Integrand size = 15, antiderivative size = 45 \[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=-\frac {2}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^3(x)}}{\sqrt {a}}\right )+\frac {2}{3} \sqrt {a+b \cosh ^3(x)} \] Output:
-2/3*a^(1/2)*arctanh((a+b*cosh(x)^3)^(1/2)/a^(1/2))+2/3*(a+b*cosh(x)^3)^(1 /2)
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=-\frac {2}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^3(x)}}{\sqrt {a}}\right )+\frac {2}{3} \sqrt {a+b \cosh ^3(x)} \] Input:
Integrate[Sqrt[a + b*Cosh[x]^3]*Tanh[x],x]
Output:
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]])/3 + (2*Sqrt[a + b*Cosh [x]^3])/3
Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 26, 3709, 798, 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.
\(\displaystyle \int \tanh (x) \sqrt {a+b \cosh ^3(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sqrt {a+b \sin \left (\frac {\pi }{2}+i x\right )^3}}{\tan \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sqrt {b \sin \left (i x+\frac {\pi }{2}\right )^3+a}}{\tan \left (i x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3709 |
\(\displaystyle \int \text {sech}(x) \sqrt {a+b \cosh ^3(x)}d\cosh (x)\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \sqrt {b \cosh ^3(x)+a} \text {sech}(x)d\cosh ^3(x)\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{3} \left (a \int \frac {\text {sech}(x)}{\sqrt {b \cosh ^3(x)+a}}d\cosh ^3(x)+2 \sqrt {a+b \cosh ^3(x)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {2 a \int \frac {1}{\frac {\cosh ^6(x)}{b}-\frac {a}{b}}d\sqrt {b \cosh ^3(x)+a}}{b}+2 \sqrt {a+b \cosh ^3(x)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (2 \sqrt {a+b \cosh ^3(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \cosh ^3(x)}}{\sqrt {a}}\right )\right )\) |
Input:
Int[Sqrt[a + b*Cosh[x]^3]*Tanh[x],x]
Output:
(-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]] + 2*Sqrt[a + b*Cosh[x]^ 3])/3
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_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff^(m + 1)/f Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
\[\int \sqrt {a +b \cosh \left (x \right )^{3}}\, \tanh \left (x \right )d x\]
Input:
int((a+b*cosh(x)^3)^(1/2)*tanh(x),x)
Output:
int((a+b*cosh(x)^3)^(1/2)*tanh(x),x)
Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (33) = 66\).
Time = 0.66 (sec) , antiderivative size = 1842, normalized size of antiderivative = 40.93 \[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=\text {Too large to display} \] Input:
integrate((a+b*cosh(x)^3)^(1/2)*tanh(x),x, algorithm="fricas")
Output:
[1/6*(sqrt(a)*(cosh(x) + sinh(x))*log(-(b^2*cosh(x)^12 + 12*b^2*cosh(x)*si nh(x)^11 + b^2*sinh(x)^12 + 6*b^2*cosh(x)^10 + 64*a*b*cosh(x)^9 + 6*(11*b^ 2*cosh(x)^2 + b^2)*sinh(x)^10 + 15*b^2*cosh(x)^8 + 4*(55*b^2*cosh(x)^3 + 1 5*b^2*cosh(x) + 16*a*b)*sinh(x)^9 + 192*a*b*cosh(x)^7 + 3*(165*b^2*cosh(x) ^4 + 90*b^2*cosh(x)^2 + 192*a*b*cosh(x) + 5*b^2)*sinh(x)^8 + 24*(33*b^2*co sh(x)^5 + 30*b^2*cosh(x)^3 + 96*a*b*cosh(x)^2 + 5*b^2*cosh(x) + 8*a*b)*sin h(x)^7 + 192*a*b*cosh(x)^5 + 4*(128*a^2 + 5*b^2)*cosh(x)^6 + 4*(231*b^2*co sh(x)^6 + 315*b^2*cosh(x)^4 + 1344*a*b*cosh(x)^3 + 105*b^2*cosh(x)^2 + 336 *a*b*cosh(x) + 128*a^2 + 5*b^2)*sinh(x)^6 + 15*b^2*cosh(x)^4 + 24*(33*b^2* cosh(x)^7 + 63*b^2*cosh(x)^5 + 336*a*b*cosh(x)^4 + 35*b^2*cosh(x)^3 + 168* a*b*cosh(x)^2 + 8*a*b + (128*a^2 + 5*b^2)*cosh(x))*sinh(x)^5 + 64*a*b*cosh (x)^3 + 3*(165*b^2*cosh(x)^8 + 420*b^2*cosh(x)^6 + 2688*a*b*cosh(x)^5 + 35 0*b^2*cosh(x)^4 + 2240*a*b*cosh(x)^3 + 320*a*b*cosh(x) + 20*(128*a^2 + 5*b ^2)*cosh(x)^2 + 5*b^2)*sinh(x)^4 + 6*b^2*cosh(x)^2 + 4*(55*b^2*cosh(x)^9 + 180*b^2*cosh(x)^7 + 1344*a*b*cosh(x)^6 + 210*b^2*cosh(x)^5 + 1680*a*b*cos h(x)^4 + 480*a*b*cosh(x)^2 + 20*(128*a^2 + 5*b^2)*cosh(x)^3 + 15*b^2*cosh( x) + 16*a*b)*sinh(x)^3 + 6*(11*b^2*cosh(x)^10 + 45*b^2*cosh(x)^8 + 384*a*b *cosh(x)^7 + 70*b^2*cosh(x)^6 + 672*a*b*cosh(x)^5 + 320*a*b*cosh(x)^3 + 10 *(128*a^2 + 5*b^2)*cosh(x)^4 + 15*b^2*cosh(x)^2 + 32*a*b*cosh(x) + b^2)*si nh(x)^2 + b^2 - 16*(b*cosh(x)^8 + 8*b*cosh(x)*sinh(x)^7 + b*sinh(x)^8 +...
\[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=\int \sqrt {a + b \cosh ^{3}{\left (x \right )}} \tanh {\left (x \right )}\, dx \] Input:
integrate((a+b*cosh(x)**3)**(1/2)*tanh(x),x)
Output:
Integral(sqrt(a + b*cosh(x)**3)*tanh(x), x)
\[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=\int { \sqrt {b \cosh \left (x\right )^{3} + a} \tanh \left (x\right ) \,d x } \] Input:
integrate((a+b*cosh(x)^3)^(1/2)*tanh(x),x, algorithm="maxima")
Output:
integrate(sqrt(b*cosh(x)^3 + a)*tanh(x), x)
\[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=\int { \sqrt {b \cosh \left (x\right )^{3} + a} \tanh \left (x\right ) \,d x } \] Input:
integrate((a+b*cosh(x)^3)^(1/2)*tanh(x),x, algorithm="giac")
Output:
integrate(sqrt(b*cosh(x)^3 + a)*tanh(x), x)
Timed out. \[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=\int \mathrm {tanh}\left (x\right )\,\sqrt {b\,{\mathrm {cosh}\left (x\right )}^3+a} \,d x \] Input:
int(tanh(x)*(a + b*cosh(x)^3)^(1/2),x)
Output:
int(tanh(x)*(a + b*cosh(x)^3)^(1/2), x)
\[ \int \sqrt {a+b \cosh ^3(x)} \tanh (x) \, dx=\int \sqrt {\cosh \left (x \right )^{3} b +a}\, \tanh \left (x \right )d x \] Input:
int((a+b*cosh(x)^3)^(1/2)*tanh(x),x)
Output:
int(sqrt(cosh(x)**3*b + a)*tanh(x),x)