Integrand size = 25, antiderivative size = 284 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {15}{64} b^2 d x \sqrt {d+c^2 d x^2}+\frac {1}{32} b^2 x \left (d+c^2 d x^2\right )^{3/2}-\frac {9 b^2 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 c \sqrt {1+c^2 x^2}}-\frac {3 b c d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}-\frac {b d \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{8 b c \sqrt {1+c^2 x^2}} \] Output:
15/64*b^2*d*x*(c^2*d*x^2+d)^(1/2)+1/32*b^2*x*(c^2*d*x^2+d)^(3/2)-9/64*b^2* d*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)/c/(c^2*x^2+1)^(1/2)-3/8*b*c*d*x^2*(c^2* d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-1/8*b*d*(c^2*x^2+1)^(3 /2)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))/c+3/8*d*x*(c^2*d*x^2+d)^(1/2)*( a+b*arcsinh(c*x))^2+1/4*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2+1/8*d*( c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^3/b/c/(c^2*x^2+1)^(1/2)
Time = 2.53 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.16 \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {96 a^2 c d x \sqrt {1+c^2 x^2} \left (5+2 c^2 x^2\right ) \sqrt {d+c^2 d x^2}+288 a^2 d^{3/2} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+32 b^2 d \sqrt {d+c^2 d x^2} \left (4 \text {arcsinh}(c x)^3-6 \text {arcsinh}(c x) \cosh (2 \text {arcsinh}(c x))+\left (3+6 \text {arcsinh}(c x)^2\right ) \sinh (2 \text {arcsinh}(c x))\right )-192 a b d \sqrt {d+c^2 d x^2} (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))-12 a b d \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )-b^2 d \sqrt {d+c^2 d x^2} \left (32 \text {arcsinh}(c x)^3+12 \text {arcsinh}(c x) \cosh (4 \text {arcsinh}(c x))-3 \left (1+8 \text {arcsinh}(c x)^2\right ) \sinh (4 \text {arcsinh}(c x))\right )}{768 c \sqrt {1+c^2 x^2}} \] Input:
Integrate[(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(96*a^2*c*d*x*Sqrt[1 + c^2*x^2]*(5 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2] + 288* a^2*d^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 3 2*b^2*d*Sqrt[d + c^2*d*x^2]*(4*ArcSinh[c*x]^3 - 6*ArcSinh[c*x]*Cosh[2*ArcS inh[c*x]] + (3 + 6*ArcSinh[c*x]^2)*Sinh[2*ArcSinh[c*x]]) - 192*a*b*d*Sqrt[ d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh [2*ArcSinh[c*x]])) - 12*a*b*d*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh [4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]) - b^2*d*Sqrt[d + c ^2*d*x^2]*(32*ArcSinh[c*x]^3 + 12*ArcSinh[c*x]*Cosh[4*ArcSinh[c*x]] - 3*(1 + 8*ArcSinh[c*x]^2)*Sinh[4*ArcSinh[c*x]]))/(768*c*Sqrt[1 + c^2*x^2])
Time = 1.65 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 222}
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.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \int \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \int x (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \int x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \int \left (c^2 x^2+1\right )^{3/2}dx}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \int \sqrt {c^2 x^2+1}dx+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle -\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {c^2 x^2+1}}dx+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b \left (\frac {3}{4} \left (\frac {\text {arcsinh}(c x)}{2 c}+\frac {1}{2} x \sqrt {c^2 x^2+1}\right )+\frac {1}{4} x \left (c^2 x^2+1\right )^{3/2}\right )}{4 c}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{\sqrt {c^2 x^2+1}}\right )\) |
Input:
Int[(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]
Output:
(x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*d*((x*Sqrt[d + c^2 *d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c* x])^3)/(6*b*c*Sqrt[1 + c^2*x^2]) - (b*c*Sqrt[d + c^2*d*x^2]*((x^2*(a + b*A rcSinh[c*x]))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^ 3)))/2))/Sqrt[1 + c^2*x^2]))/4 - (b*c*d*Sqrt[d + c^2*d*x^2]*(((1 + c^2*x^2 )^2*(a + b*ArcSinh[c*x]))/(4*c^2) - (b*((x*(1 + c^2*x^2)^(3/2))/4 + (3*((x *Sqrt[1 + c^2*x^2])/2 + ArcSinh[c*x]/(2*c)))/4))/(4*c)))/(2*Sqrt[1 + c^2*x ^2])
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(958\) vs. \(2(244)=488\).
Time = 1.15 (sec) , antiderivative size = 959, normalized size of antiderivative = 3.38
| method | result | size |
| default | \(\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} d x \sqrt {c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3} d}{8 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}-4 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{512 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}+4 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{512 \left (c^{2} x^{2}+1\right ) c}\right )+2 a b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} d}{16 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}\right )\) | \(959\) |
| parts | \(\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} d x \sqrt {c^{2} d \,x^{2}+d}}{8}+\frac {3 a^{2} d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{3} d}{8 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}-4 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{512 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (x c \right )^{2}+4 \,\operatorname {arcsinh}\left (x c \right )+1\right ) d}{512 \left (c^{2} x^{2}+1\right ) c}\right )+2 a b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} d}{16 \sqrt {c^{2} x^{2}+1}\, c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{16 \left (c^{2} x^{2}+1\right ) c}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d}{256 \left (c^{2} x^{2}+1\right ) c}\right )\) | \(959\) |
Input:
int((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/4*a^2*x*(c^2*d*x^2+d)^(3/2)+3/8*a^2*d*x*(c^2*d*x^2+d)^(1/2)+3/8*a^2*d^2* ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b^2*(1/8*(d*(c ^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*arcsinh(x*c)^3*d+1/512*(d*(c^2*x^2+1) )^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x ^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(8*arcsinh(x*c)^2-4*arcsinh(x*c)+1)*d /(c^2*x^2+1)/c+1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1) ^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2-2*arcsinh(x*c)+1)*d/(c^2 *x^2+1)/c+1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2 )+2*x*c-(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2+2*arcsinh(x*c)+1)*d/(c^2*x^2+ 1)/c+1/512*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^4*(c^2*x^2+1)^(1/2)+12 *x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+1)^(1/2))*(8*arcsinh(x *c)^2+4*arcsinh(x*c)+1)*d/(c^2*x^2+1)/c)+2*a*b*(3/16*(d*(c^2*x^2+1))^(1/2) /(c^2*x^2+1)^(1/2)/c*arcsinh(x*c)^2*d+1/256*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c ^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x* c+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(x*c))*d/(c^2*x^2+1)/c+1/16*(d*(c^2*x^2+ 1))^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))* (-1+2*arcsinh(x*c))*d/(c^2*x^2+1)/c+1/16*(d*(c^2*x^2+1))^(1/2)*(2*x^3*c^3- 2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(x*c))*d/ (c^2*x^2+1)/c+1/256*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5-8*x^4*c^4*(c^2*x^2+1) ^(1/2)+12*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c-(c^2*x^2+1)^(1/2))*...
\[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \] Input:
integrate((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
integral((a^2*c^2*d*x^2 + a^2*d + (b^2*c^2*d*x^2 + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*c^2*d*x^2 + a*b*d)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
\[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate((c**2*d*x**2+d)**(3/2)*(a+b*asinh(c*x))**2,x)
Output:
Integral((d*(c**2*x**2 + 1))**(3/2)*(a + b*asinh(c*x))**2, x)
Exception generated. \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \] Input:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(3/2),x)
Output:
int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(3/2), x)
\[ \int \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, d \left (2 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+5 \sqrt {c^{2} x^{2}+1}\, a^{2} c x +16 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+16 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )d x \right ) a b c +8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+8 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}d x \right ) b^{2} c +3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2}\right )}{8 c} \] Input:
int((c^2*d*x^2+d)^(3/2)*(a+b*asinh(c*x))^2,x)
Output:
(sqrt(d)*d*(2*sqrt(c**2*x**2 + 1)*a**2*c**3*x**3 + 5*sqrt(c**2*x**2 + 1)*a **2*c*x + 16*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*a*b*c**3 + 16*int( sqrt(c**2*x**2 + 1)*asinh(c*x),x)*a*b*c + 8*int(sqrt(c**2*x**2 + 1)*asinh( c*x)**2*x**2,x)*b**2*c**3 + 8*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2,x)*b** 2*c + 3*log(sqrt(c**2*x**2 + 1) + c*x)*a**2))/(8*c)