Integrand size = 24, antiderivative size = 84 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \] Output:
x*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(1/2)-1/2*b*(c*x-1)^(1/2)*(c*x+1)^(1 /2)*ln(-c^2*x^2+1)/c/d/(-c^2*d*x^2+d)^(1/2)
Time = 0.02 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a c x+2 b c x \text {arccosh}(c x)-b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(3/2),x]
Output:
(2*a*c*x + 2*b*c*x*ArcCosh[c*x] - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c ^2*x^2])/(2*c*d*Sqrt[d - c^2*d*x^2])
Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6314, 240}
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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6314 |
\(\displaystyle \frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(3/2),x]
Output:
(x*(a + b*ArcCosh[c*x]))/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[-1 + c*x]*Sqrt[ 1 + c*x]*Log[1 - c^2*x^2])/(2*c*d*Sqrt[d - c^2*d*x^2])
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp [b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])] Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(74)=148\).
Time = 0.34 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.14
method | result | size |
default | \(\frac {a x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}\) | \(180\) |
parts | \(\frac {a x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}\) | \(180\) |
Input:
int((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
Output:
a*x/d/(-c^2*d*x^2+d)^(1/2)-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^ (1/2)/d^2/c/(c^2*x^2-1)*arccosh(c*x)-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x) /d^2/(c^2*x^2-1)*x+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^ 2/c/(c^2*x^2-1)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*x^4 - 2*c^2*d^ 2*x^2 + d^2), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*acosh(c*x))/(-c**2*d*x**2+d)**(3/2),x)
Output:
Integral((a + b*acosh(c*x))/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, d} + \frac {b x \operatorname {arcosh}\left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a x}{\sqrt {-c^{2} d x^{2} + d} d} \] Input:
integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxima")
Output:
-1/2*b*c*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + b*x*arccosh(c*x)/(sqrt(-c^2 *d*x^2 + d)*d) + a*x/(sqrt(-c^2*d*x^2 + d)*d)
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)/(-c^2*d*x^2 + d)^(3/2), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:
int((a + b*acosh(c*x))/(d - c^2*d*x^2)^(3/2),x)
Output:
int((a + b*acosh(c*x))/(d - c^2*d*x^2)^(3/2), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-\sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b +a x}{\sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d} \] Input:
int((a+b*acosh(c*x))/(-c^2*d*x^2+d)^(3/2),x)
Output:
( - sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**2*x** 2 - sqrt( - c**2*x**2 + 1)),x)*b + a*x)/(sqrt(d)*sqrt( - c**2*x**2 + 1)*d)