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

Optimal result

Integrand size = 10, antiderivative size = 103 \[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\frac {6 \cosh (a+b x)}{5 b \sqrt {\sinh (a+b x)}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{5 b \sqrt {i \sinh (a+b x)}} \]

[Out]

-2/5*cosh(b*x+a)/b/sinh(b*x+a)^(5/2)+6/5*cosh(b*x+a)/b/sinh(b*x+a)^(1/2)-6/5*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*x)^
2)^(1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b*x)*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*x),2^(1/2))*sinh(b*x+a)^(1/2)/b/(I
*sinh(b*x+a))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2716, 2721, 2719} \[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\frac {6 \cosh (a+b x)}{5 b \sqrt {\sinh (a+b x)}}+\frac {6 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh (a+b x)}} \]

[In]

Int[Sinh[a + b*x]^(-7/2),x]

[Out]

(-2*Cosh[a + b*x])/(5*b*Sinh[a + b*x]^(5/2)) + (6*Cosh[a + b*x])/(5*b*Sqrt[Sinh[a + b*x]]) + (((6*I)/5)*Ellipt
icE[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[I*Sinh[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]

Rule 2721

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {3}{5} \int \frac {1}{\sinh ^{\frac {3}{2}}(a+b x)} \, dx \\ & = -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\frac {6 \cosh (a+b x)}{5 b \sqrt {\sinh (a+b x)}}-\frac {3}{5} \int \sqrt {\sinh (a+b x)} \, dx \\ & = -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\frac {6 \cosh (a+b x)}{5 b \sqrt {\sinh (a+b x)}}-\frac {\left (3 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{5 \sqrt {i \sinh (a+b x)}} \\ & = -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\frac {6 \cosh (a+b x)}{5 b \sqrt {\sinh (a+b x)}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{5 b \sqrt {i \sinh (a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {-2 \coth (a+b x)+6 i E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) (i \sinh (a+b x))^{3/2}+3 \sinh (2 (a+b x))}{5 b \sinh ^{\frac {3}{2}}(a+b x)} \]

[In]

Integrate[Sinh[a + b*x]^(-7/2),x]

[Out]

(-2*Coth[a + b*x] + (6*I)*EllipticE[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*(I*Sinh[a + b*x])^(3/2) + 3*Sinh[2*(a +
b*x)])/(5*b*Sinh[a + b*x]^(3/2))

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.86

method	result	size
default	\(-\frac {6 \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \sinh \left (b x +a \right )^{2} \operatorname {EllipticE}\left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \sinh \left (b x +a \right )^{2} \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \sinh \left (b x +a \right )^{4}-4 \sinh \left (b x +a \right )^{2}+2}{5 \sinh \left (b x +a \right )^{\frac {5}{2}} \cosh \left (b x +a \right ) b}\)	\(192\)

[In]

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

[Out]

-1/5/sinh(b*x+a)^(5/2)*(6*(-I*(sinh(b*x+a)+I))^(1/2)*2^(1/2)*(-I*(-sinh(b*x+a)+I))^(1/2)*(I*sinh(b*x+a))^(1/2)
*sinh(b*x+a)^2*EllipticE((-I*(sinh(b*x+a)+I))^(1/2),1/2*2^(1/2))-3*(-I*(sinh(b*x+a)+I))^(1/2)*2^(1/2)*(-I*(-si
nh(b*x+a)+I))^(1/2)*(I*sinh(b*x+a))^(1/2)*sinh(b*x+a)^2*EllipticF((-I*(sinh(b*x+a)+I))^(1/2),1/2*2^(1/2))-6*si
nh(b*x+a)^4-4*sinh(b*x+a)^2+2)/cosh(b*x+a)/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 = 621, normalized size of antiderivative = 6.03 \[ \int \frac {1}{\sinh ^{\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 {\sinh \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/sinh(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) + sinh(
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*cos
h(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(sinh(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]

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

[In]

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

[Out]

Integral(sinh(a + b*x)**(-7/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sinh(b*x + a)^(-7/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(sinh(b*x + a)^(-7/2), x)

Mupad [F(-1)]

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

[In]

int(1/sinh(a + b*x)^(7/2),x)

[Out]

int(1/sinh(a + b*x)^(7/2), x)