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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2716, 2719} \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {6 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{5 b}+\frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}} \]

[In]

Int[Cosh[a + b*x]^(-7/2),x]

[Out]

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

Rule 2716

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] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

Rule 2719

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sinh (a+b x)}{5 b \cosh ^{\frac {5}{2}}(a+b x)}+\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)}+\frac {6 \sinh (a+b x)}{5 b \sqrt {\cosh (a+b x)}}-\frac {3}{5} \int \sqrt {\cosh (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)}} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[Cosh[a + b*x]^(-7/2),x]

[Out]

((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))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(362\) vs. \(2(85)=170\).

Time = 0.96 (sec) , antiderivative size = 363, normalized size of antiderivative = 5.26

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

[In]

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

[Out]

2/5*((-1+2*cosh(1/2*b*x+1/2*a)^2)*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*(-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)^(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*(-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)^(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*EllipticE(cosh(1/2*b*x+1/2*a),2^(1/2))*(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*sinh(1/2*b*x+1/2*a)^2-1)^(1
/2))*(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/(-1+2*cosh(1/2*b*x+1/2*a)^2)^(1/2)/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 613, normalized size of antiderivative = 8.88 \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{6} + 6 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sqrt {2} \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (b x + a\right )^{2} + \sqrt {2}\right )} \sinh \left (b x + a\right )^{4} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (b x + a\right )^{4} + 6 \, \sqrt {2} \cosh \left (b x + a\right )^{2} + \sqrt {2}\right )} \sinh \left (b x + a\right )^{2} + 3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{5} + 2 \, \sqrt {2} \cosh \left (b x + a\right )^{3} + \sqrt {2} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + \sqrt {2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right ) + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{6} + 18 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, \sinh \left (b x + a\right )^{6} + {\left (45 \, \cosh \left (b x + a\right )^{2} + 8\right )} \sinh \left (b x + a\right )^{4} + 8 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (15 \, \cosh \left (b x + a\right )^{3} + 8 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (45 \, \cosh \left (b x + a\right )^{4} + 48 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + \cosh \left (b x + a\right )^{2} + 2 \, {\left (9 \, \cosh \left (b x + a\right )^{5} + 16 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sqrt {\cosh \left (b x + a\right )}\right )}}{5 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} + 3 \, b \cosh \left (b x + a\right )^{4} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{4} + 6 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (b \cosh \left (b x + a\right )^{5} + 2 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]

[In]

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

[Out]

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*sqr
t(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
*sqrt(2)*cosh(b*x + a))*sinh(b*x + a)^3 + 3*(5*sqrt(2)*cosh(b*x + a)^4 + 6*sqrt(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)
*cosh(b*x + a))*sinh(b*x + a) + sqrt(2))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(b*x + a) + sin
h(b*x + a))) + 2*(3*cosh(b*x + a)^6 + 18*cosh(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*c
osh(b*x + a)^4 + 48*cosh(b*x + a)^2 + 1)*sinh(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. \[ \int \frac {1}{\cosh ^{\frac {7}{2}}(a+b x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

integrate(cosh(b*x + a)^(-7/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(cosh(b*x + a)^(-7/2), x)

Mupad [F(-1)]

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

[In]

int(1/cosh(a + b*x)^(7/2),x)

[Out]

int(1/cosh(a + b*x)^(7/2), x)