\(\int (a+a \text {sech}(c+d x))^{5/2} \, dx\) [78]

Optimal result

Integrand size = 14, antiderivative size = 98 \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}+\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a+a \text {sech}(c+d x)}}+\frac {2 a^2 \sqrt {a+a \text {sech}(c+d x)} \tanh (c+d x)}{3 d} \] Output:

2*a^(5/2)*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/d+14/3*a^3* 
tanh(d*x+c)/d/(a+a*sech(d*x+c))^(1/2)+2/3*a^2*(a+a*sech(d*x+c))^(1/2)*tanh 
(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\frac {a^2 \text {sech}\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) \sqrt {a (1+\text {sech}(c+d x))} \left (3 \sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh ^{\frac {3}{2}}(c+d x)-6 \sinh \left (\frac {1}{2} (c+d x)\right )+8 \sinh \left (\frac {3}{2} (c+d x)\right )\right )}{3 d} \] Input:

Integrate[(a + a*Sech[c + d*x])^(5/2),x]
 

Output:

(a^2*Sech[(c + d*x)/2]*Sech[c + d*x]*Sqrt[a*(1 + Sech[c + d*x])]*(3*Sqrt[2 
]*ArcSinh[Sqrt[2]*Sinh[(c + d*x)/2]]*Cosh[c + d*x]^(3/2) - 6*Sinh[(c + d*x 
)/2] + 8*Sinh[(3*(c + d*x))/2]))/(3*d)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 4262, 27, 3042, 4403, 3042, 4261, 216, 4279}

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.

\(\displaystyle \int (a \text {sech}(c+d x)+a)^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a+a \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{5/2}dx\)

\(\Big \downarrow \) 4262

\(\displaystyle \frac {2}{3} a \int \frac {1}{2} \sqrt {\text {sech}(c+d x) a+a} (7 \text {sech}(c+d x) a+3 a)dx+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} a \int \sqrt {\text {sech}(c+d x) a+a} (7 \text {sech}(c+d x) a+3 a)dx+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \int \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a} \left (7 \csc \left (i c+i d x+\frac {\pi }{2}\right ) a+3 a\right )dx\)

\(\Big \downarrow \) 4403

\(\displaystyle \frac {1}{3} a \left (3 a \int \sqrt {\text {sech}(c+d x) a+a}dx+7 a \int \text {sech}(c+d x) \sqrt {\text {sech}(c+d x) a+a}dx\right )+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (3 a \int \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx+7 a \int \csc \left (i c+i d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\right )\)

\(\Big \downarrow \) 4261

\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (\frac {6 i a^2 \int \frac {1}{a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}+7 a \int \csc \left (i c+i d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\right )\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (\frac {6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+7 a \int \csc \left (i c+i d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\right )\)

\(\Big \downarrow \) 4279

\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (\frac {6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {14 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}\right )\)

Input:

Int[(a + a*Sech[c + d*x])^(5/2),x]
 

Output:

(2*a^2*Sqrt[a + a*Sech[c + d*x]]*Tanh[c + d*x])/(3*d) + (a*((6*a^(3/2)*Arc 
Tanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d + (14*a^2*Tanh[ 
c + d*x])/(d*Sqrt[a + a*Sech[c + d*x]])))/3
 

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]
 

rule 4279

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*b*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]])), x] /; Free 
Q[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
 

rule 4403

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_ 
.) + (c_)), x_Symbol] :> Simp[c   Int[Sqrt[a + b*Csc[e + f*x]], x], x] + Si 
mp[d   Int[Sqrt[a + b*Csc[e + f*x]]*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]
 

Maple [F]

\[\int \left (a +a \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]

Input:

int((a+a*sech(d*x+c))^(5/2),x)
 

Output:

int((a+a*sech(d*x+c))^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (84) = 168\).

Time = 0.11 (sec) , antiderivative size = 924, normalized size of antiderivative = 9.43 \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a+a*sech(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

1/6*(3*(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh 
(d*x + c)^2 + a^2)*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)*si 
nh(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)*si 
nh(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)) + 3*(a^2*cosh(d*x + c)^2 
 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 + a^2)*sqrt(a)* 
log((a*cosh(d*x + c)^2 + a*sinh(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)...
 

Sympy [F]

\[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\int \left (a \operatorname {sech}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \] Input:

integrate((a+a*sech(d*x+c))**(5/2),x)
 

Output:

Integral((a*sech(c + d*x) + a)**(5/2), x)
 

Maxima [F]

\[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\int { {\left (a \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((a+a*sech(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate((a*sech(d*x + c) + a)^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+a*sech(d*x+c))^(5/2),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. \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\int {\left (a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{5/2} \,d x \] Input:

int((a + a/cosh(c + d*x))^(5/2),x)
 

Output:

int((a + a/cosh(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\sqrt {a}\, a^{2} \left (\int \sqrt {\mathrm {sech}\left (d x +c \right )+1}d x +\int \sqrt {\mathrm {sech}\left (d x +c \right )+1}\, \mathrm {sech}\left (d x +c \right )^{2}d x +2 \left (\int \sqrt {\mathrm {sech}\left (d x +c \right )+1}\, \mathrm {sech}\left (d x +c \right )d x \right )\right ) \] Input:

int((a+a*sech(d*x+c))^(5/2),x)
 

Output:

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