Integrand size = 14, antiderivative size = 66 \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a+a \text {sech}(c+d x)}} \]
2*a^(3/2)*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/d+2*a^2*tan h(d*x+c)/d/(a+a*sech(d*x+c))^(1/2)
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14 \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\frac {a \text {sech}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\text {sech}(c+d x))} \left (\sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cosh (c+d x)}+2 \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{d} \]
(a*Sech[(c + d*x)/2]*Sqrt[a*(1 + Sech[c + d*x])]*(Sqrt[2]*ArcSinh[Sqrt[2]* Sinh[(c + d*x)/2]]*Sqrt[Cosh[c + d*x]] + 2*Sinh[(c + d*x)/2]))/d
Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4262, 27, 3042, 4261, 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.
| \(\displaystyle \int (a \text {sech}(c+d x)+a)^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+a \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4262 |
| \(\displaystyle 2 a \int \frac {1}{2} \sqrt {\text {sech}(c+d x) a+a}dx+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle a \int \sqrt {\text {sech}(c+d x) a+a}dx+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}+a \int \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\) |
| \(\Big \downarrow \) 4261 |
| \(\displaystyle \frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}+\frac {2 i a^2 \int \frac {1}{a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {2 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}\) |
(2*a^(3/2)*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d + (2*a^2*Tanh[c + d*x])/(d*Sqrt[a + a*Sech[c + d*x]])
3.1.79.3.1 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[(-b^2)*C ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[a/(n - 1) Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && Inte gerQ[2*n]
\[\int \left (a +\operatorname {sech}\left (d x +c \right ) a \right )^{\frac {3}{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (58) = 116\).
Time = 0.27 (sec) , antiderivative size = 697, normalized size of antiderivative = 10.56 \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\frac {a^{\frac {3}{2}} \log \left (-\frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} - 3 \, a \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right )^{3} + 5 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 5 \, a\right )} \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{5} + {\left (5 \, \cosh \left (d x + c\right ) - 3\right )} \sinh \left (d x + c\right )^{4} + \sinh \left (d x + c\right )^{5} - 3 \, \cosh \left (d x + c\right )^{4} + {\left (10 \, \cosh \left (d x + c\right )^{2} - 12 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{3} + 5 \, \cosh \left (d x + c\right )^{3} + {\left (10 \, \cosh \left (d x + c\right )^{3} - 18 \, \cosh \left (d x + c\right )^{2} + 15 \, \cosh \left (d x + c\right ) - 7\right )} \sinh \left (d x + c\right )^{2} - 7 \, \cosh \left (d x + c\right )^{2} + {\left (5 \, \cosh \left (d x + c\right )^{4} - 12 \, \cosh \left (d x + c\right )^{3} + 15 \, \cosh \left (d x + c\right )^{2} - 14 \, \cosh \left (d x + c\right ) + 4\right )} \sinh \left (d x + c\right ) + 4 \, \cosh \left (d x + c\right ) - 4\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} - 4 \, a \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} - 9 \, a \cosh \left (d x + c\right )^{2} + 10 \, a \cosh \left (d x + c\right ) - 4 \, a\right )} \sinh \left (d x + c\right ) + 4 \, a}{\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}}\right ) + a^{\frac {3}{2}} \log \left (\frac {a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + \cosh \left (d x + c\right )^{2} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right ) + 1\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} + a \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}\right ) + 4 \, {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - a\right )} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}}{2 \, d} \]
1/2*(a^(3/2)*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*cosh(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*cosh(d*x + c) - 4*a)*sinh(d*x + c) + 4*a)/(cosh(d*x + c)^3 + 3*cosh(d*x + c)^2*sinh(d*x + c) + 3*cosh(d*x + c)* sinh(d*x + c)^2 + sinh(d*x + c)^3)) + a^(3/2)*log((a*cosh(d*x + c)^2 + a*s inh(d*x + c)^2 + (cosh(d*x + c)^3 + (3*cosh(d*x + c) + 1)*sinh(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/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) + a*cosh(d*x + c) + (2*a*cosh(d*x + c) + a)*sinh(d*x + c) + a)/(cosh(d*x + c) + sinh(d*x + c) )) + 4*(a*cosh(d*x + c) + a*sinh(d*x + c) - a)*sqrt(a/(cosh(d*x + c)^2 + 2 *cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)))/d
\[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\int \left (a \operatorname {sech}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
\[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\int { {\left (a \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (58) = 116\).
Time = 0.38 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.79 \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\frac {\frac {2 \, a^{2} \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - a^{\frac {3}{2}} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) + \frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} - a^{2}\right )}}{\sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}}{d} \]
(2*a^2*arctan(-(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))/sqrt(-a ))/sqrt(-a) - a^(3/2)*log(abs(-sqrt(a)*e^(d*x + c) + sqrt(a*e^(2*d*x + 2*c ) + a))) + 2*(a^2*e^(d*x + c) - a^2)/sqrt(a*e^(2*d*x + 2*c) + a))/d
Timed out. \[ \int (a+a \text {sech}(c+d x))^{3/2} \, dx=\int {\left (a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2} \,d x \]