Optimal. Leaf size=145 \[ -\frac {192 b^3 \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2+2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^4-\frac {8 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2+2}}+384 b^4 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5880, 8} \[ -\frac {192 b^3 \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2+2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2-\frac {8 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^4+384 b^4 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 5880
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4 \, dx &=-\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4+\left (48 b^2\right ) \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2 \, dx\\ &=-\frac {192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {2+d x^2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2-\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx\\ &=384 b^4 x-\frac {192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {2+d x^2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2-\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 264, normalized size = 1.82 \[ \frac {-8 a b \left (a^2+24 b^2\right ) \sqrt {d x^2} \sqrt {d x^2+2}+6 b^2 \cosh ^{-1}\left (d x^2+1\right )^2 \left (a^2 d x^2-4 a b \sqrt {d x^2} \sqrt {d x^2+2}+8 b^2 d x^2\right )+d x^2 \left (a^4+48 a^2 b^2+384 b^4\right )+4 b \cosh ^{-1}\left (d x^2+1\right ) \left (a^3 d x^2-6 a^2 b \sqrt {d x^2} \sqrt {d x^2+2}+24 a b^2 d x^2-48 b^3 \sqrt {d x^2} \sqrt {d x^2+2}\right )+4 b^3 \cosh ^{-1}\left (d x^2+1\right )^3 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {d x^2+2}\right )+b^4 d x^2 \cosh ^{-1}\left (d x^2+1\right )^4}{d x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.59, size = 298, normalized size = 2.06 \[ \frac {b^{4} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{4} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 4 \, {\left (a b^{3} d x^{2} - 2 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{3} - 6 \, {\left (4 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x^{2} - 6 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} {\left (a^{2} b^{2} + 8 \, b^{4}\right )}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right ) - 8 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} {\left (a^{3} b + 24 \, a b^{3}\right )}}{d x} \]
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 = 0.14, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}+1\right )\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b^{4} x \log \left (d x^{2} + \sqrt {d x^{2} + 2} \sqrt {d} x + 1\right )^{4} + 6 \, a^{2} b^{2} x \operatorname {arcosh}\left (d x^{2} + 1\right )^{2} + 24 \, a^{2} b^{2} d {\left (\frac {2 \, x}{d} - \frac {{\left (d^{\frac {3}{2}} x^{2} + 2 \, \sqrt {d}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2} \sqrt {d x^{2}} + 1\right )}{\sqrt {d x^{2} + 2} d^{2}}\right )} + 4 \, {\left (x \operatorname {arcosh}\left (d x^{2} + 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2} d}\right )} a^{3} b + a^{4} x + \int \frac {4 \, {\left ({\left (a b^{3} d^{2} - 2 \, b^{4} d^{2}\right )} x^{4} + 2 \, a b^{3} + {\left (3 \, a b^{3} d - 4 \, b^{4} d\right )} x^{2} + {\left ({\left (a b^{3} d - 2 \, b^{4} d\right )} \sqrt {d} x^{3} + 2 \, {\left (a b^{3} - b^{4}\right )} \sqrt {d} x\right )} \sqrt {d x^{2} + 2}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2} \sqrt {d} x + 1\right )^{3}}{d^{2} x^{4} + 3 \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} + 2 \, \sqrt {d} x\right )} \sqrt {d x^{2} + 2} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________