Optimal. Leaf size=157 \[ -\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (c^2 d+2 e\right ) x \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d e^{3/2} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {270, 5347, 12,
457, 79, 65, 211} \begin {gather*} \frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c x \left (c^2 d+2 e\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d e^{3/2} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}-\frac {b c x \sqrt {c^2 x^2-1}}{8 e \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 79
Rule 211
Rule 270
Rule 457
Rule 5347
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {(b c x) \int \frac {x^3}{4 d \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {(b c x) \int \frac {x^3}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d \sqrt {c^2 x^2}}\\ &=\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {(b c x) \text {Subst}\left (\int \frac {x}{\sqrt {-1+c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 d \sqrt {c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {\left (b c \left (c^2 d+2 e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{16 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {\left (b \left (c^2 d+2 e\right ) x\right ) \text {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 e \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1+c^2 x^2}}{8 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}+\frac {x^4 \left (a+b \csc ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (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 e^{3/2} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.74, size = 390, normalized size = 2.48 \begin {gather*} \frac {\frac {4 a d}{\left (d+e x^2\right )^2}-\frac {8 a}{d+e x^2}-\frac {2 b c e \sqrt {1-\frac {1}{c^2 x^2}} x}{\left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \left (d+2 e x^2\right ) \csc ^{-1}(c x)}{\left (d+e x^2\right )^2}+\frac {4 b \text {ArcSin}\left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \log \left (\frac {16 d \sqrt {-c^2 d-e} e^{3/2} \left (i \sqrt {e}+c \left (c \sqrt {d}-i \sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (c^2 d+2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \left (-c^2 d-e\right )^{3/2}}+\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \log \left (-\frac {16 d \sqrt {-c^2 d-e} e^{3/2} \left (-\sqrt {e}+c \left (-i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d \left (-c^2 d-e\right )^{3/2}}}{16 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1830\) vs.
\(2(135)=270\).
time = 1.48, size = 1831, normalized size = 11.66
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1831\) |
default | \(\text {Expression too large to display}\) | \(1831\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 496 vs.
\(2 (138) = 276\).
time = 0.60, size = 1021, normalized size = 6.50 \begin {gather*} \left [-\frac {4 \, a c^{4} d^{4} + 8 \, a d x^{2} e^{3} + {\left (b c^{2} d^{3} + 2 \, b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 4 \, b d x^{2}\right )} e^{2} + 2 \, {\left (b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {-c^{2} d e - e^{2}} \log \left (-\frac {c^{2} d - {\left (c^{2} x^{2} - 2\right )} e + 2 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} d e - e^{2}}}{x^{2} e + d}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b d x^{2} e^{3} + {\left (4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) + 8 \, {\left (b c^{4} d^{4} + b x^{4} e^{4} + 2 \, {\left (b c^{2} d x^{4} + b d x^{2}\right )} e^{3} + {\left (b c^{4} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, {\left (4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} e^{2} + 8 \, {\left (a c^{4} d^{3} x^{2} + a c^{2} d^{3}\right )} e + 2 \, {\left (b c^{2} d^{3} e + b d x^{2} e^{3} + {\left (b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{16 \, {\left (c^{4} d^{5} e^{2} + d x^{4} e^{6} + 2 \, {\left (c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} e^{5} + {\left (c^{4} d^{3} x^{4} + 4 \, c^{2} d^{3} x^{2} + d^{3}\right )} e^{4} + 2 \, {\left (c^{4} d^{4} x^{2} + c^{2} d^{4}\right )} e^{3}\right )}}, -\frac {2 \, a c^{4} d^{4} + 4 \, a d x^{2} e^{3} - {\left (b c^{2} d^{3} + 2 \, b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 4 \, b d x^{2}\right )} e^{2} + 2 \, {\left (b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {c^{2} d e + e^{2}} \arctan \left (\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {c^{2} d e + e^{2}}}{c^{2} d + e}\right ) + 2 \, {\left (b c^{4} d^{4} + 2 \, b d x^{2} e^{3} + {\left (4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) + 4 \, {\left (b c^{4} d^{4} + b x^{4} e^{4} + 2 \, {\left (b c^{2} d x^{4} + b d x^{2}\right )} e^{3} + {\left (b c^{4} d^{2} x^{4} + 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} e^{2} + 4 \, {\left (a c^{4} d^{3} x^{2} + a c^{2} d^{3}\right )} e + {\left (b c^{2} d^{3} e + b d x^{2} e^{3} + {\left (b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, {\left (c^{4} d^{5} e^{2} + d x^{4} e^{6} + 2 \, {\left (c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} e^{5} + {\left (c^{4} d^{3} x^{4} + 4 \, c^{2} d^{3} x^{2} + d^{3}\right )} e^{4} + 2 \, {\left (c^{4} d^{4} x^{2} + c^{2} d^{4}\right )} e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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