Optimal. Leaf size=144 \[ \frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b c}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b c}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b c}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 b c}+\frac {3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5699, 3312, 3303, 3298, 3301} \[ \frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}+\frac {3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 3301
Rule 3303
Rule 3312
Rule 5699
Rubi steps
\begin {align*} \int \frac {\left (1+c^2 x^2\right )^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {3}{8 (a+b x)}+\frac {\cosh (2 x)}{2 (a+b x)}+\frac {\cosh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c}\\ &=\frac {3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}\\ &=\frac {3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c}+\frac {\cosh \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}+\frac {\cosh \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c}-\frac {\sinh \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c}-\frac {\sinh \left (\frac {4 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c}\\ &=\frac {\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}+\frac {\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}+\frac {3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c}-\frac {\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c}-\frac {\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 109, normalized size = 0.76 \[ \frac {4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+3 \log \left (a+b \sinh ^{-1}(c x)\right )}{8 b c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arsinh}\left (c x\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arsinh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.31, size = 139, normalized size = 0.97 \[ \frac {3 \ln \left (a +b \arcsinh \left (c x \right )\right )}{8 b c}-\frac {{\mathrm e}^{\frac {4 a}{b}} \Ei \left (1, 4 \arcsinh \left (c x \right )+\frac {4 a}{b}\right )}{16 c b}-\frac {{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right )}{4 c b}-\frac {{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right )}{4 c b}-\frac {{\mathrm e}^{-\frac {4 a}{b}} \Ei \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right )}{16 c b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b \operatorname {arsinh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c^2\,x^2+1\right )}^{3/2}}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________