Integrand size = 23, antiderivative size = 51 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c \pi ^{3/2}} \] Output:
x*(a+b*arcsinh(c*x))/Pi/(Pi*c^2*x^2+Pi)^(1/2)-1/2*b*ln(c^2*x^2+1)/c/Pi^(3/ 2)
Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {2 a c x+2 b c x \text {arcsinh}(c x)-b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c \pi ^{3/2} \sqrt {1+c^2 x^2}} \] Input:
Integrate[(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(3/2),x]
Output:
(2*a*c*x + 2*b*c*x*ArcSinh[c*x] - b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2 *c*Pi^(3/2)*Sqrt[1 + c^2*x^2])
Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6202, 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.
\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi c^2 x^2+\pi \right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6202 |
\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b c \int \frac {x}{c^2 x^2+1}dx}{\pi ^{3/2}}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c}\) |
Input:
Int[(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(3/2),x]
Output:
(x*(a + b*ArcSinh[c*x]))/(Pi*Sqrt[Pi + c^2*Pi*x^2]) - (b*Log[1 + c^2*x^2]) /(2*c*Pi^(3/2))
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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp [b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSinh[ c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(109\) vs. \(2(45)=90\).
Time = 1.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.16
method | result | size |
default | \(\frac {a x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+b \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c}-\frac {\left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\pi ^{\frac {3}{2}} c}\right )\) | \(110\) |
parts | \(\frac {a x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+b \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c}-\frac {\left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \operatorname {arcsinh}\left (x c \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\pi ^{\frac {3}{2}} c}\right )\) | \(110\) |
Input:
int((a+b*arcsinh(x*c))/(Pi*c^2*x^2+Pi)^(3/2),x,method=_RETURNVERBOSE)
Output:
a/Pi*x/(Pi*c^2*x^2+Pi)^(1/2)+b*(2/Pi^(3/2)/c*arcsinh(x*c)-1/Pi^(3/2)*(c^2* x^2-(c^2*x^2+1)^(1/2)*x*c+1)*arcsinh(x*c)/c/(c^2*x^2+1)-1/Pi^(3/2)/c*ln(1+ (x*c+(c^2*x^2+1)^(1/2))^2))
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^2*c^4*x^4 + 2*pi^2 *c^2*x^2 + pi^2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \] Input:
integrate((a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(3/2),x)
Output:
(Integral(a/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + In tegral(b*asinh(c*x)/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x))/pi**(3/2)
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {b x \operatorname {arsinh}\left (c x\right )}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {a x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} - \frac {b \log \left (x^{2} + \frac {1}{c^{2}}\right )}{2 \, \pi ^{\frac {3}{2}} c} \] Input:
integrate((a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="maxima")
Output:
b*x*arcsinh(c*x)/(pi*sqrt(pi + pi*c^2*x^2)) + a*x/(pi*sqrt(pi + pi*c^2*x^2 )) - 1/2*b*log(x^2 + 1/c^2)/(pi^(3/2)*c)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)/(pi + pi*c^2*x^2)^(3/2), x)
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \] Input:
int((a + b*asinh(c*x))/(Pi + Pi*c^2*x^2)^(3/2),x)
Output:
int((a + b*asinh(c*x))/(Pi + Pi*c^2*x^2)^(3/2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a c x +\left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3} x^{2}+\left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b c +a \,c^{2} x^{2}+a}{\sqrt {\pi }\, c \pi \left (c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))/(Pi*c^2*x^2+Pi)^(3/2),x)
Output:
(sqrt(c**2*x**2 + 1)*a*c*x + int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c**3*x**2 + int(asinh(c*x)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b*c + a*c**2*x**2 + a)/(sqrt(pi)*c* pi*(c**2*x**2 + 1))