Integrand size = 23, antiderivative size = 76 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arcsinh}(c x))}{d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c d \sqrt {d+c^2 d x^2}} \] Output:
x*(a+b*arcsinh(c*x))/d/(c^2*d*x^2+d)^(1/2)-1/2*b*(c^2*x^2+1)^(1/2)*ln(c^2* x^2+1)/c/d/(c^2*d*x^2+d)^(1/2)
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.32 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (2 a c x \sqrt {1+c^2 x^2}+2 b c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-\left (b+b c^2 x^2\right ) \log \left (1+c^2 x^2\right )\right )}{2 c d^2 \left (1+c^2 x^2\right )^{3/2}} \] Input:
Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2),x]
Output:
(Sqrt[d + c^2*d*x^2]*(2*a*c*x*Sqrt[1 + c^2*x^2] + 2*b*c*x*Sqrt[1 + c^2*x^2 ]*ArcSinh[c*x] - (b + b*c^2*x^2)*Log[1 + c^2*x^2]))/(2*c*d^2*(1 + c^2*x^2) ^(3/2))
Time = 0.25 (sec) , antiderivative size = 76, 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.
\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 d x^2+d\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6202 |
\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1} \int \frac {x}{c^2 x^2+1}dx}{d \sqrt {c^2 d x^2+d}}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {x (a+b \text {arcsinh}(c x))}{d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c d \sqrt {c^2 d x^2+d}}\) |
Input:
Int[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2),x]
Output:
(x*(a + b*ArcSinh[c*x]))/(d*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*Lo g[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_.) + 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. \(142\) vs. \(2(68)=136\).
Time = 0.77 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {a x}{d \sqrt {c^{2} d \,x^{2}+d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}\) | \(143\) |
parts | \(\frac {a x}{d \sqrt {c^{2} d \,x^{2}+d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, d^{2} c}\) | \(143\) |
Input:
int((a+b*arcsinh(x*c))/(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^2*x^2+1)^(1/2)/d^2/c* arcsinh(x*c)+b*(d*(c^2*x^2+1))^(1/2)*arcsinh(x*c)/d^2/(c^2*x^2+1)*x-b*(d*( c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^2/c*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(c^2*d*x^2 + d)*(b*arcsinh(c*x) + a)/(c^4*d^2*x^4 + 2*c^2*d^2 *x^2 + d^2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*asinh(c*x))/(c**2*d*x**2+d)**(3/2),x)
Output:
Integral((a + b*asinh(c*x))/(d*(c**2*x**2 + 1))**(3/2), x)
Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {b x \operatorname {arsinh}\left (c x\right )}{\sqrt {c^{2} d x^{2} + d} d} + \frac {a x}{\sqrt {c^{2} d x^{2} + d} d} - \frac {b \log \left (x^{2} + \frac {1}{c^{2}}\right )}{2 \, c d^{\frac {3}{2}}} \] Input:
integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(3/2),x, algorithm="maxima")
Output:
b*x*arcsinh(c*x)/(sqrt(c^2*d*x^2 + d)*d) + a*x/(sqrt(c^2*d*x^2 + d)*d) - 1 /2*b*log(x^2 + 1/c^2)/(c*d^(3/2))
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/(c^2*d*x^2+d)^(3/2),x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)/(c^2*d*x^2 + d)^(3/2), x)
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:
int((a + b*asinh(c*x))/(d + c^2*d*x^2)^(3/2),x)
Output:
int((a + b*asinh(c*x))/(d + c^2*d*x^2)^(3/2), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {\sqrt {c^{2} x^{2}+1}\, a c x +\left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b \,c^{3} x^{2}+\left (\int \frac {\mathit {asinh} \left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b c +a \,c^{2} x^{2}+a}{\sqrt {d}\, c d \left (c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))/(c^2*d*x^2+d)^(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(d)*c*d *(c**2*x**2 + 1))