Integrand size = 21, antiderivative size = 142 \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{(a-b)^{3/2} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}+\frac {2 b^2}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}} \]
2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-arctanh((a+b*sech(d*x +c))^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/d-arctanh((a+b*sech(d*x+c))^(1/2)/(a+b )^(1/2))/(a+b)^(3/2)/d+2*b^2/a/(a^2-b^2)/d/(a+b*sech(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.05 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.49 \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=-\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}-\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \text {sech}(c+d x)}{a-b}\right )}{(a-b) \sqrt {a+b \text {sech}(c+d x)}}+\frac {a \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \text {sech}(c+d x)}{a+b}\right )}{(a+b) \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \text {sech}(c+d x)}{a}\right )}{a \sqrt {a+b \text {sech}(c+d x)}}}{b d} \]
-((-(ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/Sqrt[a - b]) + ArcTanh [Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/Sqrt[a + b] - (a*Hypergeometric2F1 [-1/2, 1, 1/2, (a + b*Sech[c + d*x])/(a - b)])/((a - b)*Sqrt[a + b*Sech[c + d*x]]) + (a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sech[c + d*x])/(a + b )])/((a + b)*Sqrt[a + b*Sech[c + d*x]]) + (2*b*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sech[c + d*x])/a])/(a*Sqrt[a + b*Sech[c + d*x]]))/(b*d))
Time = 0.42 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4373, 561, 1610, 2009}
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.
\(\displaystyle \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\cot \left (i c+i d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{\cot \left (\frac {1}{2} (2 i c+\pi )+i d x\right ) \left (a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4373 |
\(\displaystyle -\frac {b^2 \int \frac {\cosh (c+d x)}{b (a+b \text {sech}(c+d x))^{3/2} \left (b^2-b^2 \text {sech}^2(c+d x)\right )}d(b \text {sech}(c+d x))}{d}\) |
\(\Big \downarrow \) 561 |
\(\displaystyle -\frac {2 b^2 \int \frac {\cosh ^2(c+d x)}{b^2 \left (a-b^2 \text {sech}^2(c+d x)\right ) \left (b^4 \text {sech}^4(c+d x)-2 a b^2 \text {sech}^2(c+d x)+a^2-b^2\right )}d\sqrt {a+b \text {sech}(c+d x)}}{d}\) |
\(\Big \downarrow \) 1610 |
\(\displaystyle -\frac {2 b^2 \int \left (\frac {\cosh ^2(c+d x)}{a b^2 \left (a^2-b^2\right )}-\frac {1}{a b^2 \left (a-b^2 \text {sech}^2(c+d x)\right )}+\frac {1}{2 (a-b) b^2 \left (-b^2 \text {sech}^2(c+d x)+a-b\right )}+\frac {1}{2 b^2 (a+b) \left (-b^2 \text {sech}^2(c+d x)+a+b\right )}\right )d\sqrt {a+b \text {sech}(c+d x)}}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 b^2 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} b^2}-\frac {\cosh (c+d x)}{a b \left (a^2-b^2\right )}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a-b}}\right )}{2 b^2 (a-b)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )}{2 b^2 (a+b)^{3/2}}\right )}{d}\) |
(-2*b^2*(-(ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]]/(a^(3/2)*b^2)) + Arc Tanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a - b]]/(2*(a - b)^(3/2)*b^2) + ArcTan h[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]]/(2*b^2*(a + b)^(3/2)) - Cosh[c + d*x]/(a*b*(a^2 - b^2))))/d
3.2.45.3.1 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[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k/d Subst[Int[x^(k*(n + 1) - 1)*(-c /d + x^k/d)^m*Simp[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^k/d^2) + b*(x^(2*k)/d^2), x]^p, x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, m, p}, x] && Frac tionQ[n] && IntegerQ[p] && IntegerQ[m]
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^2 - 4 *a*c, 0] && IntegerQ[q] && IntegerQ[m]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
\[\int \frac {\coth \left (d x +c \right )}{\left (a +b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (122) = 244\).
Time = 3.80 (sec) , antiderivative size = 14412, normalized size of antiderivative = 101.49 \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\coth {\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\coth \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\coth (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]