Integrand size = 10, antiderivative size = 76 \[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right )}{b \sqrt {\text {csch}(a+b x)} \sqrt {i \sinh (a+b x)}} \] Output:
-2*cosh(b*x+a)*csch(b*x+a)^(1/2)/b+2*I*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I* b*x),2^(1/2))/b/csch(b*x+a)^(1/2)/(I*sinh(b*x+a))^(1/2)
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75 \[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \sqrt {\text {csch}(a+b x)} \left (\cosh (a+b x)-E\left (\left .\frac {1}{4} (-2 i a+\pi -2 i b x)\right |2\right ) \sqrt {i \sinh (a+b x)}\right )}{b} \] Input:
Integrate[Csch[a + b*x]^(3/2),x]
Output:
(-2*Sqrt[Csch[a + b*x]]*(Cosh[a + b*x] - EllipticE[((-2*I)*a + Pi - (2*I)* b*x)/4, 2]*Sqrt[I*Sinh[a + b*x]]))/b
Time = 0.36 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4255, 3042, 4258, 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.
\(\displaystyle \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (i \csc (i a+i b x))^{3/2}dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \int \frac {1}{\sqrt {\text {csch}(a+b x)}}dx-\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}+\int \frac {1}{\sqrt {i \csc (i a+i b x)}}dx\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle -\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}+\frac {\int \sqrt {i \sinh (a+b x)}dx}{\sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}+\frac {\int \sqrt {\sin (i a+i b x)}dx}{\sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {2 \cosh (a+b x) \sqrt {\text {csch}(a+b x)}}{b}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{b \sqrt {i \sinh (a+b x)} \sqrt {\text {csch}(a+b x)}}\) |
Input:
Int[Csch[a + b*x]^(3/2),x]
Output:
(-2*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/b - ((2*I)*EllipticE[(I*a - Pi/2 + I*b*x)/2, 2])/(b*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (63 ) = 126\).
Time = 0.16 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.03
method | result | size |
default | \(\frac {2 \sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (b x +a \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (b x +a \right )}\, \sqrt {i \sinh \left (b x +a \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (b x +a \right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (b x +a \right )^{2}}{\cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}\, b}\) | \(154\) |
Input:
int(csch(b*x+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
(2*(1-I*sinh(b*x+a))^(1/2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+a)) ^(1/2)*EllipticE((1-I*sinh(b*x+a))^(1/2),1/2*2^(1/2))-(1-I*sinh(b*x+a))^(1 /2)*2^(1/2)*(1+I*sinh(b*x+a))^(1/2)*(I*sinh(b*x+a))^(1/2)*EllipticF((1-I*s inh(b*x+a))^(1/2),1/2*2^(1/2))-2*cosh(b*x+a)^2)/cosh(b*x+a)/sinh(b*x+a)^(1 /2)/b
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.26 \[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=-\frac {2 \, {\left (\sqrt {2} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + \sqrt {2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )\right )\right )}}{b} \] Input:
integrate(csch(b*x+a)^(3/2),x, algorithm="fricas")
Output:
-2*(sqrt(2)*sqrt((cosh(b*x + a) + sinh(b*x + a))/(cosh(b*x + a)^2 + 2*cosh (b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1))*(cosh(b*x + a) + sinh(b*x + a)) + sqrt(2)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(b*x + a) + sinh(b*x + a))))/b
\[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int \operatorname {csch}^{\frac {3}{2}}{\left (a + b x \right )}\, dx \] Input:
integrate(csch(b*x+a)**(3/2),x)
Output:
Integral(csch(a + b*x)**(3/2), x)
\[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(csch(b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate(csch(b*x + a)^(3/2), x)
\[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int { \operatorname {csch}\left (b x + a\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(csch(b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(csch(b*x + a)^(3/2), x)
Timed out. \[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int {\left (\frac {1}{\mathrm {sinh}\left (a+b\,x\right )}\right )}^{3/2} \,d x \] Input:
int((1/sinh(a + b*x))^(3/2),x)
Output:
int((1/sinh(a + b*x))^(3/2), x)
\[ \int \text {csch}^{\frac {3}{2}}(a+b x) \, dx=\int \sqrt {\mathrm {csch}\left (b x +a \right )}\, \mathrm {csch}\left (b x +a \right )d x \] Input:
int(csch(b*x+a)^(3/2),x)
Output:
int(sqrt(csch(a + b*x))*csch(a + b*x),x)