Optimal. Leaf size=163 \[ -\frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {2 b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2}} \]
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Rubi [A]
time = 0.16, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {272, 45, 5347,
12, 587, 157, 95, 210} \begin {gather*} -\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}+\frac {2 b c x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2}}-\frac {b c x \sqrt {c^2 x^2-1}}{3 e \sqrt {c^2 x^2} \left (c^2 d+e\right ) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 95
Rule 157
Rule 210
Rule 272
Rule 587
Rule 5347
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {-2 d-3 e x^2}{3 e^2 x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{\sqrt {c^2 x^2}}\\ &=\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {-2 d-3 e x^2}{x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2 \sqrt {c^2 x^2}}\\ &=\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {(b c x) \text {Subst}\left (\int \frac {-2 d-3 e x}{x \sqrt {-1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \text {Subst}\left (\int \frac {d \left (c^2 d+e\right )}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d e^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {(2 b c x) \text {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{3 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b c x \sqrt {-1+c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \csc ^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {2 b c x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 173, normalized size = 1.06 \begin {gather*} \frac {-b c e \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+e x^2\right )-a \left (c^2 d+e\right ) \left (2 d+3 e x^2\right )-b \left (c^2 d+e\right ) \left (2 d+3 e x^2\right ) \csc ^{-1}(c x)}{3 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac {2 b c \sqrt {1-\frac {1}{c^2 x^2}} x \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{3 \sqrt {d} e^2 \sqrt {-1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
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. 332 vs.
\(2 (138) = 276\).
time = 0.48, size = 690, normalized size = 4.23 \begin {gather*} \left [-\frac {{\left (b c^{2} d^{3} + b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 2 \, b d x^{2}\right )} e^{2} + {\left (2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {-d} \log \left (\frac {c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} e^{2} + 4 \, {\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {-d} + 8 \, d^{2} - 2 \, {\left (3 \, c^{2} d x^{4} - 4 \, d x^{2}\right )} e}{x^{4}}\right ) + 2 \, {\left (2 \, a c^{2} d^{3} + 3 \, a d x^{2} e^{2} + {\left (2 \, b c^{2} d^{3} + 3 \, b d x^{2} e^{2} + {\left (3 \, b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) + {\left (3 \, a c^{2} d^{2} x^{2} + 2 \, a d^{2}\right )} e + {\left (b d x^{2} e^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}}{6 \, {\left (c^{2} d^{4} e^{2} + d x^{4} e^{5} + {\left (c^{2} d^{2} x^{4} + 2 \, d^{2} x^{2}\right )} e^{4} + {\left (2 \, c^{2} d^{3} x^{2} + d^{3}\right )} e^{3}\right )}}, \frac {{\left (b c^{2} d^{3} + b x^{4} e^{3} + {\left (b c^{2} d x^{4} + 2 \, b d x^{2}\right )} e^{2} + {\left (2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e\right )} \sqrt {d} \arctan \left (-\frac {{\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {d}}{2 \, {\left (c^{2} d^{2} x^{2} - d^{2} + {\left (c^{2} d x^{4} - d x^{2}\right )} e\right )}}\right ) - {\left (2 \, a c^{2} d^{3} + 3 \, a d x^{2} e^{2} + {\left (2 \, b c^{2} d^{3} + 3 \, b d x^{2} e^{2} + {\left (3 \, b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} e\right )} \operatorname {arccsc}\left (c x\right ) + {\left (3 \, a c^{2} d^{2} x^{2} + 2 \, a d^{2}\right )} e + {\left (b d x^{2} e^{2} + b d^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}}{3 \, {\left (c^{2} d^{4} e^{2} + d x^{4} e^{5} + {\left (c^{2} d^{2} x^{4} + 2 \, d^{2} x^{2}\right )} e^{4} + {\left (2 \, c^{2} d^{3} x^{2} + d^{3}\right )} e^{3}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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