Optimal. Leaf size=175 \[ -\frac {\text {Int}\left (\frac {c^2 x^2+1}{x^2 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b c}+\frac {9 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2}-\frac {9 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{4 b^2}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{4 b^2}-\frac {\left (c^2 x^2+1\right )^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, 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 = {} \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {\left (1+c^2 x^2\right )^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}-\frac {\int \frac {1+c^2 x^2}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}+\frac {(3 c) \int \frac {1+c^2 x^2}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac {\left (1+c^2 x^2\right )^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh ^3(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac {\int \frac {1+c^2 x^2}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {\left (1+c^2 x^2\right )^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \operatorname {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 (a+b x)}+\frac {\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b}-\frac {\int \frac {1+c^2 x^2}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {\left (1+c^2 x^2\right )^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b}-\frac {\int \frac {1+c^2 x^2}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {\left (1+c^2 x^2\right )^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}-\frac {\int \frac {1+c^2 x^2}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}+\frac {\left (9 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b}+\frac {\left (3 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b}-\frac {\left (9 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b}-\frac {\left (3 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b}\\ &=-\frac {\left (1+c^2 x^2\right )^2}{b c x \left (a+b \sinh ^{-1}(c x)\right )}+\frac {9 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2}+\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2}-\frac {9 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b^2}-\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b^2}-\frac {\int \frac {1+c^2 x^2}{x^2 \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 7.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b x \operatorname {arsinh}\left (c x\right ) + a^{2} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (c^{4} x^{4} + 2 \, c^{2} x^{2} + 1\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (c^{5} x^{5} + 2 \, c^{3} x^{3} + c x\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{3} x^{3} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x^{2} + a b c x + {\left (b^{2} c^{3} x^{3} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x^{2} + b^{2} c x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (3 \, c^{5} x^{5} + c^{3} x^{3} - 2 \, c x\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + {\left (6 \, c^{6} x^{6} + 7 \, c^{4} x^{4} - 1\right )} {\left (c^{2} x^{2} + 1\right )} + 3 \, {\left (c^{7} x^{7} + 2 \, c^{5} x^{5} + c^{3} x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} x^{6} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{4} + 2 \, a b c^{3} x^{4} + a b c x^{2} + {\left (b^{2} c^{5} x^{6} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{4} + 2 \, b^{2} c^{3} x^{4} + b^{2} c x^{2} + 2 \, {\left (b^{2} c^{4} x^{5} + b^{2} c^{2} x^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{5} + a b c^{2} x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c^2\,x^2+1\right )}^{3/2}}{x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________