\(\int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx\) [14]

Optimal result

Integrand size = 10, antiderivative size = 69 \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}} \] Output:

6/5*I*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))/b+2/5*sinh(b*x+a)/b/cosh(b* 
x+a)^(5/2)+6/5*sinh(b*x+a)/b/cosh(b*x+a)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {6 i \cosh ^{\frac {3}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+3 \sinh (2 (a+b x))+2 \tanh (a+b x)}{5 b \cosh ^{\frac {3}{2}}(a+b x)} \] Input:

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

Output:

((6*I)*Cosh[a + b*x]^(3/2)*EllipticE[(I/2)*(a + b*x), 2] + 3*Sinh[2*(a + b 
*x)] + 2*Tanh[a + b*x])/(5*b*Cosh[a + b*x]^(3/2))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3116, 3042, 3116, 3042, 3119}

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.

\(\displaystyle \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin \left (i a+i b x+\frac {\pi }{2}\right )^{7/2}}dx\)

\(\Big \downarrow \) 3116

\(\displaystyle \frac {3}{5} \int \frac {1}{\cosh ^{\frac {3}{2}}(a+b x)}dx+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {3}{5} \int \frac {1}{\sin \left (i a+i b x+\frac {\pi }{2}\right )^{3/2}}dx\)

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {3}{5} \left (\frac {2 \sinh (a+b x)}{b \sqrt {\cosh (a+b x)}}-\int \sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}dx\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {3}{5} \left (\frac {2 \sinh (a+b x)}{b \sqrt {\cosh (a+b x)}}+\frac {2 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b}\right )\)

Input:

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

Output:

(2*Sinh[a + b*x])/(5*b*Cosh[a + b*x]^(5/2)) + (3*(((2*I)*EllipticE[(I/2)*( 
a + b*x), 2])/b + (2*Sinh[a + b*x])/(b*Sqrt[Cosh[a + b*x]])))/5
 

Defintions of rubi rules used

rule 3042

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

rule 3116

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1))   I 
nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && 
 IntegerQ[2*n]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(362\) vs. \(2(61)=122\).

Time = 1.46 (sec) , antiderivative size = 363, normalized size of antiderivative = 5.26

method	result	size
default	\(\frac {2 \sqrt {\left (2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (24 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{6} \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+24 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4} \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )+12 \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+8 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+3 \sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right ) \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}}{5 \left (8 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}+12 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+6 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \sqrt {2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}-1}\, b}\)	\(363\)

Input:

int(1/cosh(b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

2/5*((2*cosh(1/2*b*x+1/2*a)^2-1)*sinh(1/2*b*x+1/2*a)^2)^(1/2)/(8*sinh(1/2* 
b*x+1/2*a)^6+12*sinh(1/2*b*x+1/2*a)^4+6*sinh(1/2*b*x+1/2*a)^2+1)/sinh(1/2* 
b*x+1/2*a)^3*(24*sinh(1/2*b*x+1/2*a)^6*cosh(1/2*b*x+1/2*a)+12*(-sinh(1/2*b 
*x+1/2*a)^2)^(1/2)*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticE(cosh(1/2*b 
*x+1/2*a),2^(1/2))*sinh(1/2*b*x+1/2*a)^4+24*sinh(1/2*b*x+1/2*a)^4*cosh(1/2 
*b*x+1/2*a)+12*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*sinh(1/2*b*x+1/2*a)^2-1) 
^(1/2)*EllipticE(cosh(1/2*b*x+1/2*a),2^(1/2))*sinh(1/2*b*x+1/2*a)^2+8*cosh 
(1/2*b*x+1/2*a)*sinh(1/2*b*x+1/2*a)^2+3*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2 
*sinh(1/2*b*x+1/2*a)^2-1)^(1/2)*EllipticE(cosh(1/2*b*x+1/2*a),2^(1/2)))*(2 
*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/(2*cosh(1/2*b*x+1/2*a) 
^2-1)^(1/2)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (59) = 118\).

Time = 0.11 (sec) , antiderivative size = 613, normalized size of antiderivative = 8.88 \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx =\text {Too large to display} \] Input:

integrate(1/cosh(b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

2/5*(3*(sqrt(2)*cosh(b*x + a)^6 + 6*sqrt(2)*cosh(b*x + a)*sinh(b*x + a)^5 
+ sqrt(2)*sinh(b*x + a)^6 + 3*(5*sqrt(2)*cosh(b*x + a)^2 + sqrt(2))*sinh(b 
*x + a)^4 + 3*sqrt(2)*cosh(b*x + a)^4 + 4*(5*sqrt(2)*cosh(b*x + a)^3 + 3*s 
qrt(2)*cosh(b*x + a))*sinh(b*x + a)^3 + 3*(5*sqrt(2)*cosh(b*x + a)^4 + 6*s 
qrt(2)*cosh(b*x + a)^2 + sqrt(2))*sinh(b*x + a)^2 + 3*sqrt(2)*cosh(b*x + a 
)^2 + 6*(sqrt(2)*cosh(b*x + a)^5 + 2*sqrt(2)*cosh(b*x + a)^3 + sqrt(2)*cos 
h(b*x + a))*sinh(b*x + a) + sqrt(2))*weierstrassZeta(-4, 0, weierstrassPIn 
verse(-4, 0, cosh(b*x + a) + sinh(b*x + a))) + 2*(3*cosh(b*x + a)^6 + 18*c 
osh(b*x + a)*sinh(b*x + a)^5 + 3*sinh(b*x + a)^6 + (45*cosh(b*x + a)^2 + 8 
)*sinh(b*x + a)^4 + 8*cosh(b*x + a)^4 + 4*(15*cosh(b*x + a)^3 + 8*cosh(b*x 
 + a))*sinh(b*x + a)^3 + (45*cosh(b*x + a)^4 + 48*cosh(b*x + a)^2 + 1)*sin 
h(b*x + a)^2 + cosh(b*x + a)^2 + 2*(9*cosh(b*x + a)^5 + 16*cosh(b*x + a)^3 
 + cosh(b*x + a))*sinh(b*x + a))*sqrt(cosh(b*x + a)))/(b*cosh(b*x + a)^6 + 
 6*b*cosh(b*x + a)*sinh(b*x + a)^5 + b*sinh(b*x + a)^6 + 3*b*cosh(b*x + a) 
^4 + 3*(5*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^4 + 4*(5*b*cosh(b*x + a)^3 
+ 3*b*cosh(b*x + a))*sinh(b*x + a)^3 + 3*b*cosh(b*x + a)^2 + 3*(5*b*cosh(b 
*x + a)^4 + 6*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^2 + 6*(b*cosh(b*x + a)^ 
5 + 2*b*cosh(b*x + a)^3 + b*cosh(b*x + a))*sinh(b*x + a) + b)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\text {Timed out} \] Input:

integrate(1/cosh(b*x+a)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/cosh(b*x+a)^(7/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\cosh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/cosh(b*x+a)^(7/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^{7/2}} \,d x \] Input:

int(1/cosh(a + b*x)^(7/2),x)
 

Output:

int(1/cosh(a + b*x)^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {\sqrt {\cosh \left (b x +a \right )}}{\cosh \left (b x +a \right )^{4}}d x \] Input:

int(1/cosh(b*x+a)^(7/2),x)
                                                                                    
                                                                                    
 

Output:

int(sqrt(cosh(a + b*x))/cosh(a + b*x)**4,x)