Optimal. Leaf size=193 \[ -\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{4 d^2 e \sqrt {c^2 x^2}}+\frac {b c x \left (3 c^2 d+2 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}+\frac {b c x \sqrt {c^2 x^2-1}}{8 d \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]
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Rubi [A] time = 0.19, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5237, 446, 103, 156, 63, 205} \[ -\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c x \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{4 d^2 e \sqrt {c^2 x^2}}+\frac {b c x \left (3 c^2 d+2 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}+\frac {b c x \sqrt {c^2 x^2-1}}{8 d \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]
Antiderivative was successfully verified.
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Rule 63
Rule 103
Rule 156
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 )^3} \, dx &=-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b c x) \int \frac {1}{x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e \sqrt {c^2 x^2}}\\ &=-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {-c^2 d-e+\frac {1}{2} c^2 e x}{x \sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{8 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b c x) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{8 d^2 e \sqrt {c^2 x^2}}+\frac {\left (b c \left (3 c^2 d+2 e\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{16 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^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 )}{4 c d^2 e \sqrt {c^2 x^2}}+\frac {\left (b \left (3 c^2 d+2 e\right ) x\right ) \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 )}{8 c d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c x \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{4 d^2 e \sqrt {c^2 x^2}}+\frac {b c \left (3 c^2 d+2 e\right ) x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.94, size = 385, normalized size = 1.99 \[ \frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}+\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 d^2 \sqrt {e} \sqrt {c^2 (-d)-e} \left (c x \left (c \sqrt {d}-i \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 (-d)-e}\right )+i \sqrt {e}\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d^2 \sqrt {e} \left (c^2 (-d)-e\right )^{3/2}}+\frac {b \left (3 c^2 d+2 e\right ) \log \left (-\frac {16 d^2 \sqrt {e} \sqrt {c^2 (-d)-e} \left (-\sqrt {e}+c x \left (\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {c^2 (-d)-e}-i c \sqrt {d}\right )\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt {e} x+i \sqrt {d}\right )}\right )}{d^2 \sqrt {e} \left (c^2 (-d)-e\right )^{3/2}}+\frac {2 b c x \sqrt {1-\frac {1}{c^2 x^2}}}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac {4 b \sin ^{-1}\left (\frac {1}{c x}\right )}{d^2 e}-\frac {4 b \csc ^{-1}(c x)}{e \left (d+e x^2\right )^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 890, normalized size = 4.61 \[ \left [-\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + {\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d e - e^{2}} \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 ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{16 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} + 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} - {\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d e + e^{2}} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} d + e}\right ) + 2 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\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 = 1840, normalized size = 9.53 \[ \text {result too large to display} \]
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 \[ -\frac {{\left ({\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{2} + c^{2} d^{2} e\right )} \int \frac {x e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{3} x^{6} + {\left (2 \, c^{2} d e^{2} - e^{3}\right )} x^{4} + {\left (c^{2} e^{3} x^{6} + {\left (2 \, c^{2} d e^{2} - e^{3}\right )} x^{4} - d^{2} e + {\left (c^{2} d^{2} e - 2 \, d e^{2}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - d^{2} e + {\left (c^{2} d^{2} e - 2 \, d e^{2}\right )} x^{2}}\,{d x} + \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b}{4 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}} - \frac {a}{4 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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