3.1.27 \(\int \frac {x}{(a+b \text {sech}(c+d x^2))^2} \, dx\) [27]

Optimal. Leaf size=123 \[ \frac {x^2}{2 a^2}-\frac {b \left (2 a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d}+\frac {b^2 \tanh \left (c+d x^2\right )}{2 a \left (a^2-b^2\right ) d \left (a+b \text {sech}\left (c+d x^2\right )\right )} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5544, 3870, 4004, 3916, 2738, 214} \begin {gather*} -\frac {b \left (2 a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a+b}}\right )}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {b^2 \tanh \left (c+d x^2\right )}{2 a d \left (a^2-b^2\right ) \left (a+b \text {sech}\left (c+d x^2\right )\right )}+\frac {x^2}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sech[c + d*x^2])^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.33, size = 220, normalized size = 1.79 \begin {gather*} \frac {a \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^2\right )+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \cosh \left (c+d x^2\right )+b \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^2\right )+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2-b^2}}\right )+a b \sqrt {a^2-b^2} \sinh \left (c+d x^2\right )\right )}{2 a^2 (a-b) (a+b) \sqrt {a^2-b^2} d \left (b+a \cosh \left (c+d x^2\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.91, size = 177, normalized size = 1.44

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (-\frac {a b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}}{2 d}\)	\(177\)
default	\(\frac {-\frac {2 b \left (-\frac {a b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )+a +b \right )}+\frac {\left (2 a^{2}-b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}+\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}}{2 d}\)	\(177\)
risch	\(\frac {x^{2}}{2 a^{2}}-\frac {b^{2} \left (b \,{\mathrm e}^{d \,x^{2}+c}+a \right )}{a^{2} \left (a^{2}-b^{2}\right ) d \left (a \,{\mathrm e}^{2 d \,x^{2}+2 c}+2 b \,{\mathrm e}^{d \,x^{2}+c}+a \right )}-\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {b^{3} \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}+\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {b^{3} \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{2}}\)	\(385\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*(-2*b/a^2*(-1/(a^2-b^2)*a*b*tanh(1/2*d*x^2+1/2*c)/(a*tanh(1/2*d*x^2+1/2*c)^2-b*tanh(1/2*d*x^2+1/2*c)^2+a
+b)+(2*a^2-b^2)/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*d*x^2+1/2*c)/((a+b)*(a-b))^(1/2)))-1/a^2
*ln(tanh(1/2*d*x^2+1/2*c)-1)+1/a^2*ln(tanh(1/2*d*x^2+1/2*c)+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(x/(a+b*sech(d*x^2+c))^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. 587 vs. \(2 (110) = 220\).
time = 0.40, size = 1314, normalized size = 10.68 \begin {gather*} \left [\frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cosh \left (d x^{2} + c\right )^{2} + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sinh \left (d x^{2} + c\right )^{2} - 2 \, a^{3} b^{2} + 2 \, a b^{4} + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} - {\left (2 \, a^{3} b - a b^{3} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x^{2} + c\right )^{2} + {\left (2 \, a^{3} b - a b^{3}\right )} \sinh \left (d x^{2} + c\right )^{2} + 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (d x^{2} + c\right ) + 2 \, {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x^{2} + c\right )\right )} \sinh \left (d x^{2} + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (d x^{2} + c\right )^{2} + a^{2} \sinh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (d x^{2} + c\right ) + a b\right )} \sinh \left (d x^{2} + c\right ) + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b\right )}}{a \cosh \left (d x^{2} + c\right )^{2} + a \sinh \left (d x^{2} + c\right )^{2} + 2 \, b \cosh \left (d x^{2} + c\right ) + 2 \, {\left (a \cosh \left (d x^{2} + c\right ) + b\right )} \sinh \left (d x^{2} + c\right ) + a}\right ) - 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2}\right )} \cosh \left (d x^{2} + c\right ) - 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cosh \left (d x^{2} + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2}\right )} \sinh \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x^{2} + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d x^{2} + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d x^{2} + 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 x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d x^{2} + c\right )\right )}}, \frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cosh \left (d x^{2} + c\right )^{2} + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \sinh \left (d x^{2} + c\right )^{2} - 2 \, a^{3} b^{2} + 2 \, a b^{4} + {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} + 2 \, {\left (2 \, a^{3} b - a b^{3} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x^{2} + c\right )^{2} + {\left (2 \, a^{3} b - a b^{3}\right )} \sinh \left (d x^{2} + c\right )^{2} + 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (d x^{2} + c\right ) + 2 \, {\left (2 \, a^{2} b^{2} - b^{4} + {\left (2 \, a^{3} b - a b^{3}\right )} \cosh \left (d x^{2} + c\right )\right )} \sinh \left (d x^{2} + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \left (d x^{2} + c\right ) + a \sinh \left (d x^{2} + c\right ) + b}{\sqrt {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 x^{2}\right )} \cosh \left (d x^{2} + c\right ) - 2 \, {\left (a^{2} b^{3} - b^{5} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d x^{2} \cosh \left (d x^{2} + c\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d x^{2}\right )} \sinh \left (d x^{2} + c\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cosh \left (d x^{2} + c\right )^{2} + {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \sinh \left (d x^{2} + c\right )^{2} + 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d \cosh \left (d x^{2} + 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 x^{2} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )} \sinh \left (d x^{2} + c\right )\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((a^5 - 2*a^3*b^2 + a*b^4)*d*x^2*cosh(d*x^2 + c)^2 + (a^5 - 2*a^3*b^2 + a*b^4)*d*x^2*sinh(d*x^2 + c)^2 -
2*a^3*b^2 + 2*a*b^4 + (a^5 - 2*a^3*b^2 + a*b^4)*d*x^2 - (2*a^3*b - a*b^3 + (2*a^3*b - a*b^3)*cosh(d*x^2 + c)^2
+ (2*a^3*b - a*b^3)*sinh(d*x^2 + c)^2 + 2*(2*a^2*b^2 - b^4)*cosh(d*x^2 + c) + 2*(2*a^2*b^2 - b^4 + (2*a^3*b -
a*b^3)*cosh(d*x^2 + c))*sinh(d*x^2 + c))*sqrt(-a^2 + b^2)*log((a^2*cosh(d*x^2 + c)^2 + a^2*sinh(d*x^2 + c)^2
+ 2*a*b*cosh(d*x^2 + c) - a^2 + 2*b^2 + 2*(a^2*cosh(d*x^2 + c) + a*b)*sinh(d*x^2 + c) + 2*sqrt(-a^2 + b^2)*(a*
cosh(d*x^2 + c) + a*sinh(d*x^2 + c) + b))/(a*cosh(d*x^2 + c)^2 + a*sinh(d*x^2 + c)^2 + 2*b*cosh(d*x^2 + c) + 2
*(a*cosh(d*x^2 + c) + b)*sinh(d*x^2 + c) + a)) - 2*(a^2*b^3 - b^5 - (a^4*b - 2*a^2*b^3 + b^5)*d*x^2)*cosh(d*x^
2 + c) - 2*(a^2*b^3 - b^5 - (a^5 - 2*a^3*b^2 + a*b^4)*d*x^2*cosh(d*x^2 + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*x^2)
*sinh(d*x^2 + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x^2 + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*sinh(d*x^2
+ c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*cosh(d*x^2 + 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*x^2 + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d)*sinh(d*x^2 + c)), 1/2*((a^5 - 2*a^3*b^2 + a
*b^4)*d*x^2*cosh(d*x^2 + c)^2 + (a^5 - 2*a^3*b^2 + a*b^4)*d*x^2*sinh(d*x^2 + c)^2 - 2*a^3*b^2 + 2*a*b^4 + (a^5
- 2*a^3*b^2 + a*b^4)*d*x^2 + 2*(2*a^3*b - a*b^3 + (2*a^3*b - a*b^3)*cosh(d*x^2 + c)^2 + (2*a^3*b - a*b^3)*sin
h(d*x^2 + c)^2 + 2*(2*a^2*b^2 - b^4)*cosh(d*x^2 + c) + 2*(2*a^2*b^2 - b^4 + (2*a^3*b - a*b^3)*cosh(d*x^2 + c))
*sinh(d*x^2 + c))*sqrt(a^2 - b^2)*arctan(-(a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c) + b)/sqrt(a^2 - b^2)) - 2*(a^
2*b^3 - b^5 - (a^4*b - 2*a^2*b^3 + b^5)*d*x^2)*cosh(d*x^2 + c) - 2*(a^2*b^3 - b^5 - (a^5 - 2*a^3*b^2 + a*b^4)*
d*x^2*cosh(d*x^2 + c) - (a^4*b - 2*a^2*b^3 + b^5)*d*x^2)*sinh(d*x^2 + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*cosh(
d*x^2 + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*sinh(d*x^2 + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*cosh(d*x^2
+ 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*x^2 + c) + (a^6*b - 2*a^4*b^3 +
a^2*b^5)*d)*sinh(d*x^2 + c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 148, normalized size = 1.20 \begin {gather*} -\frac {{\left (2 \, a^{2} b - b^{3}\right )} \arctan \left (\frac {a e^{\left (d x^{2} + 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 {b^{3} e^{\left (d x^{2} + c\right )} + a b^{2}}{{\left (a^{4} d - a^{2} b^{2} d\right )} {\left (a e^{\left (2 \, d x^{2} + 2 \, c\right )} + 2 \, b e^{\left (d x^{2} + c\right )} + a\right )}} + \frac {d x^{2} + c}{2 \, a^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.76, size = 316, normalized size = 2.57 \begin {gather*} \frac {\frac {b^2}{d\,\left (a\,b^2-a^3\right )}+\frac {b^3\,{\mathrm {e}}^{d\,x^2+c}}{a\,d\,\left (a\,b^2-a^3\right )}}{a+2\,b\,{\mathrm {e}}^{d\,x^2+c}+a\,{\mathrm {e}}^{2\,d\,x^2+2\,c}}+\frac {x^2}{2\,a^2}+\frac {b\,\ln \left (\frac {2\,b\,x\,{\mathrm {e}}^{d\,x^2+c}\,\left (2\,a^2-b^2\right )}{a^3\,\left (a^2-b^2\right )}-\frac {2\,b\,x\,\left (a+b\,{\mathrm {e}}^{d\,x^2+c}\right )\,\left (2\,a^2-b^2\right )}{a^3\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}\right )\,\left (2\,a^2-b^2\right )}{2\,a^2\,d\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}-\frac {b\,\ln \left (\frac {2\,b\,x\,{\mathrm {e}}^{d\,x^2+c}\,\left (2\,a^2-b^2\right )}{a^3\,\left (a^2-b^2\right )}+\frac {2\,b\,x\,\left (a+b\,{\mathrm {e}}^{d\,x^2+c}\right )\,\left (2\,a^2-b^2\right )}{a^3\,{\left (a+b\right )}^{3/2}\,{\left (b-a\right )}^{3/2}}\right )\,\left (2\,a^2-b^2\right )}{2\,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(x/(a + b/cosh(c + d*x^2))^2,x)

[Out]

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

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