Integrand size = 15, antiderivative size = 79 \[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}} \] Output:
-1/4*arctan(1/2*a^(1/2)*sinh(d*x+c)*2^(1/2)/(a-a*cosh(d*x+c))^(1/2))*2^(1/ 2)/a^(3/2)/d-1/2*sinh(d*x+c)/d/(a-a*cosh(d*x+c))^(3/2)
Time = 0.26 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=\frac {\left (\text {csch}^2\left (\frac {1}{4} (c+d x)\right )-4 \log \left (\cosh \left (\frac {1}{4} (c+d x)\right )\right )+4 \log \left (\sinh \left (\frac {1}{4} (c+d x)\right )\right )+\text {sech}^2\left (\frac {1}{4} (c+d x)\right )\right ) \sinh ^3\left (\frac {1}{2} (c+d x)\right )}{4 a d (-1+\cosh (c+d x)) \sqrt {a-a \cosh (c+d x)}} \] Input:
Integrate[(a - a*Cosh[c + d*x])^(-3/2),x]
Output:
((Csch[(c + d*x)/4]^2 - 4*Log[Cosh[(c + d*x)/4]] + 4*Log[Sinh[(c + d*x)/4] ] + Sech[(c + d*x)/4]^2)*Sinh[(c + d*x)/2]^3)/(4*a*d*(-1 + Cosh[c + d*x])* Sqrt[a - a*Cosh[c + d*x]])
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3129, 3042, 3128, 219}
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.
\(\displaystyle \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a-a \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a-a \cosh (c+d x)}}dx}{4 a}-\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a-a \sin \left (i c+i d x+\frac {\pi }{2}\right )}}dx}{4 a}\) |
\(\Big \downarrow \) 3128 |
\(\displaystyle -\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}+\frac {i \int \frac {1}{\frac {a^2 \sinh ^2(c+d x)}{a-a \cosh (c+d x)}+2 a}d\frac {i a \sinh (c+d x)}{\sqrt {a-a \cosh (c+d x)}}}{2 a d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {2} \sqrt {a-a \cosh (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\sinh (c+d x)}{2 d (a-a \cosh (c+d x))^{3/2}}\) |
Input:
Int[(a - a*Cosh[c + d*x])^(-3/2),x]
Output:
-1/2*ArcTan[(Sqrt[a]*Sinh[c + d*x])/(Sqrt[2]*Sqrt[a - a*Cosh[c + d*x]])]/( Sqrt[2]*a^(3/2)*d) - Sinh[c + d*x]/(2*d*(a - a*Cosh[c + d*x])^(3/2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d*(2*n + 1))), x] + Simp[(n + 1)/(a*(2*n + 1)) Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Time = 0.44 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {-2 \cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\ln \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\ln \left (\cosh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )\right ) \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}\, d}\) | \(87\) |
Input:
int(1/(a-a*cosh(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4/a*(-2*cosh(1/2*d*x+1/2*c)+(ln(cosh(1/2*d*x+1/2*c)+1)-ln(cosh(1/2*d*x+ 1/2*c)-1))*sinh(1/2*d*x+1/2*c)^2)/sinh(1/2*d*x+1/2*c)/(-2*sinh(1/2*d*x+1/2 *c)^2*a)^(1/2)/d
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (64) = 128\).
Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 3.47 \[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2} {\left (\cosh \left (d x + c\right )^{2} + 2 \, {\left (\cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-a} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt {\frac {1}{2}} {\left (\cosh \left (d x + c\right )^{2} + {\left (2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + \cosh \left (d x + c\right )\right )} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{4 \, {\left (a^{2} d \cosh \left (d x + c\right )^{2} + a^{2} d \sinh \left (d x + c\right )^{2} - 2 \, a^{2} d \cosh \left (d x + c\right ) + a^{2} d + 2 \, {\left (a^{2} d \cosh \left (d x + c\right ) - a^{2} d\right )} \sinh \left (d x + c\right )\right )}} \] Input:
integrate(1/(a-a*cosh(d*x+c))^(3/2),x, algorithm="fricas")
Output:
-1/4*(sqrt(2)*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) - 1)*sinh(d*x + c) + sin h(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(1/2)*sqr t(-a)*sqrt(-a/(cosh(d*x + c) + sinh(d*x + c)))*(cosh(d*x + c) + sinh(d*x + c)) + a*cosh(d*x + c) + a*sinh(d*x + c) + a)/(cosh(d*x + c) + sinh(d*x + c) - 1)) + 4*sqrt(1/2)*(cosh(d*x + c)^2 + (2*cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + cosh(d*x + c))*sqrt(-a/(cosh(d*x + c) + sinh(d*x + c))))/(a^2*d*cosh(d*x + c)^2 + a^2*d*sinh(d*x + c)^2 - 2*a^2*d*cosh(d*x + c) + a^2*d + 2*(a^2*d*cosh(d*x + c) - a^2*d)*sinh(d*x + c))
\[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=\int \frac {1}{\left (- a \cosh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a-a*cosh(d*x+c))**(3/2),x)
Output:
Integral((-a*cosh(c + d*x) + a)**(-3/2), x)
\[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (-a \cosh \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a-a*cosh(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate((-a*cosh(d*x + c) + a)^(-3/2), x)
Time = 0.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.46 \[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {\sqrt {-a e^{\left (d x + c\right )}}}{\sqrt {a}}\right )}{2 \, a^{\frac {3}{2}} d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} + \frac {\sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a e^{\left (d x + c\right )} + \sqrt {2} \sqrt {-a e^{\left (d x + c\right )}} a}{2 \, {\left (a e^{\left (d x + c\right )} - a\right )}^{2} a d \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )} \] Input:
integrate(1/(a-a*cosh(d*x+c))^(3/2),x, algorithm="giac")
Output:
-1/2*sqrt(2)*arctan(sqrt(-a*e^(d*x + c))/sqrt(a))/(a^(3/2)*d*sgn(-e^(d*x + c) + 1)) + 1/2*(sqrt(2)*sqrt(-a*e^(d*x + c))*a*e^(d*x + c) + sqrt(2)*sqrt (-a*e^(d*x + c))*a)/((a*e^(d*x + c) - a)^2*a*d*sgn(-e^(d*x + c) + 1))
Timed out. \[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a-a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int(1/(a - a*cosh(c + d*x))^(3/2),x)
Output:
int(1/(a - a*cosh(c + d*x))^(3/2), x)
\[ \int \frac {1}{(a-a \cosh (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {-\cosh \left (d x +c \right )+1}}{\cosh \left (d x +c \right )^{2}-2 \cosh \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:
int(1/(a-a*cosh(d*x+c))^(3/2),x)
Output:
(sqrt(a)*int(sqrt( - cosh(c + d*x) + 1)/(cosh(c + d*x)**2 - 2*cosh(c + d*x ) + 1),x))/a**2