Optimal. Leaf size=139 \[ -\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \tan ^{-1}\left (\sqrt {-c^2 x^2-1}\right )}{2 d e \sqrt {-c^2 x^2}}+\frac {b c x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{2 d \sqrt {e} \sqrt {-c^2 x^2} \sqrt {c^2 d-e}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6300, 446, 86, 63, 205, 208} \[ -\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \tan ^{-1}\left (\sqrt {-c^2 x^2-1}\right )}{2 d e \sqrt {-c^2 x^2}}+\frac {b c x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{\sqrt {c^2 d-e}}\right )}{2 d \sqrt {e} \sqrt {-c^2 x^2} \sqrt {c^2 d-e}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 86
Rule 205
Rule 208
Rule 446
Rule 6300
Rubi steps
\begin {align*} \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b c x) \int \frac {1}{x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e \sqrt {-c^2 x^2}}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 e \sqrt {-c^2 x^2}}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} (d+e x)} \, dx,x,x^2\right )}{4 d \sqrt {-c^2 x^2}}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{4 d e \sqrt {-c^2 x^2}}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b x) \operatorname {Subst}\left (\int \frac {1}{d-\frac {e}{c^2}-\frac {e x^2}{c^2}} \, dx,x,\sqrt {-1-c^2 x^2}\right )}{2 c d \sqrt {-c^2 x^2}}-\frac {(b x) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {-1-c^2 x^2}\right )}{2 c d e \sqrt {-c^2 x^2}}\\ &=-\frac {a+b \text {csch}^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \tan ^{-1}\left (\sqrt {-1-c^2 x^2}\right )}{2 d e \sqrt {-c^2 x^2}}+\frac {b c x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{\sqrt {c^2 d-e}}\right )}{2 d \sqrt {c^2 d-e} \sqrt {e} \sqrt {-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.79, size = 271, normalized size = 1.95 \[ -\frac {\frac {2 a}{d+e x^2}+\frac {b \sqrt {e} \log \left (-\frac {4 \left (c d \sqrt {e} x \left (c \sqrt {d}+i \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {e-c^2 d}\right )+i d e\right )}{b \sqrt {e-c^2 d} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{d \sqrt {e-c^2 d}}+\frac {b \sqrt {e} \log \left (\frac {4 i \left (d e+c d \sqrt {e} x \left (\sqrt {\frac {1}{c^2 x^2}+1} \sqrt {e-c^2 d}+i c \sqrt {d}\right )\right )}{b \sqrt {e-c^2 d} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \sqrt {e-c^2 d}}+\frac {2 b \text {csch}^{-1}(c x)}{d+e x^2}-\frac {2 b \sinh ^{-1}\left (\frac {1}{c x}\right )}{d}}{4 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.94, size = 615, normalized size = 4.42 \[ \left [-\frac {2 \, a c^{2} d^{2} - 2 \, a d e + \sqrt {-c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{2} e x^{2} - c^{2} d - 2 \, \sqrt {-c^{2} d e + e^{2}} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, e}{e x^{2} + d}\right ) - 2 \, {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 2 \, {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{4 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} - a d e + \sqrt {c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (-\frac {\sqrt {c^{2} d e - e^{2}} c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{c^{2} d - e}\right ) - {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (b c^{2} d^{2} - b d e + {\left (b c^{2} d e - b e^{2}\right )} x^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + {\left (b c^{2} d^{2} - b d e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 360, normalized size = 2.59 \[ -\frac {c^{2} a}{2 e \left (c^{2} x^{2} e +c^{2} d \right )}-\frac {c^{2} b \,\mathrm {arccsch}\left (c x \right )}{2 e \left (c^{2} x^{2} e +c^{2} d \right )}+\frac {b \sqrt {c^{2} x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{2 c e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d}-\frac {b \sqrt {c^{2} x^{2}+1}\, \ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{-c x e +\sqrt {-c^{2} d e}}\right )}{4 c e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d -e}{e}}}-\frac {b \sqrt {c^{2} x^{2}+1}\, \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d -e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{c x e +\sqrt {-c^{2} d e}}\right )}{4 c e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d -e}{e}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, {\left (4 \, c^{2} \int \frac {x}{2 \, {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e + e^{2}\right )} x^{2} + d e + {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e + e^{2}\right )} x^{2} + d e\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x} - \frac {2 \, c^{2} d^{2} \log \relax (c) - 2 \, {\left (c^{2} d e - e^{2}\right )} x^{2} \log \relax (x) - 2 \, d e \log \relax (c) + {\left (c^{2} d e x^{2} + c^{2} d^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (c^{2} d^{2} - d e\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}} + \frac {\log \left (e x^{2} + d\right )}{c^{2} d^{2} - d e}\right )} b - \frac {a}{2 \, {\left (e^{2} x^{2} + d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________