Integrand size = 23, antiderivative size = 286 \[ \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx=\frac {b \left (a^2-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{\sqrt {2} \left (a^4+b^4\right ) d}-\frac {b \left (a^2-b^2\right ) \arctan \left (1+\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{\sqrt {2} \left (a^4+b^4\right ) d}+\frac {a^3 \arctan (\sinh (c+d x))}{\left (a^4+b^4\right ) d}+\frac {a b^2 \log (\cosh (c+d x))}{\left (a^4+b^4\right ) d}-\frac {2 a b^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{\left (a^4+b^4\right ) d}-\frac {b \left (a^2+b^2\right ) \log \left (1-\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )}{2 \sqrt {2} \left (a^4+b^4\right ) d}+\frac {b \left (a^2+b^2\right ) \log \left (1+\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )}{2 \sqrt {2} \left (a^4+b^4\right ) d} \]
a^3*arctan(sinh(d*x+c))/(a^4+b^4)/d+a*b^2*ln(cosh(d*x+c))/(a^4+b^4)/d-2*a* b^2*ln(a+b*sinh(d*x+c)^(1/2))/(a^4+b^4)/d-1/2*b*(a^2-b^2)*arctan(-1+2^(1/2 )*sinh(d*x+c)^(1/2))/(a^4+b^4)/d*2^(1/2)-1/2*b*(a^2-b^2)*arctan(1+2^(1/2)* sinh(d*x+c)^(1/2))/(a^4+b^4)/d*2^(1/2)-1/4*b*(a^2+b^2)*ln(1+sinh(d*x+c)-2^ (1/2)*sinh(d*x+c)^(1/2))/(a^4+b^4)/d*2^(1/2)+1/4*b*(a^2+b^2)*ln(1+sinh(d*x +c)+2^(1/2)*sinh(d*x+c)^(1/2))/(a^4+b^4)/d*2^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.19 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.80 \[ \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx=\frac {3 \left (-2 \sqrt {2} b^3 \arctan \left (1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )+2 \sqrt {2} b^3 \arctan \left (1+\sqrt {2} \sqrt {\sinh (c+d x)}\right )+4 a^3 \arctan (\sinh (c+d x))+4 a b^2 \log (\cosh (c+d x))-8 a b^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )-\sqrt {2} b^3 \log \left (1-\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )+\sqrt {2} b^3 \log \left (1+\sqrt {2} \sqrt {\sinh (c+d x)}+\sinh (c+d x)\right )\right )-8 a^2 b \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\sinh ^2(c+d x)\right ) \sinh ^{\frac {3}{2}}(c+d x)}{12 \left (a^4+b^4\right ) d} \]
(3*(-2*Sqrt[2]*b^3*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]] + 2*Sqrt[2]*b^3 *ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]] + 4*a^3*ArcTan[Sinh[c + d*x]] + 4 *a*b^2*Log[Cosh[c + d*x]] - 8*a*b^2*Log[a + b*Sqrt[Sinh[c + d*x]]] - Sqrt[ 2]*b^3*Log[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]] + Sqrt[2]*b^3* Log[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]]) - 8*a^2*b*Hypergeome tric2F1[3/4, 1, 7/4, -Sinh[c + d*x]^2]*Sinh[c + d*x]^(3/2))/(12*(a^4 + b^4 )*d)
Time = 0.76 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3702, 7267, 7276, 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.
| \(\displaystyle \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i c+i d x) \left (a+b \sqrt {-i \sin (i c+i d x)}\right )}dx\) |
| \(\Big \downarrow \) 3702 |
| \(\displaystyle \frac {\int \frac {1}{\left (a+b \sqrt {\sinh (c+d x)}\right ) \left (\sinh ^2(c+d x)+1\right )}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {\sinh (c+d x)}}{\left (a+b \sqrt {\sinh (c+d x)}\right ) \left (\sinh ^2(c+d x)+1\right )}d\sqrt {\sinh (c+d x)}}{d}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {2 \int \left (\frac {\sqrt {\sinh (c+d x)} a^3-b \sinh (c+d x) a^2+b^2 \sinh ^{\frac {3}{2}}(c+d x) a+b^3}{\left (a^4+b^4\right ) \left (\sinh ^2(c+d x)+1\right )}-\frac {a b^3}{\left (a^4+b^4\right ) \left (a+b \sqrt {\sinh (c+d x)}\right )}\right )d\sqrt {\sinh (c+d x)}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {a b^2 \log \left (\sinh ^2(c+d x)+1\right )}{4 \left (a^4+b^4\right )}-\frac {a b^2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{a^4+b^4}+\frac {a^3 \arctan (\sinh (c+d x))}{2 \left (a^4+b^4\right )}+\frac {b \left (a^2-b^2\right ) \arctan \left (1-\sqrt {2} \sqrt {\sinh (c+d x)}\right )}{2 \sqrt {2} \left (a^4+b^4\right )}-\frac {b \left (a^2-b^2\right ) \arctan \left (\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{2 \sqrt {2} \left (a^4+b^4\right )}-\frac {b \left (a^2+b^2\right ) \log \left (\sinh (c+d x)-\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{4 \sqrt {2} \left (a^4+b^4\right )}+\frac {b \left (a^2+b^2\right ) \log \left (\sinh (c+d x)+\sqrt {2} \sqrt {\sinh (c+d x)}+1\right )}{4 \sqrt {2} \left (a^4+b^4\right )}\right )}{d}\) |
(2*((b*(a^2 - b^2)*ArcTan[1 - Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(2*Sqrt[2]*(a^ 4 + b^4)) - (b*(a^2 - b^2)*ArcTan[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]]])/(2*Sqr t[2]*(a^4 + b^4)) + (a^3*ArcTan[Sinh[c + d*x]])/(2*(a^4 + b^4)) - (a*b^2*L og[a + b*Sqrt[Sinh[c + d*x]]])/(a^4 + b^4) - (b*(a^2 + b^2)*Log[1 - Sqrt[2 ]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]])/(4*Sqrt[2]*(a^4 + b^4)) + (b*(a^2 + b^2)*Log[1 + Sqrt[2]*Sqrt[Sinh[c + d*x]] + Sinh[c + d*x]])/(4*Sqrt[2]*(a ^4 + b^4)) + (a*b^2*Log[1 + Sinh[c + d*x]^2])/(4*(a^4 + b^4))))/d
3.5.14.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x _)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si mp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x ] /; !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.68
| method | result | size |
| default | \(\frac {a \left (-\frac {b^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}-a^{2}\right )}{a^{4}+b^{4}}+\frac {b^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right )+2 a^{2} \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}+b^{4}}\right )}{d}+\frac {\operatorname {`\,int/indef0`\,}\left (\frac {b \sqrt {\sinh \left (d x +c \right )}\, \left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right )}{-b^{4} \cosh \left (d x +c \right )^{4}+2 a^{2} b^{2} \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )+\left (-a^{4}+b^{4}\right ) \cosh \left (d x +c \right )^{2}}, \sinh \left (d x +c \right )\right )}{d}\) | \(195\) |
a/d*(-b^2/(a^4+b^4)*ln(tanh(1/2*d*x+1/2*c)^2*a^2+2*tanh(1/2*d*x+1/2*c)*b^2 -a^2)+2/(a^4+b^4)*(1/2*b^2*ln(tanh(1/2*d*x+1/2*c)^2+1)+a^2*arctan(tanh(1/2 *d*x+1/2*c))))+`int/indef0`(b*sinh(d*x+c)^(1/2)*(-b^2*sinh(d*x+c)+a^2)/(-b ^4*cosh(d*x+c)^4+2*a^2*b^2*cosh(d*x+c)^2*sinh(d*x+c)+(-a^4+b^4)*cosh(d*x+c )^2),sinh(d*x+c))/d
Leaf count of result is larger than twice the leaf count of optimal. 5671 vs. \(2 (256) = 512\).
Time = 1.15 (sec) , antiderivative size = 5671, normalized size of antiderivative = 19.83 \[ \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx=\int \frac {\operatorname {sech}{\left (c + d x \right )}}{a + b \sqrt {\sinh {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )}{b \sqrt {\sinh \left (d x + c\right )} + a} \,d x } \]
\[ \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )}{b \sqrt {\sinh \left (d x + c\right )} + a} \,d x } \]
Timed out. \[ \int \frac {\text {sech}(c+d x)}{a+b \sqrt {\sinh (c+d x)}} \, dx=\int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\right )} \,d x \]