\(\int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx\) [18]

Optimal result

Integrand size = 21, antiderivative size = 106 \[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {\arctan \left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}+\frac {\text {arctanh}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}} \] Output:

-arctan(1/cosh(b*x+a)^(1/2)*sinh(b*x+a)^(1/2))/b+arctanh(1/cosh(b*x+a)^(1/ 
2)*sinh(b*x+a)^(1/2))/b-2/5*cosh(b*x+a)^(5/2)/b/sinh(b*x+a)^(5/2)-2*cosh(b 
*x+a)^(1/2)/b/sinh(b*x+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.56 \[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cosh ^2(a+b x)^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {5}{4},-\frac {1}{4},-\sinh ^2(a+b x)\right )}{5 b \cosh ^{\frac {3}{2}}(a+b x) \sinh ^{\frac {5}{2}}(a+b x)} \] Input:

Integrate[Cosh[a + b*x]^(7/2)/Sinh[a + b*x]^(7/2),x]
 

Output:

(-2*(Cosh[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, -5/4, -1/4, -Sinh[a + 
b*x]^2])/(5*b*Cosh[a + b*x]^(3/2)*Sinh[a + b*x]^(5/2))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3047, 3042, 3047, 3042, 3054, 25, 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.

\(\displaystyle \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (i a+i b x)^{7/2}}{(-i \sin (i a+i b x))^{7/2}}dx\)

\(\Big \downarrow \) 3047

\(\displaystyle \int \frac {\cosh ^{\frac {3}{2}}(a+b x)}{\sinh ^{\frac {3}{2}}(a+b x)}dx-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\int \frac {\cos (i a+i b x)^{3/2}}{(-i \sin (i a+i b x))^{3/2}}dx\)

\(\Big \downarrow \) 3047

\(\displaystyle \int \frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}dx-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {-i \sin (i a+i b x)}}{\sqrt {\cos (i a+i b x)}}dx-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\)

\(\Big \downarrow \) 3054

\(\displaystyle -\frac {2 \int -\frac {\tanh (a+b x)}{1-\tanh ^2(a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}}{b}-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {\tanh (a+b x)}{1-\tanh ^2(a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}}{b}-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\)

\(\Big \downarrow \) 827

\(\displaystyle -\frac {2 \left (\frac {1}{2} \int \frac {1}{\tanh (a+b x)+1}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}-\frac {1}{2} \int \frac {1}{1-\tanh (a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {2 \left (\frac {1}{2} \arctan \left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )-\frac {1}{2} \int \frac {1}{1-\tanh (a+b x)}d\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )}{b}-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {2 \left (\frac {1}{2} \arctan \left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {\sinh (a+b x)}}{\sqrt {\cosh (a+b x)}}\right )\right )}{b}-\frac {2 \cosh ^{\frac {5}{2}}(a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {2 \sqrt {\cosh (a+b x)}}{b \sqrt {\sinh (a+b x)}}\)

Input:

Int[Cosh[a + b*x]^(7/2)/Sinh[a + b*x]^(7/2),x]
 

Output:

(-2*(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/2 - ArcTanh[Sqrt[Sinh 
[a + b*x]]/Sqrt[Cosh[a + b*x]]]/2))/b - (2*Cosh[a + b*x]^(5/2))/(5*b*Sinh[ 
a + b*x]^(5/2)) - (2*Sqrt[Cosh[a + b*x]])/(b*Sqrt[Sinh[a + b*x]])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 3047

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_), x_Symbol] :> Simp[a*(a*Cos[e + f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/ 
(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1)))   Int[(a*Cos[e + f*x] 
)^(m - 2)*(b*Sin[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 3054

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

Maple [F]

\[\int \frac {\cosh \left (b x +a \right )^{\frac {7}{2}}}{\sinh \left (b x +a \right )^{\frac {7}{2}}}d x\]

Input:

int(cosh(b*x+a)^(7/2)/sinh(b*x+a)^(7/2),x)
 

Output:

int(cosh(b*x+a)^(7/2)/sinh(b*x+a)^(7/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1001 vs. \(2 (88) = 176\).

Time = 0.11 (sec) , antiderivative size = 1001, normalized size of antiderivative = 9.44 \[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\text {Too large to display} \] Input:

integrate(cosh(b*x+a)^(7/2)/sinh(b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

-1/10*(24*cosh(b*x + a)^6 + 144*cosh(b*x + a)*sinh(b*x + a)^5 + 24*sinh(b* 
x + a)^6 + 72*(5*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^4 - 72*cosh(b*x + a)^4 
 + 96*(5*cosh(b*x + a)^3 - 3*cosh(b*x + a))*sinh(b*x + a)^3 + 72*(5*cosh(b 
*x + a)^4 - 6*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 10*(cosh(b*x + a)^6 + 
 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 
- 1)*sinh(b*x + a)^4 - 3*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 - 3*cosh(b 
*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 - 6*cosh(b*x + a)^2 + 1)*s 
inh(b*x + a)^2 + 3*cosh(b*x + a)^2 + 6*(cosh(b*x + a)^5 - 2*cosh(b*x + a)^ 
3 + cosh(b*x + a))*sinh(b*x + a) - 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) + 72*cosh(b*x + a)^2 + 5*(cosh(b* 
x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b 
*x + a)^2 - 1)*sinh(b*x + a)^4 - 3*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 
- 3*cosh(b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 - 6*cosh(b*x + a 
)^2 + 1)*sinh(b*x + a)^2 + 3*cosh(b*x + a)^2 + 6*(cosh(b*x + a)^5 - 2*cosh 
(b*x + a)^3 + cosh(b*x + a))*sinh(b*x + a) - 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) + 16*(3*cosh(b*x + a)^5 + 
15*cosh(b*x + a)*sinh(b*x + a)^4 + 3*sinh(b*x + a)^5 + 2*(15*cosh(b*x + a) 
^2 - 2)*sinh(b*x + a)^3 - 4*cosh(b*x + a)^3 + 6*(5*cosh(b*x + a)^3 - 2*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\text {Timed out} \] Input:

integrate(cosh(b*x+a)**(7/2)/sinh(b*x+a)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {\cosh \left (b x + a\right )^{\frac {7}{2}}}{\sinh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(cosh(b*x+a)^(7/2)/sinh(b*x+a)^(7/2),x, algorithm="maxima")
 

Output:

integrate(cosh(b*x + a)^(7/2)/sinh(b*x + a)^(7/2), x)
 

Giac [F]

\[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {\cosh \left (b x + a\right )^{\frac {7}{2}}}{\sinh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(cosh(b*x+a)^(7/2)/sinh(b*x+a)^(7/2),x, algorithm="giac")
 

Output:

integrate(cosh(b*x + a)^(7/2)/sinh(b*x + a)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^{7/2}}{{\mathrm {sinh}\left (a+b\,x\right )}^{7/2}} \,d x \] Input:

int(cosh(a + b*x)^(7/2)/sinh(a + b*x)^(7/2),x)
 

Output:

int(cosh(a + b*x)^(7/2)/sinh(a + b*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\cosh ^{\frac {7}{2}}(a+b x)}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {-2 \sqrt {\sinh \left (b x +a \right )}\, \sqrt {\cosh \left (b x +a \right )}\, \cosh \left (b x +a \right )^{2}-10 \sqrt {\sinh \left (b x +a \right )}\, \sqrt {\cosh \left (b x +a \right )}\, \sinh \left (b x +a \right )^{2}+5 \left (\int \frac {\sqrt {\sinh \left (b x +a \right )}\, \sqrt {\cosh \left (b x +a \right )}}{\cosh \left (b x +a \right )}d x \right ) \sinh \left (b x +a \right )^{3} b}{5 \sinh \left (b x +a \right )^{3} b} \] Input:

int(cosh(b*x+a)^(7/2)/sinh(b*x+a)^(7/2),x)
 

Output:

( - 2*sqrt(sinh(a + b*x))*sqrt(cosh(a + b*x))*cosh(a + b*x)**2 - 10*sqrt(s 
inh(a + b*x))*sqrt(cosh(a + b*x))*sinh(a + b*x)**2 + 5*int((sqrt(sinh(a + 
b*x))*sqrt(cosh(a + b*x)))/cosh(a + b*x),x)*sinh(a + b*x)**3*b)/(5*sinh(a 
+ b*x)**3*b)