Optimal. Leaf size=125 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{4 \sqrt {2} a c^{7/2}}-\frac {3 \sqrt {c-a c x}}{4 a c^4 \sqrt {1-a^2 x^2}}+\frac {1}{2 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6127, 673, 667, 661, 208} \[ -\frac {3 \sqrt {c-a c x}}{4 a c^4 \sqrt {1-a^2 x^2}}+\frac {1}{2 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a c x}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{4 \sqrt {2} a c^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 661
Rule 667
Rule 673
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-a c x)^{7/2}} \, dx &=\frac {\int \frac {1}{\sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{2 a c^3 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}+\frac {3 \int \frac {\sqrt {c-a c x}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{4 c^4}\\ &=\frac {1}{2 a c^3 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {c-a c x}}{4 a c^4 \sqrt {1-a^2 x^2}}+\frac {3 \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx}{8 c^3}\\ &=\frac {1}{2 a c^3 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {c-a c x}}{4 a c^4 \sqrt {1-a^2 x^2}}-\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )}{4 c^2}\\ &=\frac {1}{2 a c^3 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {c-a c x}}{4 a c^4 \sqrt {1-a^2 x^2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{4 \sqrt {2} a c^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 57, normalized size = 0.46 \[ -\frac {(1-a x)^{3/2} \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {1}{2} (a x+1)\right )}{2 a c^2 \sqrt {a x+1} (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 310, normalized size = 2.48 \[ \left [\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (3 \, a x - 1\right )}}{16 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}, \frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (3 \, a x - 1\right )}}{8 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 82, normalized size = 0.66 \[ -\frac {{\left (\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c} c^{2}} + \frac {2 \, {\left (3 \, a c x - c\right )}}{{\left ({\left (a c x + c\right )}^{\frac {3}{2}} - 2 \, \sqrt {a c x + c} c\right )} a c^{2}}\right )} {\left | c \right |}}{8 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 124, normalized size = 0.99 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a \sqrt {c \left (a x +1\right )}-3 \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c \left (a x +1\right )}-6 x a \sqrt {c}+2 \sqrt {c}\right )}{8 c^{\frac {9}{2}} \left (a x -1\right )^{2} \left (a x +1\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (-a c x + c\right )}^{\frac {7}{2}} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-a\,c\,x\right )}^{7/2}\,{\left (a\,x+1\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (a x - 1\right )\right )^{\frac {7}{2}} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________