d*x^
(1/2))^2+a+b)+(2*a^2-b^2)/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*c+1/2*d*x^(1/2))/((a+b)*(a-b))
^(1/2)))-1/a^2*ln(tanh(1/2*c+1/2*d*x^(1/2))-1)+1/a^2*ln(tanh(1/2*c+1/2*d*x^(1/2))+1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (110) = 220\).
time = 0.43, size = 1387, normalized size = 10.92 \begin {gather*} \left [-\frac {2 \, {\left (2 \, a^{3} b^{2} - 2 \, a b^{4} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \sinh \left (d \sqrt {x} + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x}\right )} \cosh \left (d \sqrt {x} + c\right ) + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} \cosh \left (d \sqrt {x} + c\right )^{2} + {\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} \sinh \left (d \sqrt {x} + c\right )^{2} + 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}} \cosh \left (d \sqrt {x} + c\right ) + 2 \, {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} \cosh \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}}\right )} \sinh \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}}\right )} \log \left (\frac {a b + {\left (b^{2} + \sqrt {-a^{2} + b^{2}} b\right )} \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{2} - b^{2} - \sqrt {-a^{2} + b^{2}} b\right )} \sinh \left (d \sqrt {x} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{a \cosh \left (d \sqrt {x} + c\right ) + b}\right ) + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x}\right )} \sinh \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d \sqrt {x} + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d \sqrt {x} + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d + 2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d \sqrt {x} + c\right )}, -\frac {2 \, {\left (2 \, a^{3} b^{2} - 2 \, a b^{4} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \sinh \left (d \sqrt {x} + c\right )^{2} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} - 2 \, {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} \cosh \left (d \sqrt {x} + c\right )^{2} + {\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} \sinh \left (d \sqrt {x} + c\right )^{2} + 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}} \cosh \left (d \sqrt {x} + c\right ) + 2 \, {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} \cosh \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}}\right )} \sinh \left (d \sqrt {x} + c\right ) + {\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}}\right )} \arctan \left (-\frac {\sqrt {a^{2} - b^{2}} a \cosh \left (d \sqrt {x} + c\right ) + \sqrt {a^{2} - b^{2}} a \sinh \left (d \sqrt {x} + c\right ) + \sqrt {a^{2} - b^{2}} b}{a^{2} - b^{2}}\right ) + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x}\right )} \cosh \left (d \sqrt {x} + c\right ) + 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d \sqrt {x} \cosh \left (d \sqrt {x} + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sqrt {x}\right )} \sinh \left (d \sqrt {x} + c\right )\right )}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d \sqrt {x} + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d \sqrt {x} + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d + 2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d \sqrt {x} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d \sqrt {x} + c\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-2*(2*a^3*b^2 - 2*a*b^4 - (a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x)*cosh(d*sqrt(x) + c)^2 - (a^5 - 2*a^3*b^2 + a*b^
4)*d*sqrt(x)*sinh(d*sqrt(x) + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x) + 2*(a^2*b^3 - b^5 - (a^4*b - 2*a^2*b
^3 + b^5)*d*sqrt(x))*cosh(d*sqrt(x) + c) + ((2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*cosh(d*sqrt(x) + c)^2 + (2*a^3*
b - a*b^3)*sqrt(-a^2 + b^2)*sinh(d*sqrt(x) + c)^2 + 2*(2*a^2*b^2 - b^4)*sqrt(-a^2 + b^2)*cosh(d*sqrt(x) + c) +
2*((2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*cosh(d*sqrt(x) + c) + (2*a^2*b^2 - b^4)*sqrt(-a^2 + b^2))*sinh(d*sqrt(x
) + c) + (2*a^3*b - a*b^3)*sqrt(-a^2 + b^2))*log((a*b + (b^2 + sqrt(-a^2 + b^2)*b)*cosh(d*sqrt(x) + c) + (a^2
- b^2 - sqrt(-a^2 + b^2)*b)*sinh(d*sqrt(x) + c) + sqrt(-a^2 + b^2)*a)/(a*cosh(d*sqrt(x) + c) + b)) + 2*(a^2*b^
3 - b^5 - (a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x)*cosh(d*sqrt(x) + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*sqrt(x))*sinh(
d*sqrt(x) + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cosh(d*sqrt(x) + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*sinh(d*sq
rt(x) + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*cosh(d*sqrt(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*sqrt(x) + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d)*sinh(d*sqrt(x) + c)), -2*(2*a
^3*b^2 - 2*a*b^4 - (a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x)*cosh(d*sqrt(x) + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d*sqr
t(x)*sinh(d*sqrt(x) + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x) - 2*((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*cosh(d
*sqrt(x) + c)^2 + (2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*sinh(d*sqrt(x) + c)^2 + 2*(2*a^2*b^2 - b^4)*sqrt(a^2 - b^2
)*cosh(d*sqrt(x) + c) + 2*((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*cosh(d*sqrt(x) + c) + (2*a^2*b^2 - b^4)*sqrt(a^2
- b^2))*sinh(d*sqrt(x) + c) + (2*a^3*b - a*b^3)*sqrt(a^2 - b^2))*arctan(-(sqrt(a^2 - b^2)*a*cosh(d*sqrt(x) + c
) + sqrt(a^2 - b^2)*a*sinh(d*sqrt(x) + c) + sqrt(a^2 - b^2)*b)/(a^2 - b^2)) + 2*(a^2*b^3 - b^5 - (a^4*b - 2*a^
2*b^3 + b^5)*d*sqrt(x))*cosh(d*sqrt(x) + c) + 2*(a^2*b^3 - b^5 - (a^5 - 2*a^3*b^2 + a*b^4)*d*sqrt(x)*cosh(d*sq
rt(x) + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*sqrt(x))*sinh(d*sqrt(x) + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cosh(d*s
qrt(x) + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*sinh(d*sqrt(x) + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*cosh(d
*sqrt(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*sqrt(x) + c) + (a^6*b
- 2*a^4*b^3 + a^2*b^5)*d)*sinh(d*sqrt(x) + c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(x)*(a + b*sech(c + d*sqrt(x)))**2), x)

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Giac [A]
time = 0.39, size = 148, normalized size = 1.17 \begin {gather*} -\frac {4 \, {\left (2 \, a^{2} b - b^{3}\right )} \arctan \left (\frac {a e^{\left (d \sqrt {x} + c\right )} + b}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{4} d - a^{2} b^{2} d\right )} \sqrt {a^{2} - b^{2}}} - \frac {4 \, {\left (b^{3} e^{\left (d \sqrt {x} + c\right )} + a b^{2}\right )}}{{\left (a^{4} d - a^{2} b^{2} d\right )} {\left (a e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + 2 \, b e^{\left (d \sqrt {x} + c\right )} + a\right )}} + \frac {2 \, {\left (d \sqrt {x} + c\right )}}{a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(2*a^2*b - b^3)*arctan((a*e^(d*sqrt(x) + c) + b)/sqrt(a^2 - b^2))/((a^4*d - a^2*b^2*d)*sqrt(a^2 - b^2)) - 4
*(b^3*e^(d*sqrt(x) + c) + a*b^2)/((a^4*d - a^2*b^2*d)*(a*e^(2*d*sqrt(x) + 2*c) + 2*b*e^(d*sqrt(x) + c) + a)) +
2*(d*sqrt(x) + c)/(a^2*d)

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Mupad [B]
time = 1.85, size = 344, normalized size = 2.71 \begin {gather*} \frac {2\,\sqrt {x}}{a^2}-\frac {\frac {4\,b^2\,\sqrt {x}}{d\,\left (a^3\,\sqrt {x}-a\,b^2\,\sqrt {x}\right )}+\frac {4\,b^3\,\sqrt {x}\,{\mathrm {e}}^{c+d\,\sqrt {x}}}{a\,d\,\left (a^3\,\sqrt {x}-a\,b^2\,\sqrt {x}\right )}}{a+2\,b\,{\mathrm {e}}^{c+d\,\sqrt {x}}+a\,{\mathrm {e}}^{2\,c+2\,d\,\sqrt {x}}}+\frac {\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,\sqrt {x}}\,\left (2\,a^2\,b-b^3\right )}{a^3\,\sqrt {x}\,\left (a^2-b^2\right )}-\frac {\left (4\,a^2\,b-2\,b^3\right )\,\left (a+b\,{\mathrm {e}}^{c+d\,\sqrt {x}}\right )}{a^3\,\sqrt {x}\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}\right )\,\left (4\,a^2\,b-2\,b^3\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}-\frac {2\,b\,\ln \left (\frac {2\,{\mathrm {e}}^{c+d\,\sqrt {x}}\,\left (2\,a^2\,b-b^3\right )}{a^3\,\sqrt {x}\,\left (a^2-b^2\right )}+\frac {2\,b\,\left (a+b\,{\mathrm {e}}^{c+d\,\sqrt {x}}\right )\,\left (2\,a^2-b^2\right )}{a^3\,\sqrt {x}\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}\right )\,\left (2\,a^2-b^2\right )}{a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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