Optimal. Leaf size=81 \[ \frac {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}+\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {14, 4687, 12, 446, 78, 63, 208} \[ \frac {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}+\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 63
Rule 78
Rule 208
Rule 446
Rule 4687
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d 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 {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-(b c) \int \frac {d \left (-1+3 c^2 x^2\right )}{3 x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{3} (b c d) \int \frac {-1+3 c^2 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 {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{6} (b c d) \operatorname {Subst}\left (\int \frac {-1+3 c^2 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 {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{12} \left (5 b c^3 d\right ) \operatorname {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 {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {1}{6} (5 b c d) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 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 {c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 93, normalized size = 1.15 \[ \frac {a c^2 d}{x}-\frac {a d}{3 x^3}-\frac {b c d \sqrt {1-c^2 x^2}}{6 x^2}+\frac {b c^2 d \sin ^{-1}(c x)}{x}+\frac {5}{6} b c^3 d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {b d \sin ^{-1}(c x)}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 109, normalized size = 1.35 \[ \frac {5 \, b c^{3} d x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - 5 \, b c^{3} d x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 12 \, a c^{2} d x^{2} - 2 \, \sqrt {-c^{2} x^{2} + 1} b c d x - 4 \, a d + 4 \, {\left (3 \, b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.09, size = 296, normalized size = 3.65 \[ -\frac {b c^{6} d x^{3} \arcsin \left (c x\right )}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {a c^{6} d x^{3}}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {b c^{5} d x^{2}}{24 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {3 \, b c^{4} d x \arcsin \left (c x\right )}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} + \frac {3 \, a c^{4} d x}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {5}{6} \, b c^{3} d \log \left ({\left | c \right |} {\left | x \right |}\right ) + \frac {5}{6} \, b c^{3} d \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) + \frac {3 \, b c^{2} d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )} \arcsin \left (c x\right )}{8 \, x} + \frac {3 \, a c^{2} d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}}{8 \, x} - \frac {b c d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}}{24 \, x^{2}} - \frac {b d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3} \arcsin \left (c x\right )}{24 \, x^{3}} - \frac {a d {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}{24 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 1.12 \[ c^{3} \left (-d a \left (\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )-d b \left (\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arcsin \left (c x \right )}{c x}-\frac {5 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 123, normalized size = 1.52 \[ {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b c^{2} d - \frac {1}{6} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac {2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d + \frac {a c^{2} d}{x} - \frac {a d}{3 \, x^{3}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.01, size = 178, normalized size = 2.20 \[ \frac {a c^{2} d}{x} - \frac {a d}{3 x^{3}} - b c^{3} d \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) + \frac {b c^{2} d \operatorname {asin}{\left (c x \right )}}{x} + \frac {b c d \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} - \frac {c \sqrt {-1 + \frac {1}{c^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c}{2 x \sqrt {1 - \frac {1}{c^{2} x^{2}}}} + \frac {i}{2 c x^{3} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{3} - \frac {b d \operatorname {asin}{\left (c x \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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