\(\int (a+a \text {sech}(c+d x))^{3/2} \, dx\) [79]

Optimal result

Integrand size = 14, antiderivative size = 66 \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a+a \text {sech}(c+d x)}} \] Output:

2*a^(3/2)*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/d+2*a^2*tan 
h(d*x+c)/d/(a+a*sech(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\frac {a \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\text {sech}(c+d x))} \left (\sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cosh (c+d x)}+2 \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{d} \] Input:

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

Output:

(a*Sech[(c + d*x)/2]*Sqrt[a*(1 + Sech[c + d*x])]*(Sqrt[2]*ArcSinh[Sqrt[2]* 
Sinh[(c + d*x)/2]]*Sqrt[Cosh[c + d*x]] + 2*Sinh[(c + d*x)/2]))/d
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4262, 27, 3042, 4261, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4262

\(\displaystyle 2 a \int \frac {1}{2} \sqrt {\text {sech}(c+d x) a+a}dx+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4261

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

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 4262

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C 
ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[a/(n - 1) 
   Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*x]), 
 x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && Inte 
gerQ[2*n]
 

Maple [F]

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

Input:

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

Output:

int((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. 697 vs. \(2 (58) = 116\).

Time = 0.12 (sec) , antiderivative size = 697, normalized size of antiderivative = 10.56 \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*(a^(3/2)*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)*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*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)) + a^(3/2)*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*(a*cosh(d*x + c) + a*sinh(d*x + c) - a)*sqrt(a/(cosh(d*x + c)^2 + 2 
*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)))/d
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(sech(c + d*x) + 1),x) + int(sqrt(sech(c + d*x) + 1)*se 
ch(c + d*x),x))