Optimal. Leaf size=255 \[ \frac {c \sqrt {-1+c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b \sqrt {1-c x}}-\frac {15 c \sqrt {-1+c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt {1-c x}}-\frac {c \sqrt {-1+c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b \sqrt {1-c x}}+\frac {c \sqrt {-1+c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b \sqrt {1-c x}}+\text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.66, 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 = {}
\begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {3 c^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {3 c^4 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {c^6 x^4}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c^4 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^6 \sqrt {1-c^2 x^2}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 c \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^4(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 c \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {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,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 (a+b x)}+\frac {\cosh (2 x)}{2 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15 c \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15 c \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c \sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c \sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c \sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c \sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (c \sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {c \sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c \sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 c \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{8 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c \sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c \sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
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 [A]
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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
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]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________