Optimal. Leaf size=214 \[ \frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^2}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}+\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.67, antiderivative size = 210, normalized size of antiderivative = 0.98, number of steps used = 22, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4721, 4661, 3312, 3303, 3299, 3302, 4723, 4406} \[ \frac {\cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3312
Rule 4406
Rule 4661
Rule 4721
Rule 4723
Rubi steps
\begin {align*} \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\int \frac {1-c^2 x^2}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac {(5 c) \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\cos ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac {5 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 (a+b x)}+\frac {\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}-\frac {5 \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{8 (a+b x)}-\frac {\cos (3 x)}{16 (a+b x)}-\frac {\cos (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}+\frac {5 \operatorname {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}-\frac {5 \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (5 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\cos \left (\frac {3 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\left (5 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}+\frac {\left (5 \cos \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}-\frac {\left (5 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^2}+\frac {\left (3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\sin \left (\frac {3 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^2}+\frac {\left (5 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}+\frac {\left (5 \sin \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^2}\\ &=-\frac {x \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^2}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^2}+\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 295, normalized size = 1.38 \[ \frac {2 \cos \left (\frac {a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+9 \cos \left (\frac {3 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \cos \left (\frac {5 a}{b}\right ) \sin ^{-1}(c x) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+2 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+2 b \sin \left (\frac {a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+9 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+9 b \sin \left (\frac {3 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \sin \left (\frac {5 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-16 b c^5 x^5+32 b c^3 x^3-16 b c x}{16 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{2} x^{3} - x\right )} \sqrt {-c^{2} x^{2} + 1}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.11, size = 1215, normalized size = 5.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 341, normalized size = 1.59 \[ \frac {9 \arcsin \left (c x \right ) \Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +9 \arcsin \left (c x \right ) \Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b +2 \arcsin \left (c x \right ) \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +2 \arcsin \left (c x \right ) \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +5 \arcsin \left (c x \right ) \Si \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +5 \arcsin \left (c x \right ) \Ci \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b +9 \Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +9 \Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +2 \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +2 \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +5 \Si \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +5 \Ci \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -2 x b c -3 \sin \left (3 \arcsin \left (c x \right )\right ) b -\sin \left (5 \arcsin \left (c x \right )\right ) b}{16 c^{2} \left (a +b \arcsin \left (c x \right )\right ) b^{2}} \]
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 \[ -\frac {c^{4} x^{5} - 2 \, c^{2} x^{3} + x - \frac {{\left (5 \, c^{4} \int \frac {x^{4}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x} - 6 \, c^{2} \int \frac {x^{2}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x} + \int \frac {1}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x}\right )} {\left (b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c\right )}}{b c}}{b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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