\(\int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx\) [70]

Optimal result

Integrand size = 17, antiderivative size = 87 \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {i \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}} \]

[Out]

1/2*I*cosh(d*x+c)/d/(a+I*a*sinh(d*x+c))^(3/2)+1/4*I*arctanh(1/2*cosh(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*sinh(d*x+c)
)^(1/2))/a^(3/2)/d*2^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2729, 2728, 212} \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {i \text {arctanh}\left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {2} \sqrt {a+i a \sinh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}} \]

[In]

Int[(a + I*a*Sinh[c + d*x])^(-3/2),x]

[Out]

((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)*Co
sh[c + d*x])/(d*(a + I*a*Sinh[c + d*x])^(3/2))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2728

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, b*(C
os[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

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] + Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a+i a \sinh (c+d x)}} \, dx}{4 a} \\ & = \frac {i \cosh (c+d x)}{2 d (a+i a \sinh (c+d x))^{3/2}}+\frac {i \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cosh (c+d x)}{\sqrt {a+i a \sinh (c+d x)}}\right )}{2 a d} \\ & = \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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \left ((1-i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1-i \tanh \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2 a d (-i+\sinh (c+d x)) \sqrt {a+i a \sinh (c+d x)}} \]

[In]

Integrate[(a + I*a*Sinh[c + d*x])^(-3/2),x]

[Out]

((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]])

Maple [F]

\[\int \frac {1}{\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

[In]

int(1/(a+I*a*sinh(d*x+c))^(3/2),x)

[Out]

int(1/(a+I*a*sinh(d*x+c))^(3/2),x)

Fricas [B] (verification not implemented)

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 = 0.28 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (i \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} d e^{\left (d x + c\right )} - i \, a^{2} d\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (\sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) + \sqrt {\frac {1}{2}} {\left (-i \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a^{2} d e^{\left (d x + c\right )} + i \, a^{2} d\right )} \sqrt {\frac {1}{a^{3} d^{2}}} \log \left (-\sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}}\right ) - 2 \, \sqrt {\frac {1}{2} i \, a e^{\left (-d x - c\right )}} {\left (i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )}\right )}}{2 \, {\left (a^{2} d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a^{2} d e^{\left (d x + c\right )} - a^{2} d\right )}} \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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)*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))) + 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)

Sympy [F]

\[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (i a \sinh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))**(3/2),x)

[Out]

Integral((I*a*sinh(c + d*x) + a)**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((I*a*sinh(d*x + c) + a)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(a+I*a*sinh(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((I*a*sinh(d*x + c) + a)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+i a \sinh (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]

[In]

int(1/(a + a*sinh(c + d*x)*1i)^(3/2),x)

[Out]

int(1/(a + a*sinh(c + d*x)*1i)^(3/2), x)