Integrand size = 14, antiderivative size = 98 \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\frac {2 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a+a \text {sech}(c+d x)}}\right )}{d}+\frac {14 a^3 \tanh (c+d x)}{3 d \sqrt {a+a \text {sech}(c+d x)}}+\frac {2 a^2 \sqrt {a+a \text {sech}(c+d x)} \tanh (c+d x)}{3 d} \]
2*a^(5/2)*arctanh(a^(1/2)*tanh(d*x+c)/(a+a*sech(d*x+c))^(1/2))/d+14/3*a^3* tanh(d*x+c)/d/(a+a*sech(d*x+c))^(1/2)+2/3*a^2*(a+a*sech(d*x+c))^(1/2)*tanh (d*x+c)/d
Time = 0.46 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\frac {a^2 \text {sech}\left (\frac {1}{2} (c+d x)\right ) \text {sech}(c+d x) \sqrt {a (1+\text {sech}(c+d x))} \left (3 \sqrt {2} \text {arcsinh}\left (\sqrt {2} \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \cosh ^{\frac {3}{2}}(c+d x)-6 \sinh \left (\frac {1}{2} (c+d x)\right )+8 \sinh \left (\frac {3}{2} (c+d x)\right )\right )}{3 d} \]
(a^2*Sech[(c + d*x)/2]*Sech[c + d*x]*Sqrt[a*(1 + Sech[c + d*x])]*(3*Sqrt[2 ]*ArcSinh[Sqrt[2]*Sinh[(c + d*x)/2]]*Cosh[c + d*x]^(3/2) - 6*Sinh[(c + d*x )/2] + 8*Sinh[(3*(c + d*x))/2]))/(3*d)
Time = 0.56 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3042, 4262, 27, 3042, 4403, 3042, 4261, 216, 4279}
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.
\(\displaystyle \int (a \text {sech}(c+d x)+a)^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+a \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
\(\Big \downarrow \) 4262 |
\(\displaystyle \frac {2}{3} a \int \frac {1}{2} \sqrt {\text {sech}(c+d x) a+a} (7 \text {sech}(c+d x) a+3 a)dx+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} a \int \sqrt {\text {sech}(c+d x) a+a} (7 \text {sech}(c+d x) a+3 a)dx+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \int \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a} \left (7 \csc \left (i c+i d x+\frac {\pi }{2}\right ) a+3 a\right )dx\) |
\(\Big \downarrow \) 4403 |
\(\displaystyle \frac {1}{3} a \left (3 a \int \sqrt {\text {sech}(c+d x) a+a}dx+7 a \int \text {sech}(c+d x) \sqrt {\text {sech}(c+d x) a+a}dx\right )+\frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (3 a \int \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx+7 a \int \csc \left (i c+i d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\right )\) |
\(\Big \downarrow \) 4261 |
\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (\frac {6 i a^2 \int \frac {1}{a-\frac {a^2 \tanh ^2(c+d x)}{\text {sech}(c+d x) a+a}}d\left (-\frac {i a \tanh (c+d x)}{\sqrt {\text {sech}(c+d x) a+a}}\right )}{d}+7 a \int \csc \left (i c+i d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (\frac {6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+7 a \int \csc \left (i c+i d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (i c+i d x+\frac {\pi }{2}\right ) a+a}dx\right )\) |
\(\Big \downarrow \) 4279 |
\(\displaystyle \frac {2 a^2 \tanh (c+d x) \sqrt {a \text {sech}(c+d x)+a}}{3 d}+\frac {1}{3} a \left (\frac {6 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a \text {sech}(c+d x)+a}}\right )}{d}+\frac {14 a^2 \tanh (c+d x)}{d \sqrt {a \text {sech}(c+d x)+a}}\right )\) |
(2*a^2*Sqrt[a + a*Sech[c + d*x]]*Tanh[c + d*x])/(3*d) + (a*((6*a^(3/2)*Arc Tanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/d + (14*a^2*Tanh[ c + d*x])/(d*Sqrt[a + a*Sech[c + d*x]])))/3
3.1.78.3.1 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[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*b*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]])), x] /; Free Q[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_ .) + (c_)), x_Symbol] :> Simp[c Int[Sqrt[a + b*Csc[e + f*x]], x], x] + Si mp[d Int[Sqrt[a + b*Csc[e + f*x]]*Csc[e + f*x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
\[\int \left (a +\operatorname {sech}\left (d x +c \right ) a \right )^{\frac {5}{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (84) = 168\).
Time = 0.29 (sec) , antiderivative size = 924, normalized size of antiderivative = 9.43 \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\text {Too large to display} \]
1/6*(3*(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh (d*x + c)^2 + a^2)*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*cosh(d*x + c) + 5)*si nh(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)*si nh(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)) + 3*(a^2*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c)*sinh(d*x + c) + a^2*sinh(d*x + c)^2 + a^2)*sqrt(a)* log((a*cosh(d*x + c)^2 + a*sinh(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)...
\[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\int \left (a \operatorname {sech}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx \]
\[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\int { {\left (a \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Time = 0.40 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.54 \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\frac {\frac {6 \, a^{3} \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - 3 \, a^{\frac {5}{2}} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) - \frac {4 \, {\left (4 \, a^{4} - {\left (3 \, a^{4} e^{c} + {\left (4 \, a^{4} e^{\left (d x + 3 \, c\right )} - 3 \, a^{4} e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )}\right )} e^{\left (d x\right )}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{\frac {3}{2}}}}{3 \, d} \]
1/3*(6*a^3*arctan(-(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))/sqr t(-a))/sqrt(-a) - 3*a^(5/2)*log(abs(-sqrt(a)*e^(d*x + c) + sqrt(a*e^(2*d*x + 2*c) + a))) - 4*(4*a^4 - (3*a^4*e^c + (4*a^4*e^(d*x + 3*c) - 3*a^4*e^(2 *c))*e^(d*x))*e^(d*x))/(a*e^(2*d*x + 2*c) + a)^(3/2))/d
Timed out. \[ \int (a+a \text {sech}(c+d x))^{5/2} \, dx=\int {\left (a+\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{5/2} \,d x \]