Optimal. Leaf size=85 \[ -\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d (a+b \text {ArcSin}(c x))}{3 x^3}-\frac {e (a+b \text {ArcSin}(c x))}{x}-\frac {1}{6} b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {14, 4815, 12,
457, 79, 65, 214} \begin {gather*} -\frac {d (a+b \text {ArcSin}(c x))}{3 x^3}-\frac {e (a+b \text {ArcSin}(c x))}{x}-\frac {1}{6} b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 65
Rule 79
Rule 214
Rule 457
Rule 4815
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{x}-(b c) \int \frac {-d-3 e x^2}{3 x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{3} (b c) \int \frac {-d-3 e x^2}{x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {-d-3 e x}{x^2 \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {1}{12} \left (b c \left (c^2 d+6 e\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {\left (b \left (c^2 d+6 e\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{6 c}\\ &=-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{6} b c \left (c^2 d+6 e\right ) \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 109, normalized size = 1.28 \begin {gather*} -\frac {a d}{3 x^3}-\frac {a e}{x}-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}-\frac {b d \text {ArcSin}(c x)}{3 x^3}-\frac {b e \text {ArcSin}(c x)}{x}-\frac {1}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-b c e \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 120, normalized size = 1.41
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\arcsin \left (c x \right ) d}{3 c \,x^{3}}-\frac {\arcsin \left (c x \right ) e}{c x}+\frac {d \,c^{2} \left (-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c^{2} x^{2}}-\frac {\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2}\right )}{3}-e \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(120\) |
default | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\arcsin \left (c x \right ) d}{3 c \,x^{3}}-\frac {\arcsin \left (c x \right ) e}{c x}+\frac {d \,c^{2} \left (-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c^{2} x^{2}}-\frac {\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2}\right )}{3}-e \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 121, normalized size = 1.42 \begin {gather*} -\frac {1}{6} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac {2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d - {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b e - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.20, size = 125, normalized size = 1.47 \begin {gather*} -\frac {2 \, \sqrt {-c^{2} x^{2} + 1} b c d x + 12 \, a x^{2} e + 4 \, a d + 4 \, {\left (3 \, b x^{2} e + b d\right )} \arcsin \left (c x\right ) + {\left (b c^{3} d x^{3} + 6 \, b c x^{3} e\right )} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - {\left (b c^{3} d x^{3} + 6 \, b c x^{3} e\right )} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right )}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.21, size = 168, normalized size = 1.98 \begin {gather*} - \frac {a d}{3 x^{3}} - \frac {a e}{x} + \frac {b c d \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} + \frac {c}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} - \frac {1}{2 c x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{3} + b c e \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d \operatorname {asin}{\left (c x \right )}}{3 x^{3}} - \frac {b e \operatorname {asin}{\left (c x \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 424 vs.
\(2 (75) = 150\).
time = 121.54, size = 424, normalized size = 4.99 \begin {gather*} -\frac {b c^{6} d x^{3} \arcsin \left (c x\right )}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {a c^{6} d x^{3}}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {b c^{5} d x^{2}}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {b c^{4} d x \arcsin \left (c x\right )}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {a c^{4} d x}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} + \frac {1}{6} \, b c^{3} d \log \left ({\left | c \right |} {\left | x \right |}\right ) - \frac {1}{6} \, b c^{3} d \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - \frac {b c^{2} e x \arcsin \left (c x\right )}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c^{2} d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )} \arcsin \left (c x\right )}{8 \, x} - \frac {a c^{2} e x}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {a c^{2} d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}}{8 \, x} + b c e \log \left ({\left | c \right |} {\left | x \right |}\right ) - b c e \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - \frac {b c d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}}{24 \, x^{2}} - \frac {b e {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )} \arcsin \left (c x\right )}{2 \, x} - \frac {b d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3} \arcsin \left (c x\right )}{24 \, x^{3}} - \frac {a e {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}}{2 \, x} - \frac {a d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}{24 \, x^{3}} \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 {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (e\,x^2+d\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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