Integrand size = 10, antiderivative size = 103 \[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=-\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}+\frac {6 \cosh (a+b x)}{5 b \sqrt {\sinh (a+b x)}}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{5 b \sqrt {i \sinh (a+b x)}} \] Output:
-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*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)
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\frac {-2 \coth (a+b x)+6 i E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) (i \sinh (a+b x))^{3/2}+3 \sinh (2 (a+b x))}{5 b \sinh ^{\frac {3}{2}}(a+b x)} \] Input:
Integrate[Sinh[a + b*x]^(-7/2),x]
Output:
(-2*Coth[a + b*x] + (6*I)*EllipticE[((-2*I)*a + Pi - (2*I)*b*x)/4, 2]*(I*S inh[a + b*x])^(3/2) + 3*Sinh[2*(a + b*x)])/(5*b*Sinh[a + b*x]^(3/2))
Time = 0.41 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3116, 3042, 3116, 3042, 3121, 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.
\(\displaystyle \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(-i \sin (i a+i b x))^{7/2}}dx\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle -\frac {3}{5} \int \frac {1}{\sinh ^{\frac {3}{2}}(a+b x)}dx-\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {3}{5} \int \frac {1}{(-i \sin (i a+i b x))^{3/2}}dx\) |
\(\Big \downarrow \) 3116 |
\(\displaystyle -\frac {3}{5} \left (\int \sqrt {\sinh (a+b x)}dx-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}\right )-\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}+\int \sqrt {-i \sin (i a+i b x)}dx\right )\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}+\frac {\sqrt {\sinh (a+b x)} \int \sqrt {i \sinh (a+b x)}dx}{\sqrt {i \sinh (a+b x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}+\frac {\sqrt {\sinh (a+b x)} \int \sqrt {\sin (i a+i b x)}dx}{\sqrt {i \sinh (a+b x)}}\right )\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {2 \cosh (a+b x)}{5 b \sinh ^{\frac {5}{2}}(a+b x)}-\frac {3}{5} \left (-\frac {2 \cosh (a+b x)}{b \sqrt {\sinh (a+b x)}}-\frac {2 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)}}\right )\) |
Input:
Int[Sinh[a + b*x]^(-7/2),x]
Output:
(-3*((-2*Cosh[a + b*x])/(b*Sqrt[Sinh[a + b*x]]) - ((2*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2]*Sqrt[Sinh[a + b*x]])/(b*Sqrt[I*Sinh[a + b*x]])))/5 - ( 2*Cosh[a + b*x])/(5*b*Sinh[a + b*x]^(5/2))
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]
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]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (82 ) = 164\).
Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.86
method | result | size |
default | \(-\frac {6 \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \sinh \left (b x +a \right )^{2} \operatorname {EllipticE}\left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (b x +a \right )+i\right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \sinh \left (b x +a \right )^{2} \operatorname {EllipticF}\left (\sqrt {-i \left (\sinh \left (b x +a \right )+i\right )}, \frac {\sqrt {2}}{2}\right )-6 \sinh \left (b x +a \right )^{4}-4 \sinh \left (b x +a \right )^{2}+2}{5 \sinh \left (b x +a \right )^{\frac {5}{2}} \cosh \left (b x +a \right ) b}\) | \(192\) |
Input:
int(1/sinh(b*x+a)^(7/2),x,method=_RETURNVERBOSE)
Output:
-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*(-sinh (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*sinh(b*x+a)^4-4*sinh(b*x+a)^2+2)/cosh(b*x+ a)/b
Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (78) = 156\).
Time = 0.10 (sec) , antiderivative size = 621, normalized size of antiderivative = 6.03 \[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx =\text {Too large to display} \] Input:
integrate(1/sinh(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, weierstrassPInv erse(4, 0, cosh(b*x + a) + sinh(b*x + a))) + 2*(3*cosh(b*x + a)^6 + 18*cos h(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)*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)
\[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{\sinh ^{\frac {7}{2}}{\left (a + b x \right )}}\, dx \] Input:
integrate(1/sinh(b*x+a)**(7/2),x)
Output:
Integral(sinh(a + b*x)**(-7/2), x)
\[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\sinh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/sinh(b*x+a)^(7/2),x, algorithm="maxima")
Output:
integrate(sinh(b*x + a)^(-7/2), x)
\[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int { \frac {1}{\sinh \left (b x + a\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/sinh(b*x+a)^(7/2),x, algorithm="giac")
Output:
integrate(sinh(b*x + a)^(-7/2), x)
Timed out. \[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {1}{{\mathrm {sinh}\left (a+b\,x\right )}^{7/2}} \,d x \] Input:
int(1/sinh(a + b*x)^(7/2),x)
Output:
int(1/sinh(a + b*x)^(7/2), x)
\[ \int \frac {1}{\sinh ^{\frac {7}{2}}(a+b x)} \, dx=\int \frac {\sqrt {\sinh \left (b x +a \right )}}{\sinh \left (b x +a \right )^{4}}d x \] Input:
int(1/sinh(b*x+a)^(7/2),x)
Output:
int(sqrt(sinh(a + b*x))/sinh(a + b*x)**4,x)