Integrand size = 17, antiderivative size = 87 \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {i \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}} \] Output:
1/4*I*arctanh(1/2*a^(1/2)*cosh(d*x+c)*2^(1/2)/(a+I*a*sinh(d*x+c))^(1/2))*2 ^(1/2)/a^(3/2)/d+1/2*I*cosh(d*x+c)/d/(a+I*a*sinh(d*x+c))^(3/2)
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \left ((1-i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1-i \tanh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2 a d (-i+\sinh (c+d x)) \sqrt {a+i a \sinh (c+d x)}} \] Input:
Integrate[(a + I*a*Sinh[c + d*x])^(-3/2),x]
Output:
((Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2])*(Cosh[(c + d*x)/2] - I*((1 - I) *(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*(1 - I*Tanh[(c + d*x)/4])]*(Cosh [(c + d*x)/2] + I*Sinh[(c + d*x)/2])^2 + Sinh[(c + d*x)/2])))/(2*a*d*(-I + Sinh[c + d*x])*Sqrt[a + I*a*Sinh[c + d*x]])
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 3129, 3042, 3128, 219}
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.
\(\displaystyle \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+a \sin (i c+i d x))^{3/2}}dx\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {i \sinh (c+d x) a+a}}dx}{4 a}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {\sin (i c+i d x) a+a}}dx}{4 a}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle \frac {i \int \frac {1}{2 a-\frac {a^2 \cosh ^2(c+d x)}{i \sinh (c+d x) a+a}}d\frac {a \cosh (c+d x)}{\sqrt {i \sinh (c+d x) a+a}}}{2 a d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {i \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}\) |
Input:
Int[(a + I*a*Sinh[c + d*x])^(-3/2),x]
Output:
((I/2)*ArcTanh[(Sqrt[a]*Cosh[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Sinh[c + d*x] ])])/(Sqrt[2]*a^(3/2)*d) + ((I/2)*Cosh[c + d*x])/(d*(a + I*a*Sinh[c + d*x] )^(3/2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[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. 175 vs. \(2 (68 ) = 136\).
Time = 0.26 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.02
method | result | size |
default | \(-\frac {\sqrt {\sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )^{2} a}\, \left (\ln \left (\frac {2 \sqrt {-a}\, \sqrt {\sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )^{2} a}-2 a}{\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}\right ) a \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )^{2}-\sqrt {-a}\, \sqrt {\sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )^{2} a}\right ) \sqrt {2}}{4 a^{2} \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {-a}\, \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )^{2}}\, d}\) | \(176\) |
Input:
int(1/(a+I*a*sinh(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4*(sinh(1/2*c+1/4*I*Pi+1/2*d*x)^2*a)^(1/2)*(ln(2*((-a)^(1/2)*(sinh(1/2* c+1/4*I*Pi+1/2*d*x)^2*a)^(1/2)-a)/cosh(1/2*c+1/4*I*Pi+1/2*d*x))*a*cosh(1/2 *c+1/4*I*Pi+1/2*d*x)^2-(-a)^(1/2)*(sinh(1/2*c+1/4*I*Pi+1/2*d*x)^2*a)^(1/2) )/a^2/cosh(1/2*c+1/4*I*Pi+1/2*d*x)/(-a)^(1/2)/sinh(1/2*c+1/4*I*Pi+1/2*d*x) *2^(1/2)/(a*cosh(1/2*c+1/4*I*Pi+1/2*d*x)^2)^(1/2)/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (64) = 128\).
Time = 0.09 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} d e^{\left (d x + c\right )} - i \, a^{2} d\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (\sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) + \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{2} d e^{\left (d x + c\right )} + i \, a^{2} d\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (-\sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) - 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )}\right )}}{2 \, {\left (a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a^{2} d e^{\left (d x + c\right )} - a^{2} d\right )}} \] Input:
integrate(1/(a+I*a*sinh(d*x+c))^(3/2),x, algorithm="fricas")
Output:
1/2*(sqrt(1/2)*(I*a^2*d*e^(2*d*x + 2*c) + 2*a^2*d*e^(d*x + c) - I*a^2*d)*s qrt(1/(a^3*d^2))*log(sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2)) + sqrt(1/2*I*a*e^(- d*x - c))) + sqrt(1/2)*(-I*a^2*d*e^(2*d*x + 2*c) - 2*a^2*d*e^(d*x + c) + I *a^2*d)*sqrt(1/(a^3*d^2))*log(-sqrt(1/2)*a^2*d*sqrt(1/(a^3*d^2)) + sqrt(1/ 2*I*a*e^(-d*x - c))) - 2*sqrt(1/2*I*a*e^(-d*x - c))*(I*e^(2*d*x + 2*c) - e ^(d*x + c)))/(a^2*d*e^(2*d*x + 2*c) - 2*I*a^2*d*e^(d*x + c) - a^2*d)
\[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (i a \sinh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a+I*a*sinh(d*x+c))**(3/2),x)
Output:
Integral((I*a*sinh(c + d*x) + a)**(-3/2), x)
\[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a+I*a*sinh(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((I*a*sinh(d*x + c) + a)^(-3/2), x)
\[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a+I*a*sinh(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate((I*a*sinh(d*x + c) + a)^(-3/2), x)
Timed out. \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:
int(1/(a + a*sinh(c + d*x)*1i)^(3/2),x)
Output:
int(1/(a + a*sinh(c + d*x)*1i)^(3/2), x)
\[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sinh \left (d x +c \right ) i +1}}{\sinh \left (d x +c \right )^{3} i +\sinh \left (d x +c \right )^{2}+\sinh \left (d x +c \right ) i +1}d x -\left (\int \frac {\sqrt {\sinh \left (d x +c \right ) i +1}\, \sinh \left (d x +c \right )}{\sinh \left (d x +c \right )^{3} i +\sinh \left (d x +c \right )^{2}+\sinh \left (d x +c \right ) i +1}d x \right ) i \right )}{a^{2}} \] Input:
int(1/(a+I*a*sinh(d*x+c))^(3/2),x)
Output:
(sqrt(a)*(int(sqrt(sinh(c + d*x)*i + 1)/(sinh(c + d*x)**3*i + sinh(c + d*x )**2 + sinh(c + d*x)*i + 1),x) - int((sqrt(sinh(c + d*x)*i + 1)*sinh(c + d *x))/(sinh(c + d*x)**3*i + sinh(c + d*x)**2 + sinh(c + d*x)*i + 1),x)*i))/ a**2