\(\int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx\) [13]

Optimal result

Integrand size = 21, antiderivative size = 79 \[ \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx=-\frac {\arctan \left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}+\frac {\text {arctanh}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}-\frac {2 \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}} \] Output:

-arctan(cosh(b*x+a)^(1/2)/sinh(b*x+a)^(1/2))/b+arctanh(cosh(b*x+a)^(1/2)/s 
inh(b*x+a)^(1/2))/b-2*sinh(b*x+a)^(1/2)/b/cosh(b*x+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \sqrt [4]{\cosh ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},-\sinh ^2(a+b x)\right ) \sinh ^{\frac {5}{2}}(a+b x)}{5 b \sqrt {\cosh (a+b x)}} \] Input:

Integrate[Sinh[a + b*x]^(3/2)/Cosh[a + b*x]^(3/2),x]
 

Output:

(2*(Cosh[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, -Sinh[a + b*x] 
^2]*Sinh[a + b*x]^(5/2))/(5*b*Sqrt[Cosh[a + b*x]])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3046, 3042, 3055, 827, 216, 219}

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.

\(\displaystyle \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(-i \sin (i a+i b x))^{3/2}}{\cos (i a+i b x)^{3/2}}dx\)

\(\Big \downarrow \) 3046

\(\displaystyle \int \frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}dx-\frac {2 \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}}+\int \frac {\sqrt {\cos (i a+i b x)}}{\sqrt {-i \sin (i a+i b x)}}dx\)

\(\Big \downarrow \) 3055

\(\displaystyle \frac {2 \int \frac {\coth (a+b x)}{1-\coth ^2(a+b x)}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}}{b}-\frac {2 \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (a+b x)}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}-\frac {1}{2} \int \frac {1}{\coth (a+b x)+1}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )}{b}-\frac {2 \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth (a+b x)}d\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}-\frac {1}{2} \arctan \left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )\right )}{b}-\frac {2 \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt {\cosh (a+b x)}}{\sqrt {\sinh (a+b x)}}\right )\right )}{b}-\frac {2 \sqrt {\sinh (a+b x)}}{b \sqrt {\cosh (a+b x)}}\)

Input:

Int[Sinh[a + b*x]^(3/2)/Cosh[a + b*x]^(3/2),x]
 

Output:

(2*(-1/2*ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]] + ArcTanh[Sqrt[Co 
sh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/2))/b - (2*Sqrt[Sinh[a + b*x]])/(b*Sqrt[ 
Cosh[a + b*x]])
 

Defintions of rubi rules used

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 219

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

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 3042

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

rule 3046

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 
1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1)))   Int[(a*Sin[e + f 
*x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && 
GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
 

rule 3055

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> With[{k = Denominator[m]}, Simp[(-k)*a*(b/f)   Subst[Int[x 
^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Cos[e + f*x])^(1/k)/(b*Sin[ 
e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 
0] && LtQ[m, 1]
 

Maple [F]

\[\int \frac {\sinh \left (b x +a \right )^{\frac {3}{2}}}{\cosh \left (b x +a \right )^{\frac {3}{2}}}d x\]

Input:

int(sinh(b*x+a)^(3/2)/cosh(b*x+a)^(3/2),x)
 

Output:

int(sinh(b*x+a)^(3/2)/cosh(b*x+a)^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (67) = 134\).

Time = 0.11 (sec) , antiderivative size = 310, normalized size of antiderivative = 3.92 \[ \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \arctan \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - 4 \, \cosh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1\right )} \log \left (-\cosh \left (b x + a\right )^{2} + 2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - \sinh \left (b x + a\right )^{2}\right ) - 8 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )} \sqrt {\sinh \left (b x + a\right )} - 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - 4 \, \sinh \left (b x + a\right )^{2} - 4}{2 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} + b\right )}} \] Input:

integrate(sinh(b*x+a)^(3/2)/cosh(b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

1/2*(2*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 
+ 1)*arctan(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh(b*x + a))*sqrt(cosh 
(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh(b*x + a) - sinh(b*x 
+ a)^2) - 4*cosh(b*x + a)^2 - (cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x 
+ a) + sinh(b*x + a)^2 + 1)*log(-cosh(b*x + a)^2 + 2*(cosh(b*x + a) + sinh 
(b*x + a))*sqrt(cosh(b*x + a))*sqrt(sinh(b*x + a)) - 2*cosh(b*x + a)*sinh( 
b*x + a) - sinh(b*x + a)^2) - 8*(cosh(b*x + a) + sinh(b*x + a))*sqrt(cosh( 
b*x + a))*sqrt(sinh(b*x + a)) - 8*cosh(b*x + a)*sinh(b*x + a) - 4*sinh(b*x 
 + a)^2 - 4)/(b*cosh(b*x + a)^2 + 2*b*cosh(b*x + a)*sinh(b*x + a) + b*sinh 
(b*x + a)^2 + b)
 

Sympy [F]

\[ \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {\sinh ^{\frac {3}{2}}{\left (a + b x \right )}}{\cosh ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \] Input:

integrate(sinh(b*x+a)**(3/2)/cosh(b*x+a)**(3/2),x)
 

Output:

Integral(sinh(a + b*x)**(3/2)/cosh(a + b*x)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {\sinh \left (b x + a\right )^{\frac {3}{2}}}{\cosh \left (b x + a\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(sinh(b*x+a)^(3/2)/cosh(b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate(sinh(b*x + a)^(3/2)/cosh(b*x + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {\sinh \left (b x + a\right )^{\frac {3}{2}}}{\cosh \left (b x + a\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(sinh(b*x+a)^(3/2)/cosh(b*x+a)^(3/2),x, algorithm="giac")
 

Output:

integrate(sinh(b*x + a)^(3/2)/cosh(b*x + a)^(3/2), x)
 

Mupad [F(-1)]

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

int(sinh(a + b*x)^(3/2)/cosh(a + b*x)^(3/2),x)
 

Output:

int(sinh(a + b*x)^(3/2)/cosh(a + b*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\sinh ^{\frac {3}{2}}(a+b x)}{\cosh ^{\frac {3}{2}}(a+b x)} \, dx=\frac {\cosh \left (b x +a \right ) \left (\int \frac {\sqrt {\sinh \left (b x +a \right )}\, \sqrt {\cosh \left (b x +a \right )}}{\sinh \left (b x +a \right )}d x \right ) b -2 \sqrt {\sinh \left (b x +a \right )}\, \sqrt {\cosh \left (b x +a \right )}}{\cosh \left (b x +a \right ) b} \] Input:

int(sinh(b*x+a)^(3/2)/cosh(b*x+a)^(3/2),x)
 

Output:

(cosh(a + b*x)*int((sqrt(sinh(a + b*x))*sqrt(cosh(a + b*x)))/sinh(a + b*x) 
,x)*b - 2*sqrt(sinh(a + b*x))*sqrt(cosh(a + b*x)))/(cosh(a + b*x)*b)