Optimal. Leaf size=134 \[ \frac {-a-b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{2 d e \sqrt {c^2 x^2}}+\frac {b c x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 x^2} \sqrt {c^2 d+e}} \]
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Rubi [A] time = 0.12, antiderivative size = 131, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5237, 446, 86, 63, 205} \[ -\frac {a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{2 d e \sqrt {c^2 x^2}}+\frac {b c x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 x^2} \sqrt {c^2 d+e}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 86
Rule 205
Rule 446
Rule 5237
Rubi steps
\begin {align*} \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\frac {a+b \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.78, size = 286, normalized size = 2.13 \[ -\frac {\frac {2 a}{d+e x^2}+\frac {b \sqrt {e} \log \left (\frac {4 i d e-4 c d \sqrt {e} x \left (c \sqrt {d}+i \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 (-d)-e}\right )}{b \sqrt {c^2 (-d)-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{d \sqrt {c^2 (-d)-e}}+\frac {b \sqrt {e} \log \left (\frac {4 i \left (-d e+c d \sqrt {e} x \left (\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 (-d)-e}+i c \sqrt {d}\right )\right )}{b \sqrt {c^2 (-d)-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \sqrt {c^2 (-d)-e}}+\frac {2 b \csc ^{-1}(c x)}{d+e x^2}-\frac {2 b \sin ^{-1}\left (\frac {1}{c x}\right )}{d}}{4 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 385, normalized size = 2.87 \[ \left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d e + \sqrt {-c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{2} e x^{2} - c^{2} d - 2 \, \sqrt {-c^{2} d e - e^{2}} \sqrt {c^{2} x^{2} - 1} - 2 \, e}{e x^{2} + d}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} + a d e - \sqrt {c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} d + e}\right ) + {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 354, normalized size = 2.64 \[ -\frac {c^{2} a}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c^{2} b \,\mathrm {arccsc}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{2 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d}-\frac {b \sqrt {c^{2} x^{2}-1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} e d}\, c x -2 e}{c e x +\sqrt {-c^{2} e d}}\right )}{4 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d +e}{e}}}-\frac {b \sqrt {c^{2} x^{2}-1}\, \ln \left (-\frac {2 \left (\sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +\sqrt {-c^{2} e d}\, c x -e \right )}{-c e x +\sqrt {-c^{2} e d}}\right )}{4 c e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d +e}{e}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left ({\left (c^{2} e^{2} x^{2} + c^{2} d e\right )} \int \frac {x e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{2} x^{4} + {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e - e^{2}\right )} x^{2} - d e\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + {\left (c^{2} d e - e^{2}\right )} x^{2} - d e}\,{d x} + \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{2 \, {\left (e^{2} x^{2} + d e\right )}} - \frac {a}{2 \, {\left (e^{2} x^{2} + d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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