Optimal. Leaf size=205 \[ -\frac {3 a x^2 \sqrt {c+a^2 c x^2}}{8 \sqrt {1+a^2 x^2}}+\frac {3}{4} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {3 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{8 a \sqrt {1+a^2 x^2}}-\frac {3 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{4 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{8 a \sqrt {1+a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5785, 5783,
5776, 5812, 30} \begin {gather*} -\frac {3 a x^2 \sqrt {a^2 c x^2+c}}{8 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^4}{8 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^3-\frac {3 a x^2 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^2}{4 \sqrt {a^2 x^2+1}}-\frac {3 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^2}{8 a \sqrt {a^2 x^2+1}}+\frac {3}{4} x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 5776
Rule 5783
Rule 5785
Rule 5812
Rubi steps
\begin {align*} \int \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3 \, dx &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {\sqrt {c+a^2 c x^2} \int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {1+a^2 x^2}}-\frac {\left (3 a \sqrt {c+a^2 c x^2}\right ) \int x \sinh ^{-1}(a x)^2 \, dx}{2 \sqrt {1+a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{4 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{8 a \sqrt {1+a^2 x^2}}+\frac {\left (3 a^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {1+a^2 x^2}}\\ &=\frac {3}{4} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {3 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{4 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{8 a \sqrt {1+a^2 x^2}}-\frac {\left (3 \sqrt {c+a^2 c x^2}\right ) \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {1+a^2 x^2}}-\frac {\left (3 a \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{4 \sqrt {1+a^2 x^2}}\\ &=-\frac {3 a x^2 \sqrt {c+a^2 c x^2}}{8 \sqrt {1+a^2 x^2}}+\frac {3}{4} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {3 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{8 a \sqrt {1+a^2 x^2}}-\frac {3 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{4 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{8 a \sqrt {1+a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 86, normalized size = 0.42 \begin {gather*} \frac {\sqrt {c \left (1+a^2 x^2\right )} \left (-3 \left (1+2 \sinh ^{-1}(a x)^2\right ) \cosh \left (2 \sinh ^{-1}(a x)\right )+2 \sinh ^{-1}(a x) \left (\sinh ^{-1}(a x)^3+\left (3+2 \sinh ^{-1}(a x)^2\right ) \sinh \left (2 \sinh ^{-1}(a x)\right )\right )\right )}{16 a \sqrt {1+a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.46, size = 231, normalized size = 1.13
method | result | size |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{4}}{8 \sqrt {a^{2} x^{2}+1}\, a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}+2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+2 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (4 \arcsinh \left (a x \right )^{3}-6 \arcsinh \left (a x \right )^{2}+6 \arcsinh \left (a x \right )-3\right )}{32 a \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}-2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+2 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (4 \arcsinh \left (a x \right )^{3}+6 \arcsinh \left (a x \right )^{2}+6 \arcsinh \left (a x \right )+3\right )}{32 a \left (a^{2} x^{2}+1\right )}\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {asinh}^{3}{\left (a x \right )}\, 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: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {asinh}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________