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

Optimal result

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

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

Mathematica [A] (verified)

Time = 3.12 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\frac {\cosh ^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) \left (4 \left (1+e^{c+d x}\right ) \text {arcsinh}\left (e^{c+d x}\right )+5 \sqrt {2} \left (1+e^{c+d x}\right ) \text {arctanh}\left (\frac {1-e^{c+d x}}{\sqrt {2} \sqrt {1+e^{2 (c+d x)}}}\right )-4 \left (1+e^{c+d x}\right ) \text {arctanh}\left (\sqrt {1+e^{2 (c+d x)}}\right )-2 \sqrt {1+e^{2 (c+d x)}} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d \sqrt {1+e^{2 (c+d x)}} (a (1+\text {sech}(c+d x)))^{3/2}} \] Input:

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

Output:

(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))*ArcTanh[(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))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4264, 27, 3042, 4408, 3042, 4261, 216, 4282, 216}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+a \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4264

\(\displaystyle -\frac {\int -\frac {4 a-a \text {sech}(c+d x)}{2 \sqrt {\text {sech}(c+d x) a+a}}dx}{2 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {4 a-a \text {sech}(c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}dx}{4 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\int \frac {4 a-a \csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\)

\(\Big \downarrow \) 4408

\(\displaystyle \frac {4 \int \sqrt {\text {sech}(c+d x) a+a}dx-5 a \int \frac {\text {sech}(c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}dx}{4 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {4 \int \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx-5 a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\)

\(\Big \downarrow \) 4261

\(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\frac {8 i a \int \frac {1}{a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}-5 a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}-5 a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\)

\(\Big \downarrow \) 4282

\(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}-\frac {10 i a \int \frac {1}{2 a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}}{4 a^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}-\frac {5 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a \text {sech}(c+d x)+a}}\right )}{d}}{4 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\)

Input:

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

Output:

((8*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d 
- (5*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/(Sqrt[2]*Sqrt[a + a*S 
ech[c + d*x]])])/d)/(4*a^2) - Tanh[c + d*x]/(2*d*(a + a*Sech[c + d*x])^(3/ 
2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4261

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

rule 4264

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] + Simp[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] && Int 
egerQ[2*n]
 

rule 4282

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-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 4408

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

Maple [F]

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

Input:

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

Output:

int(1/(a+a*sech(d*x+c))^(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 = 0.13 (sec) , antiderivative size = 1190, normalized size of antiderivative = 10.44 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*(5*sqrt(2)*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x + c) + si 
nh(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)*si 
nh(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/(cos 
h(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)*sinh(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*c 
osh(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)*sqrt(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*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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