Integrand size = 14, antiderivative size = 61 \[ \int \sqrt {a+b \cosh (c+d x)} \, dx=-\frac {2 i \sqrt {a+b \cosh (c+d x)} E\left (\frac {1}{2} i (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cosh (c+d x)}{a+b}}} \] Output:
-2*I*(a+b*cosh(d*x+c))^(1/2)*EllipticE(I*sinh(1/2*d*x+1/2*c),2^(1/2)*(b/(a +b))^(1/2))/d/((a+b*cosh(d*x+c))/(a+b))^(1/2)
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \cosh (c+d x)} \, dx=-\frac {2 i \sqrt {a+b \cosh (c+d x)} E\left (\frac {1}{2} i (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cosh (c+d x)}{a+b}}} \] Input:
Integrate[Sqrt[a + b*Cosh[c + d*x]],x]
Output:
((-2*I)*Sqrt[a + b*Cosh[c + d*x]]*EllipticE[(I/2)*(c + d*x), (2*b)/(a + b) ])/(d*Sqrt[(a + b*Cosh[c + d*x])/(a + b)])
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3134, 3042, 3132}
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.
| \(\displaystyle \int \sqrt {a+b \cosh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a+b \sin \left (i c+i d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\sqrt {a+b \cosh (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cosh (c+d x)}{a+b}}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \cosh (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i c+i d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cosh (c+d x)}{a+b}}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {2 i \sqrt {a+b \cosh (c+d x)} E\left (\frac {1}{2} i (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cosh (c+d x)}{a+b}}}\) |
Input:
Int[Sqrt[a + b*Cosh[c + d*x]],x]
Output:
((-2*I)*Sqrt[a + b*Cosh[c + d*x]]*EllipticE[(I/2)*(c + d*x), (2*b)/(a + b) ])/(d*Sqrt[(a + b*Cosh[c + d*x])/(a + b)])
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(275\) vs. \(2(62)=124\).
Time = 4.83 (sec) , antiderivative size = 276, normalized size of antiderivative = 4.52
| method | result | size |
| default | \(\frac {2 \left (a \operatorname {EllipticF}\left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )+b \operatorname {EllipticF}\left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-2 b \operatorname {EllipticE}\left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )\right ) \sqrt {-\sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {2 b \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \sqrt {\left (2 b \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b \right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{\sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 b \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a +b}\, d}\) | \(276\) |
| risch | \(\frac {\sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right ) {\mathrm e}^{-d x -c}}}{d}+\frac {\left (\frac {2 a \left (a +\sqrt {a^{2}-b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}}\, \sqrt {-\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}-b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )}{b \sqrt {b \,{\mathrm e}^{3 d x +3 c}+2 \,{\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{d x +c} b}}+b \left (-\frac {2 \left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right )}{b \sqrt {\left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right ) {\mathrm e}^{d x +c}}}+\frac {2 \left (a +\sqrt {a^{2}-b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}}\, \sqrt {-\frac {b \,{\mathrm e}^{d x +c}}{a +\sqrt {a^{2}-b^{2}}}}\, \left (\left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )+\frac {\left (-a +\sqrt {a^{2}-b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )}{b}\right )}{b \sqrt {b \,{\mathrm e}^{3 d x +3 c}+2 \,{\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{d x +c} b}}\right )\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right ) {\mathrm e}^{-d x -c}}\, \sqrt {\left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right ) {\mathrm e}^{d x +c}}}{d \left ({\mathrm e}^{2 d x +2 c} b +2 \,{\mathrm e}^{d x +c} a +b \right )}\) | \(949\) |
Input:
int((a+b*cosh(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
2*(a*EllipticF(cosh(1/2*d*x+1/2*c)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/ 2))+b*EllipticF(cosh(1/2*d*x+1/2*c)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b))^(1 /2))-2*b*EllipticE(cosh(1/2*d*x+1/2*c)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a-b)) ^(1/2)))*(-sinh(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cosh(1/2*d*x+1/2*c)^2+a-b)/( a-b))^(1/2)*((2*b*cosh(1/2*d*x+1/2*c)^2+a-b)*sinh(1/2*d*x+1/2*c)^2)^(1/2)/ (-2*b/(a-b))^(1/2)/(2*sinh(1/2*d*x+1/2*c)^4*b+(a+b)*sinh(1/2*d*x+1/2*c)^2) ^(1/2)/sinh(1/2*d*x+1/2*c)/(2*b*sinh(1/2*d*x+1/2*c)^2+a+b)^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (60) = 120\).
Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.23 \[ \int \sqrt {a+b \cosh (c+d x)} \, dx=\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (d x + c\right ) + 3 \, b \sinh \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 6 \, \sqrt {\frac {1}{2}} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (d x + c\right ) + 3 \, b \sinh \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {b \cosh \left (d x + c\right ) + a} b\right )}}{3 \, b d} \] Input:
integrate((a+b*cosh(d*x+c))^(1/2),x, algorithm="fricas")
Output:
2/3*(2*sqrt(1/2)*a*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8 /27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(d*x + c) + 3*b*sinh(d*x + c) + 2* a)/b) - 6*sqrt(1/2)*b^(3/2)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27 *(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27 *(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(d*x + c) + 3*b*sinh(d*x + c) + 2*a)/ b)) - 3*sqrt(b*cosh(d*x + c) + a)*b)/(b*d)
\[ \int \sqrt {a+b \cosh (c+d x)} \, dx=\int \sqrt {a + b \cosh {\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*cosh(d*x+c))**(1/2),x)
Output:
Integral(sqrt(a + b*cosh(c + d*x)), x)
\[ \int \sqrt {a+b \cosh (c+d x)} \, dx=\int { \sqrt {b \cosh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*cosh(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*cosh(d*x + c) + a), x)
\[ \int \sqrt {a+b \cosh (c+d x)} \, dx=\int { \sqrt {b \cosh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*cosh(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*cosh(d*x + c) + a), x)
Timed out. \[ \int \sqrt {a+b \cosh (c+d x)} \, dx=\int \sqrt {a+b\,\mathrm {cosh}\left (c+d\,x\right )} \,d x \] Input:
int((a + b*cosh(c + d*x))^(1/2),x)
Output:
int((a + b*cosh(c + d*x))^(1/2), x)
\[ \int \sqrt {a+b \cosh (c+d x)} \, dx=\int \sqrt {\cosh \left (d x +c \right ) b +a}d x \] Input:
int((a+b*cosh(d*x+c))^(1/2),x)
Output:
int(sqrt(cosh(c + d*x)*b + a),x)