Integrand size = 17, antiderivative size = 123 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}} \]
2*arctanh(coth(d*x+c)*a^(1/2)/(a+I*a*csch(d*x+c))^(1/2))/a^(3/2)/d-1/2*cot h(d*x+c)/d/(a+I*a*csch(d*x+c))^(3/2)-5/4*arctanh(1/2*coth(d*x+c)*a^(1/2)*2 ^(1/2)/(a+I*a*csch(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(123)=246\).
Time = 3.42 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {\left (-2 \sqrt {a}-8 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+5 \sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+i \text {csch}(c+d x) \left (2 \sqrt {a}-8 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+5 \sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^{3/2} d (i+\text {csch}(c+d x)) \sqrt {a+i a \text {csch}(c+d x)} \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \]
((-2*Sqrt[a] - 8*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]]*Sqrt[I*a*(I + Csch[c + d*x])] + 5*Sqrt[2]*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt[ 2]*Sqrt[a])]*Sqrt[I*a*(I + Csch[c + d*x])] + I*Csch[c + d*x]*(2*Sqrt[a] - 8*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]]*Sqrt[I*a*(I + Csch[c + d*x ])] + 5*Sqrt[2]*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt[2]*Sqrt[a])]*Sq rt[I*a*(I + Csch[c + d*x])]))*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]))/( 4*a^(3/2)*d*(I + Csch[c + d*x])*Sqrt[a + I*a*Csch[c + d*x]]*(Cosh[(c + d*x )/2] - I*Sinh[(c + d*x)/2]))
Time = 0.58 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3042, 4264, 27, 3042, 4408, 26, 3042, 26, 4261, 216, 4282, 216}
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.
\(\displaystyle \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a-a \csc (i c+i d x))^{3/2}}dx\) |
\(\Big \downarrow \) 4264 |
\(\displaystyle -\frac {\int -\frac {4 a-i a \text {csch}(c+d x)}{2 \sqrt {i \text {csch}(c+d x) a+a}}dx}{2 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {4 a-i a \text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\csc (i c+i d x) a+4 a}{\sqrt {a-a \csc (i c+i d x)}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 4408 |
\(\displaystyle \frac {5 a \int -\frac {i \text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx+4 \int \sqrt {i \text {csch}(c+d x) a+a}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {4 \int \sqrt {i \text {csch}(c+d x) a+a}dx-5 i a \int \frac {\text {csch}(c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 \int \sqrt {a-a \csc (i c+i d x)}dx-5 i a \int \frac {i \csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {5 a \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx+4 \int \sqrt {a-a \csc (i c+i d x)}dx}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 4261 |
\(\displaystyle \frac {5 a \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx-\frac {8 i a \int \frac {1}{a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {5 a \int \frac {\csc (i c+i d x)}{\sqrt {a-a \csc (i c+i d x)}}dx+\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 4282 |
\(\displaystyle \frac {\frac {10 i a \int \frac {1}{2 a-\frac {a^2 \coth ^2(c+d x)}{i \text {csch}(c+d x) a+a}}d\frac {i a \coth (c+d x)}{\sqrt {i \text {csch}(c+d x) a+a}}}{d}+\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}-\frac {5 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}}{4 a^2}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}\) |
((8*Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/ d - (5*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[c + d*x])/(Sqrt[2]*Sqrt[a + I *a*Csch[c + d*x]])])/d)/(4*a^2) - Coth[c + d*x]/(2*d*(a + I*a*Csch[c + d*x ])^(3/2))
3.1.55.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-Cot[c + d*x])*((a + b*Csc[c + d*x])^n/(d*(2*n + 1))), x] + Simp[1/(a^2*(2*n + 1)) Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && Int egerQ[2*n]
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2/f Subst[Int[1/(2*a + x^2), x], x, b*(Cot[e + f*x]/Sqrt[ a + b*Csc[e + f*x]])], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c/a Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Simp[(b*c - a*d)/a Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; F reeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
\[\int \frac {1}{\left (a +i a \,\operatorname {csch}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 873 vs. \(2 (96) = 192\).
Time = 0.30 (sec) , antiderivative size = 873, normalized size of antiderivative = 7.10 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\text {Too large to display} \]
-1/4*(5*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)* sqrt(1/(a^3*d^2))*log(2*(2*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) - a^2*d)*sqrt( a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) + a*e^(d*x + c) - I*a)*e^(-d*x - c)) - 5*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d )*sqrt(1/(a^3*d^2))*log(-2*(2*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) - a^2*d)*sq rt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) - a*e^(d*x + c) + I*a)*e^(-d *x - c)) - 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt( 1/(a^3*d^2))*log(2*((a*d*e^(2*d*x + 2*c) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) + e^(d*x + c) + I)*e^(-d*x - c)/(a*d)) + 2*(a^2*d*e^ (2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(-2*(( a*d*e^(2*d*x + 2*c) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) - e^(d*x + c) - I)*e^(-d*x - c)/(a*d)) - 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a ^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(2*((a^2*d*e^(2*d*x + 2*c) - I*a^2*d*e^(d*x + c) - 2*a^2*d)*sqrt(1/(a^3*d^2)) + sqrt(a/(e^(2*d*x + 2* c) - 1))*(e^(3*d*x + 3*c) - 2*I*e^(2*d*x + 2*c) - e^(d*x + c) + 2*I))*e^(- 2*d*x - 2*c)/(a*d)) + 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a ^2*d)*sqrt(1/(a^3*d^2))*log(-2*((a^2*d*e^(2*d*x + 2*c) - I*a^2*d*e^(d*x + c) - 2*a^2*d)*sqrt(1/(a^3*d^2)) - sqrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) - 2*I*e^(2*d*x + 2*c) - e^(d*x + c) + 2*I))*e^(-2*d*x - 2*c)/(a*d)) + 2*sqrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) - I*e^(2*d*x + 2*c)...
\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{\left (i a \operatorname {csch}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]