Optimal. Leaf size=125 \[ 24 a b^2 x+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3-\frac {6 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2+2}}-\frac {48 b^3 \sqrt {\frac {d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+24 b^3 x \cosh ^{-1}\left (d x^2+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5880, 5901, 12, 6719, 261} \[ 24 a b^2 x-\frac {6 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3-\frac {48 b^3 \sqrt {\frac {d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+24 b^3 x \cosh ^{-1}\left (d x^2+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 261
Rule 5880
Rule 5901
Rule 6719
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3 \, dx &=-\frac {6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3+\left (24 b^2\right ) \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right ) \, dx\\ &=24 a b^2 x-\frac {6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3+\left (24 b^3\right ) \int \cosh ^{-1}\left (1+d x^2\right ) \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac {6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3-\left (24 b^3\right ) \int 2 \sqrt {\frac {d x^2}{2+d x^2}} \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac {6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3-\left (48 b^3\right ) \int \sqrt {\frac {d x^2}{2+d x^2}} \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac {6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3-\frac {\left (48 b^3 \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2}\right ) \int \frac {x}{\sqrt {2+d x^2}} \, dx}{x}\\ &=24 a b^2 x-\frac {48 b^3 \sqrt {\frac {d x^2}{2+d x^2}} \left (2+d x^2\right )}{d x}+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac {6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 171, normalized size = 1.37 \[ \frac {a d x^2 \left (a^2+24 b^2\right )-6 b \left (a^2+8 b^2\right ) \sqrt {d x^2} \sqrt {d x^2+2}+3 b \cosh ^{-1}\left (d x^2+1\right ) \left (a^2 d x^2-4 a b \sqrt {d x^2} \sqrt {d x^2+2}+8 b^2 d x^2\right )+3 b^2 \cosh ^{-1}\left (d x^2+1\right )^2 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {d x^2+2}\right )+b^3 d x^2 \cosh ^{-1}\left (d x^2+1\right )^3}{d x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 210, normalized size = 1.68 \[ \frac {b^{3} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{3} + {\left (a^{3} + 24 \, a b^{2}\right )} d x^{2} + 3 \, {\left (a b^{2} d x^{2} - 2 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} b^{3}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{2} + 3 \, {\left ({\left (a^{2} b + 8 \, b^{3}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} a b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right ) - 6 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} {\left (a^{2} b + 8 \, b^{3}\right )}}{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}+1\right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 3 \, a b^{2} x \operatorname {arcosh}\left (d x^{2} + 1\right )^{2} + 12 \, a b^{2} d {\left (\frac {2 \, x}{d} - \frac {{\left (d^{\frac {3}{2}} x^{2} + 2 \, \sqrt {d}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2} \sqrt {d x^{2}} + 1\right )}{\sqrt {d x^{2} + 2} d^{2}}\right )} + 3 \, {\left (x \operatorname {arcosh}\left (d x^{2} + 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2} d}\right )} a^{2} b + {\left (x \log \left (d x^{2} + \sqrt {d x^{2} + 2} \sqrt {d} x + 1\right )^{3} - \int \frac {6 \, {\left (d^{2} x^{4} + 2 \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} + \sqrt {d} x\right )} \sqrt {d x^{2} + 2}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2} \sqrt {d} x + 1\right )^{2}}{d^{2} x^{4} + 3 \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} + 2 \, \sqrt {d} x\right )} \sqrt {d x^{2} + 2} + 2}\,{d x}\right )} b^{3} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________