Integrand size = 15, antiderivative size = 39 \[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )+\sqrt {a+b \cosh ^2(x)} \] Output:
-a^(1/2)*arctanh((a+b*cosh(x)^2)^(1/2)/a^(1/2))+(a+b*cosh(x)^2)^(1/2)
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )+\sqrt {a+b \cosh ^2(x)} \] Input:
Integrate[Sqrt[a + b*Cosh[x]^2]*Tanh[x],x]
Output:
-(Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^2]/Sqrt[a]]) + Sqrt[a + b*Cosh[x]^2]
Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 26, 3673, 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.
\(\displaystyle \int \tanh (x) \sqrt {a+b \cosh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sqrt {a+b \sin \left (\frac {\pi }{2}+i x\right )^2}}{\tan \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sqrt {b \sin \left (i x+\frac {\pi }{2}\right )^2+a}}{\tan \left (i x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3673 |
\(\displaystyle \frac {1}{2} \int \sqrt {b \cosh ^2(x)+a} \text {sech}^2(x)d\cosh ^2(x)\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (a \int \frac {\text {sech}^2(x)}{\sqrt {b \cosh ^2(x)+a}}d\cosh ^2(x)+2 \sqrt {a+b \cosh ^2(x)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {2 a \int \frac {1}{\frac {\cosh ^4(x)}{b}-\frac {a}{b}}d\sqrt {b \cosh ^2(x)+a}}{b}+2 \sqrt {a+b \cosh ^2(x)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (2 \sqrt {a+b \cosh ^2(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^2(x)}}{\sqrt {a}}\right )\right )\) |
Input:
Int[Sqrt[a + b*Cosh[x]^2]*Tanh[x],x]
Output:
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^2]/Sqrt[a]] + 2*Sqrt[a + b*Cosh[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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[ff^((m + 1)/2)/(2*f) Subst[Int[x^((m - 1)/2)*((a + b*ff*x)^p/(1 - ff*x)^((m + 1 )/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && Integ erQ[(m - 1)/2]
Time = 0.41 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.08
method | result | size |
default | \(\sqrt {a +b \cosh \left (x \right )^{2}}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \cosh \left (x \right )^{2}}}{\cosh \left (x \right )}\right )\) | \(42\) |
Input:
int((a+b*cosh(x)^2)^(1/2)*tanh(x),x,method=_RETURNVERBOSE)
Output:
(a+b*cosh(x)^2)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(a+b*cosh(x)^2)^(1/2))/cos h(x))
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (31) = 62\).
Time = 0.26 (sec) , antiderivative size = 424, normalized size of antiderivative = 10.87 \[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx =\text {Too large to display} \] Input:
integrate((a+b*cosh(x)^2)^(1/2)*tanh(x),x, algorithm="fricas")
Output:
[1/2*(sqrt(a)*(cosh(x) + sinh(x))*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(4*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 4*a + b)*sinh( x)^2 - 4*sqrt(2)*sqrt(a)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(cosh( x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))*(cosh(x) + sinh(x)) + 4*(b*cosh(x)^ 3 + (4*a + b)*cosh(x))*sinh(x) + b)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sin h(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh (x))*sinh(x) + 1)) + sqrt(2)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(c osh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x) + sinh(x)), 1/2*(2*sq rt(-a)*(cosh(x) + sinh(x))*arctan(2*sqrt(2)*sqrt(-a)*sqrt((b*cosh(x)^2 + b *sinh(x)^2 + 2*a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))*(cosh(x ) + sinh(x))/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + ( 2*a + b)*cosh(x))*sinh(x) + b)) + sqrt(2)*sqrt((b*cosh(x)^2 + b*sinh(x)^2 + 2*a + b)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)))/(cosh(x) + sinh(x ))]
\[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx=\int \sqrt {a + b \cosh ^{2}{\left (x \right )}} \tanh {\left (x \right )}\, dx \] Input:
integrate((a+b*cosh(x)**2)**(1/2)*tanh(x),x)
Output:
Integral(sqrt(a + b*cosh(x)**2)*tanh(x), x)
\[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx=\int { \sqrt {b \cosh \left (x\right )^{2} + a} \tanh \left (x\right ) \,d x } \] Input:
integrate((a+b*cosh(x)^2)^(1/2)*tanh(x),x, algorithm="maxima")
Output:
integrate(sqrt(b*cosh(x)^2 + a)*tanh(x), x)
\[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx=\int { \sqrt {b \cosh \left (x\right )^{2} + a} \tanh \left (x\right ) \,d x } \] Input:
integrate((a+b*cosh(x)^2)^(1/2)*tanh(x),x, algorithm="giac")
Output:
integrate(sqrt(b*cosh(x)^2 + a)*tanh(x), x)
Timed out. \[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx=\int \mathrm {tanh}\left (x\right )\,\sqrt {b\,{\mathrm {cosh}\left (x\right )}^2+a} \,d x \] Input:
int(tanh(x)*(a + b*cosh(x)^2)^(1/2),x)
Output:
int(tanh(x)*(a + b*cosh(x)^2)^(1/2), x)
\[ \int \sqrt {a+b \cosh ^2(x)} \tanh (x) \, dx=\int \sqrt {\cosh \left (x \right )^{2} b +a}\, \tanh \left (x \right )d x \] Input:
int((a+b*cosh(x)^2)^(1/2)*tanh(x),x)
Output:
int(sqrt(cosh(x)**2*b + a)*tanh(x),x)