\(\int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx\) [82]

Optimal result

Integrand size = 14, antiderivative size = 114 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a+a \text {sech}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\tanh (c+d x)}{2 d (a+a \text {sech}(c+d x))^{3/2}} \]

[Out]

2*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/a^(3/2)/d-5/4*arctanh(1/2*a^(1/2)*tanh(d*x+c)*2^(1/2)/(
a+a*sech(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2)-1/2*tanh(d*x+c)/d/(a+a*sech(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3862, 4005, 3859, 209, 3880} \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a \text {sech}(c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}} \]

[In]

Int[(a + a*Sech[c + d*x])^(-3/2),x]

[Out]

(2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Tanh[c + d*x]
)/(Sqrt[2]*Sqrt[a + a*Sech[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tanh[c + d*x]/(2*d*(a + a*Sech[c + d*x])^(3/2)
)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3862

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

Rule 3880

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

Rule 4005

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c/a,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\tanh (c+d x)}{2 d (a+a \text {sech}(c+d x))^{3/2}}-\frac {\int \frac {-2 a+\frac {1}{2} a \text {sech}(c+d x)}{\sqrt {a+a \text {sech}(c+d x)}} \, dx}{2 a^2} \\ & = -\frac {\tanh (c+d x)}{2 d (a+a \text {sech}(c+d x))^{3/2}}+\frac {\int \sqrt {a+a \text {sech}(c+d x)} \, dx}{a^2}-\frac {5 \int \frac {\text {sech}(c+d x)}{\sqrt {a+a \text {sech}(c+d x)}} \, dx}{4 a} \\ & = -\frac {\tanh (c+d x)}{2 d (a+a \text {sech}(c+d x))^{3/2}}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {i a \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{a d}-\frac {(5 i) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {i a \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{2 a d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a+a \text {sech}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\tanh (c+d x)}{2 d (a+a \text {sech}(c+d x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.94 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\frac {\cosh ^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) \left (4 \left (1+e^{c+d x}\right ) \text {arcsinh}\left (e^{c+d x}\right )+5 \sqrt {2} \left (1+e^{c+d x}\right ) \text {arctanh}\left (\frac {1-e^{c+d x}}{\sqrt {2} \sqrt {1+e^{2 (c+d x)}}}\right )-4 \left (1+e^{c+d x}\right ) \text {arctanh}\left (\sqrt {1+e^{2 (c+d x)}}\right )-2 \sqrt {1+e^{2 (c+d x)}} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d \sqrt {1+e^{2 (c+d x)}} (a (1+\text {sech}(c+d x)))^{3/2}} \]

[In]

Integrate[(a + a*Sech[c + d*x])^(-3/2),x]

[Out]

(Cosh[(c + d*x)/2]^2*Sech[c + d*x]*(4*(1 + E^(c + d*x))*ArcSinh[E^(c + d*x)] + 5*Sqrt[2]*(1 + E^(c + d*x))*Arc
Tanh[(1 - E^(c + d*x))/(Sqrt[2]*Sqrt[1 + E^(2*(c + d*x))])] - 4*(1 + E^(c + d*x))*ArcTanh[Sqrt[1 + E^(2*(c + d
*x))]] - 2*Sqrt[1 + E^(2*(c + d*x))]*Tanh[(c + d*x)/2]))/(2*d*Sqrt[1 + E^(2*(c + d*x))]*(a*(1 + Sech[c + d*x])
)^(3/2))

Maple [F]

\[\int \frac {1}{\left (a +\operatorname {sech}\left (d x +c \right ) a \right )^{\frac {3}{2}}}d x\]

[In]

int(1/(a+sech(d*x+c)*a)^(3/2),x)

[Out]

int(1/(a+sech(d*x+c)*a)^(3/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (93) = 186\).

Time = 0.30 (sec) , antiderivative size = 1190, normalized size of antiderivative = 10.44 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*(5*sqrt(2)*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)
*sqrt(a)*log(-(3*a*cosh(d*x + c)^2 + 3*a*sinh(d*x + c)^2 - 2*sqrt(2)*(cosh(d*x + c)^3 + (3*cosh(d*x + c) - 1)*
sinh(d*x + c)^2 + sinh(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c)
+ cosh(d*x + c) - 1)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) -
2*a*cosh(d*x + c) + 2*(3*a*cosh(d*x + c) - a)*sinh(d*x + c) + 3*a)/(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*s
inh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)) + 4*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x +
c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sqrt(a)*log(-(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 - 3*a*cosh(d*
x + c)^3 + (4*a*cosh(d*x + c) - 3*a)*sinh(d*x + c)^3 + 5*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 - 9*a*cosh(d
*x + c) + 5*a)*sinh(d*x + c)^2 + (cosh(d*x + c)^5 + (5*cosh(d*x + c) - 3)*sinh(d*x + c)^4 + sinh(d*x + c)^5 -
3*cosh(d*x + c)^4 + (10*cosh(d*x + c)^2 - 12*cosh(d*x + c) + 5)*sinh(d*x + c)^3 + 5*cosh(d*x + c)^3 + (10*cosh
(d*x + c)^3 - 18*cosh(d*x + c)^2 + 15*cosh(d*x + c) - 7)*sinh(d*x + c)^2 - 7*cosh(d*x + c)^2 + (5*cosh(d*x + c
)^4 - 12*cosh(d*x + c)^3 + 15*cosh(d*x + c)^2 - 14*cosh(d*x + c) + 4)*sinh(d*x + c) + 4*cosh(d*x + c) - 4)*sqr
t(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) - 4*a*cosh(d*x + c) + (4*
a*cosh(d*x + c)^3 - 9*a*cosh(d*x + c)^2 + 10*a*cosh(d*x + c) - 4*a)*sinh(d*x + c) + 4*a)/(cosh(d*x + c)^3 + 3*
cosh(d*x + c)^2*sinh(d*x + c) + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3)) + 4*(cosh(d*x + c)^2 + 2*(
cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sqrt(a)*log((a*cosh(d*x + c)^2 + a*s
inh(d*x + c)^2 + (cosh(d*x + c)^3 + (3*cosh(d*x + c) + 1)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + cosh(d*x + c)^2
+ (3*cosh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) + 1)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2
+ 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) + a*cosh(d*x + c) + (2*a*cosh(d*x + c) + a)*sinh(d*x
+ c) + a)/(cosh(d*x + c) + sinh(d*x + c))) - 4*(cosh(d*x + c)^3 + (3*cosh(d*x + c) - 1)*sinh(d*x + c)^2 + sinh
(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) - 1)*s
qrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)))/(a^2*d*cosh(d*x + c)^2 + a^2*d
*sinh(d*x + c)^2 + 2*a^2*d*cosh(d*x + c) + a^2*d + 2*(a^2*d*cosh(d*x + c) + a^2*d)*sinh(d*x + c))

Sympy [F]

\[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral((a*sech(c + d*x) + a)**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate((a*sech(d*x + c) + a)^(-3/2), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (93) = 186\).

Time = 0.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.07 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=-\frac {\frac {5 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} + \sqrt {a}\right )}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2 \, {\left (3 \, {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{3} + {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{2} \sqrt {a} - {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )} a + a^{\frac {3}{2}}\right )}}{{\left ({\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{2} + 2 \, {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )} \sqrt {a} - a\right )}^{2} a}}{2 \, d} \]

[In]

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

[Out]

-1/2*(5*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a) + sqrt(a))/sqrt(-a))/(s
qrt(-a)*a) + 2*(3*(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))^3 + (sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d
*x + 2*c) + a))^2*sqrt(a) - (sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))*a + a^(3/2))/(((sqrt(a)*e^(d*x
+ c) - sqrt(a*e^(2*d*x + 2*c) + a))^2 + 2*(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))*sqrt(a) - a)^2*
a))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(1/(a + a/cosh(c + d*x))^(3/2),x)

[Out]

int(1/(a + a/cosh(c + d*x))^(3/2), x)