Optimal. Leaf size=145 \[ 384 b^4 x-\frac {192 b^3 \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {2+d x^2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2-\frac {8 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^4 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, 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 = {6001, 8}
\begin {gather*} -\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 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 6001
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 = 0.15, size = 264, normalized size = 1.82 \begin {gather*} \frac {\left (a^4+48 a^2 b^2+384 b^4\right ) d x^2-8 a b \left (a^2+24 b^2\right ) \sqrt {d x^2} \sqrt {2+d x^2}+4 b \left (a^3 d x^2+24 a b^2 d x^2-6 a^2 b \sqrt {d x^2} \sqrt {2+d x^2}-48 b^3 \sqrt {d x^2} \sqrt {2+d x^2}\right ) \cosh ^{-1}\left (1+d x^2\right )+6 b^2 \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2} \sqrt {2+d x^2}\right ) \cosh ^{-1}\left (1+d x^2\right )^2+4 b^3 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {2+d x^2}\right ) \cosh ^{-1}\left (1+d x^2\right )^3+b^4 d x^2 \cosh ^{-1}\left (1+d x^2\right )^4}{d x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 298 vs.
\(2 (137) = 274\).
time = 0.35, size = 298, normalized size = 2.06 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )^{4}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________