Optimal. Leaf size=102 \[ -\frac {4 a \sqrt {a^2 x^2+1} \text {Int}\left (\frac {x}{\left (a^2 x^2+1\right )^2 \sinh ^{-1}(a x)^{3/2}},x\right )}{3 c \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{3 a \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 \sqrt {1+a^2 x^2}}{3 a \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{3/2}}-\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^{3/2}} \, dx}{3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.73, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arcsinh \left (a x \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________