Optimal. Leaf size=127 \[ -\frac {5 \sqrt {1-a x}}{a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}-\frac {x (1-a x)}{\sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6269, 6263,
895, 862, 49, 56, 221} \begin {gather*} \frac {5 \sqrt {1-a x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {x} \sqrt {c-\frac {c}{a x}}}-\frac {x (1-a x)}{\sqrt {1-a^2 x^2} \sqrt {c-\frac {c}{a x}}}-\frac {5 \sqrt {1-a x}}{a \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 49
Rule 56
Rule 221
Rule 862
Rule 895
Rule 6263
Rule 6269
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\sqrt {c-\frac {c}{a x}}} \, dx &=\frac {\sqrt {1-a x} \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {x}}{\sqrt {1-a x}} \, dx}{\sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=\frac {\sqrt {1-a x} \int \frac {\sqrt {x} (1-a x)^{5/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{\sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=-\frac {x (1-a x)}{\sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}+\frac {\left (5 \sqrt {1-a x}\right ) \int \frac {\sqrt {x} (1-a x)^{3/2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{2 \sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=-\frac {x (1-a x)}{\sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}+\frac {\left (5 \sqrt {1-a x}\right ) \int \frac {\sqrt {x}}{(1+a x)^{3/2}} \, dx}{2 \sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=-\frac {5 \sqrt {1-a x}}{a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}-\frac {x (1-a x)}{\sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}+\frac {\left (5 \sqrt {1-a x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 a \sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=-\frac {5 \sqrt {1-a x}}{a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}-\frac {x (1-a x)}{\sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}+\frac {\left (5 \sqrt {1-a x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{a \sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ &=-\frac {5 \sqrt {1-a x}}{a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}-\frac {x (1-a x)}{\sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 86, normalized size = 0.68 \begin {gather*} \frac {\sqrt {1-a x} \left (-\sqrt {a} \sqrt {x} (5+a x)+5 \sqrt {1+a x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right )}{a^{3/2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \sqrt {1+a x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.74, size = 143, normalized size = 1.13
method | result | size |
default | \(-\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (2 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+5 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a x +10 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}+5 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right )\right ) \sqrt {-a^{2} x^{2}+1}}{2 \sqrt {a}\, c \left (a x +1\right ) \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )}\) | \(143\) |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-a c x \left (a x +1\right )}\, \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}+\frac {\left (-\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right )}{2 a \sqrt {a^{2} c}}-\frac {4 \sqrt {-a^{2} c \left (x +\frac {1}{a}\right )^{2}+\left (x +\frac {1}{a}\right ) a c}}{a^{3} c \left (x +\frac {1}{a}\right )}\right ) \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}\, \left (a x -1\right )}{\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
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 = 0.38, size = 286, normalized size = 2.25 \begin {gather*} \left [-\frac {5 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{3} c x^{2} - a c\right )}}, \frac {5 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (a^{2} x^{2} + 5 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{3} c x^{2} - a c\right )}}\right ] \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 \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{\sqrt {c-\frac {c}{a\,x}}\,{\left (a\,x+1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________