Integrand size = 14, antiderivative size = 114 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a+a \text {sech}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\tanh (c+d x)}{2 d (a+a \text {sech}(c+d x))^{3/2}} \]
2*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/a^(3/2)/d-5/4*arcta nh(1/2*a^(1/2)*tanh(d*x+c)*2^(1/2)/(a+a*sech(d*x+c))^(1/2))/a^(3/2)/d*2^(1 /2)-1/2*tanh(d*x+c)/d/(a+a*sech(d*x+c))^(3/2)
Time = 4.94 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\frac {\cosh ^2\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) \left (4 \left (1+e^{c+d x}\right ) \text {arcsinh}\left (e^{c+d x}\right )+5 \sqrt {2} \left (1+e^{c+d x}\right ) \text {arctanh}\left (\frac {1-e^{c+d x}}{\sqrt {2} \sqrt {1+e^{2 (c+d x)}}}\right )-4 \left (1+e^{c+d x}\right ) \text {arctanh}\left (\sqrt {1+e^{2 (c+d x)}}\right )-2 \sqrt {1+e^{2 (c+d x)}} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d \sqrt {1+e^{2 (c+d x)}} (a (1+\text {sech}(c+d x)))^{3/2}} \]
(Cosh[(c + d*x)/2]^2*Sech[c + d*x]*(4*(1 + E^(c + d*x))*ArcSinh[E^(c + d*x )] + 5*Sqrt[2]*(1 + E^(c + d*x))*ArcTanh[(1 - E^(c + d*x))/(Sqrt[2]*Sqrt[1 + E^(2*(c + d*x))])] - 4*(1 + E^(c + d*x))*ArcTanh[Sqrt[1 + E^(2*(c + d*x ))]] - 2*Sqrt[1 + E^(2*(c + d*x))]*Tanh[(c + d*x)/2]))/(2*d*Sqrt[1 + E^(2* (c + d*x))]*(a*(1 + Sech[c + d*x]))^(3/2))
Time = 0.58 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4264, 27, 3042, 4408, 3042, 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.
| \(\displaystyle \int \frac {1}{(a \text {sech}(c+d x)+a)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+a \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4264 |
| \(\displaystyle -\frac {\int -\frac {4 a-a \text {sech}(c+d x)}{2 \sqrt {\text {sech}(c+d x) a+a}}dx}{2 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 a-a \text {sech}(c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}dx}{4 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\int \frac {4 a-a \csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\) |
| \(\Big \downarrow \) 4408 |
| \(\displaystyle \frac {4 \int \sqrt {\text {sech}(c+d x) a+a}dx-5 a \int \frac {\text {sech}(c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}dx}{4 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {4 \int \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx-5 a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\frac {8 i a \int \frac {1}{a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}-5 a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}-5 a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a^2}\) |
| \(\Big \downarrow \) 4282 |
| \(\displaystyle -\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}+\frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}-\frac {10 i a \int \frac {1}{2 a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}}{4 a^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {8 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}-\frac {5 \sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {2} \sqrt {a \text {sech}(c+d x)+a}}\right )}{d}}{4 a^2}-\frac {\tanh (c+d x)}{2 d (a \text {sech}(c+d x)+a)^{3/2}}\) |
((8*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d - (5*Sqrt[2]*Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/(Sqrt[2]*Sqrt[a + a*S ech[c + d*x]])])/d)/(4*a^2) - Tanh[c + d*x]/(2*d*(a + a*Sech[c + d*x])^(3/ 2))
3.1.82.3.1 Defintions of rubi rules used
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]
\[\int \frac {1}{\left (a +\operatorname {sech}\left (d x +c \right ) a \right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (93) = 186\).
Time = 0.30 (sec) , antiderivative size = 1190, normalized size of antiderivative = 10.44 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\text {Too large to display} \]
1/8*(5*sqrt(2)*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x + c) + si nh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sqrt(a)*log(-(3*a*cosh(d*x + c)^2 + 3 *a*sinh(d*x + c)^2 - 2*sqrt(2)*(cosh(d*x + c)^3 + (3*cosh(d*x + c) - 1)*si nh(d*x + c)^2 + sinh(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 - 2 *cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) - 1)*sqrt(a)*sqrt(a/(cos h(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) - 2*a *cosh(d*x + c) + 2*(3*a*cosh(d*x + c) - a)*sinh(d*x + c) + 3*a)/(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d* x + c) + 1)) + 4*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sqrt(a)*log(-(a*cosh(d*x + c)^4 + a *sinh(d*x + c)^4 - 3*a*cosh(d*x + c)^3 + (4*a*cosh(d*x + c) - 3*a)*sinh(d* x + c)^3 + 5*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 - 9*a*cosh(d*x + c) + 5*a)*sinh(d*x + c)^2 + (cosh(d*x + c)^5 + (5*cosh(d*x + c) - 3)*sinh(d*x + c)^4 + sinh(d*x + c)^5 - 3*cosh(d*x + c)^4 + (10*cosh(d*x + c)^2 - 12*c osh(d*x + c) + 5)*sinh(d*x + c)^3 + 5*cosh(d*x + c)^3 + (10*cosh(d*x + c)^ 3 - 18*cosh(d*x + c)^2 + 15*cosh(d*x + c) - 7)*sinh(d*x + c)^2 - 7*cosh(d* x + c)^2 + (5*cosh(d*x + c)^4 - 12*cosh(d*x + c)^3 + 15*cosh(d*x + c)^2 - 14*cosh(d*x + c) + 4)*sinh(d*x + c) + 4*cosh(d*x + c) - 4)*sqrt(a)*sqrt(a/ (cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) - 4*a*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 - 9*a*cosh(d*x + c)^2 + 10*a*...
\[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (93) = 186\).
Time = 0.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.07 \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=-\frac {\frac {5 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} + \sqrt {a}\right )}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {2 \, {\left (3 \, {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{3} + {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{2} \sqrt {a} - {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )} a + a^{\frac {3}{2}}\right )}}{{\left ({\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )}^{2} + 2 \, {\left (\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}\right )} \sqrt {a} - a\right )}^{2} a}}{2 \, d} \]
-1/2*(5*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a) + sqrt(a))/sqrt(-a))/(sqrt(-a)*a) + 2*(3*(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))^3 + (sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))^2*sqrt(a) - (sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))*a + a^(3/2))/(((sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))^2 + 2*(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))*sqrt(a) - a)^2*a))/ d
Timed out. \[ \int \frac {1}{(a+a \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]