Integrand size = 15, antiderivative size = 57 \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )-a \sqrt {a+b \text {sech}^2(x)}-\frac {1}{3} \left (a+b \text {sech}^2(x)\right )^{3/2} \] Output:
a^(3/2)*arctanh((a+b*sech(x)^2)^(1/2)/a^(1/2))-a*(a+b*sech(x)^2)^(1/2)-1/3 *(a+b*sech(x)^2)^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14 \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=-\frac {2 b \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},-\frac {a \cosh ^2(x)}{b}\right ) \left (a+b \text {sech}^2(x)\right )^{3/2}}{3 \sqrt {1+\frac {a \cosh ^2(x)}{b}} (a+2 b+a \cosh (2 x))} \] Input:
Integrate[(a + b*Sech[x]^2)^(3/2)*Tanh[x],x]
Output:
(-2*b*Hypergeometric2F1[-3/2, -3/2, -1/2, -((a*Cosh[x]^2)/b)]*(a + b*Sech[ x]^2)^(3/2))/(3*Sqrt[1 + (a*Cosh[x]^2)/b]*(a + 2*b + a*Cosh[2*x]))
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 26, 4627, 243, 60, 60, 73, 221}
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.
\(\displaystyle \int \tanh (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan (i x) \left (a+b \sec (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \left (b \sec (i x)^2+a\right )^{3/2} \tan (i x)dx\) |
\(\Big \downarrow \) 4627 |
\(\displaystyle -\int \cosh (x) \left (b \text {sech}^2(x)+a\right )^{3/2}d\text {sech}(x)\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {1}{2} \int \cosh (x) \left (b \text {sech}^2(x)+a\right )^{3/2}d\text {sech}^2(x)\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-a \int \cosh (x) \sqrt {b \text {sech}^2(x)+a}d\text {sech}^2(x)-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-a \left (a \int \frac {\cosh (x)}{\sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)+2 \sqrt {a+b \text {sech}^2(x)}\right )-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-a \left (\frac {2 a \int \frac {1}{\frac {\text {sech}^4(x)}{b}-\frac {a}{b}}d\sqrt {b \text {sech}^2(x)+a}}{b}+2 \sqrt {a+b \text {sech}^2(x)}\right )-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-a \left (2 \sqrt {a+b \text {sech}^2(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )\right )-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
Input:
Int[(a + b*Sech[x]^2)^(3/2)*Tanh[x],x]
Output:
((-2*(a + b*Sech[x]^2)^(3/2))/3 - a*(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x] ^2]/Sqrt[a]] + 2*Sqrt[a + b*Sech[x]^2]))/2
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Si mp[1/f Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x), x] , x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[( m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || Integers Q[2*n, p])
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(-\frac {\left (a +\operatorname {sech}\left (x \right )^{2} b \right )^{\frac {3}{2}}}{3}+a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +\operatorname {sech}\left (x \right )^{2} b}}{\operatorname {sech}\left (x \right )}\right )-a \sqrt {a +\operatorname {sech}\left (x \right )^{2} b}\) | \(56\) |
default | \(-\frac {\left (a +\operatorname {sech}\left (x \right )^{2} b \right )^{\frac {3}{2}}}{3}+a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +\operatorname {sech}\left (x \right )^{2} b}}{\operatorname {sech}\left (x \right )}\right )-a \sqrt {a +\operatorname {sech}\left (x \right )^{2} b}\) | \(56\) |
Input:
int((a+sech(x)^2*b)^(3/2)*tanh(x),x,method=_RETURNVERBOSE)
Output:
-1/3*(a+sech(x)^2*b)^(3/2)+a^(3/2)*ln((2*a+2*a^(1/2)*(a+sech(x)^2*b)^(1/2) )/sech(x))-a*(a+sech(x)^2*b)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 788 vs. \(2 (45) = 90\).
Time = 0.19 (sec) , antiderivative size = 2312, normalized size of antiderivative = 40.56 \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=\text {Too large to display} \] Input:
integrate((a+b*sech(x)^2)^(3/2)*tanh(x),x, algorithm="fricas")
Output:
[1/12*(3*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*cosh(x)^ 4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*sinh (x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh(x)^2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(a)*log(((a^3 + 2*a^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^ 7 + (a^3 + 2*a^2*b + a*b^2)*sinh(x)^8 + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3 )*cosh(x)^6 + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3 + 14*(a^3 + 2*a^2*b + a*b ^2)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + 3*(2* a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x))*sinh(x)^5 + (6*a^3 + 14*a^2*b + 9* a*b^2)*cosh(x)^4 + (70*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 6*a^3 + 14*a^2* b + 9*a*b^2 + 30*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 10*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^3 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3 + 2 *(2*a^3 + 3*a^2*b)*cosh(x)^2 + 2*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^6 + 1 5*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^4 + 2*a^3 + 3*a^2*b + 3*(6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*((a^2 + 2*a*b + b^2) *cosh(x)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)^5 + (a^2 + 2*a*b + b^2) *sinh(x)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 3*(5*(a^2 + 2*a*b + b^2)*co sh(x)^2 + a^2 + 2*a*b + b^2)*sinh(x)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh(x)^ 3 + 3*(a^2 + 2*a*b + b^2)*cosh(x))*sinh(x)^3 + (3*a^2 + 4*a*b)*cosh(x)^...
\[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}} \tanh {\left (x \right )}\, dx \] Input:
integrate((a+b*sech(x)**2)**(3/2)*tanh(x),x)
Output:
Integral((a + b*sech(x)**2)**(3/2)*tanh(x), x)
\[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=\int { {\left (b \operatorname {sech}\left (x\right )^{2} + a\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \] Input:
integrate((a+b*sech(x)^2)^(3/2)*tanh(x),x, algorithm="maxima")
Output:
integrate((b*sech(x)^2 + a)^(3/2)*tanh(x), x)
Exception generated. \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*sech(x)^2)^(3/2)*tanh(x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Time = 4.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79 \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}}}{\sqrt {a}}\right )-\frac {{\left (a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2}}{3}-a\,\sqrt {a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}} \] Input:
int(tanh(x)*(a + b/cosh(x)^2)^(3/2),x)
Output:
a^(3/2)*atanh((a + b/cosh(x)^2)^(1/2)/a^(1/2)) - (a + b/cosh(x)^2)^(3/2)/3 - a*(a + b/cosh(x)^2)^(1/2)
\[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh (x) \, dx=-\frac {\sqrt {\mathrm {sech}\left (x \right )^{2} b +a}\, \mathrm {sech}\left (x \right )^{2} b}{3}-\frac {4 \sqrt {\mathrm {sech}\left (x \right )^{2} b +a}\, a}{3}+\left (\int \frac {\sqrt {\mathrm {sech}\left (x \right )^{2} b +a}\, \tanh \left (x \right )}{\mathrm {sech}\left (x \right )^{2} b +a}d x \right ) a^{2} \] Input:
int((a+b*sech(x)^2)^(3/2)*tanh(x),x)
Output:
( - sqrt(sech(x)**2*b + a)*sech(x)**2*b - 4*sqrt(sech(x)**2*b + a)*a + 3*i nt((sqrt(sech(x)**2*b + a)*tanh(x))/(sech(x)**2*b + a),x)*a**2)/3