Optimal. Leaf size=278 \[ \frac {3 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (\frac {7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^4}-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.89, antiderivative size = 274, normalized size of antiderivative = 0.99, number of steps used = 28, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4721, 4723, 4406, 3303, 3299, 3302} \[ \frac {3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rule 4721
Rule 4723
Rubi steps
\begin {align*} \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac {(7 c) \int \frac {x^4 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac {7 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \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^4}-\frac {7 \operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{64 (a+b x)}-\frac {3 \cos (3 x)}{64 (a+b x)}-\frac {\cos (5 x)}{64 (a+b x)}+\frac {\cos (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {7 \operatorname {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {7 \operatorname {Subst}\left (\int \frac {\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac {21 \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {21 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (21 \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 )}{64 b c^4}+\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 )}{8 b c^4}-\frac {\left (3 \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^4}+\frac {\left (21 \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 )}{64 b c^4}+\frac {\left (7 \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 )}{64 b c^4}-\frac {\left (3 \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^4}-\frac {\left (7 \cos \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (21 \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 )}{64 b c^4}+\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 )}{8 b c^4}-\frac {\left (3 \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^4}+\frac {\left (21 \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 )}{64 b c^4}+\frac {\left (7 \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 )}{64 b c^4}-\frac {\left (3 \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^4}-\frac {\left (7 \sin \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}\\ \end {align*}
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Mathematica [A] time = 1.22, size = 399, normalized size = 1.44 \[ -\frac {-3 \cos \left (\frac {a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-9 \cos \left (\frac {3 a}{b}\right ) \left (a+b \sin ^{-1}(c x)\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \cos \left (\frac {5 a}{b}\right ) \sin ^{-1}(c x) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 a \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 b \cos \left (\frac {7 a}{b}\right ) \sin ^{-1}(c x) \text {Ci}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-3 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-3 b \sin \left (\frac {a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-9 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-9 b \sin \left (\frac {3 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 a \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 b \sin \left (\frac {5 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 a \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+7 b \sin \left (\frac {7 a}{b}\right ) \sin ^{-1}(c x) \text {Si}\left (7 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+64 b c^7 x^7-128 b c^5 x^5+64 b c^3 x^3}{64 b^2 c^4 \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{2} x^{5} - x^{3}\right )} \sqrt {-c^{2} x^{2} + 1}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.12, size = 2065, normalized size = 7.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 455, normalized size = 1.64 \[ -\frac {7 \arcsin \left (c x \right ) \Ci \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) b -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 +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 -3 \arcsin \left (c x \right ) \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -3 \arcsin \left (c x \right ) \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +7 \arcsin \left (c x \right ) \Si \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b +7 \Ci \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a -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 +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 -3 \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -3 \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +7 \Si \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +3 x b c -\sin \left (7 \arcsin \left (c x \right )\right ) b +3 \sin \left (3 \arcsin \left (c x \right )\right ) b -\sin \left (5 \arcsin \left (c x \right )\right ) b}{64 c^{4} \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 \[ -\frac {c^{4} x^{7} - 2 \, c^{2} x^{5} + x^{3} - \frac {{\left (7 \, c^{4} \int \frac {x^{6}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x} - 10 \, c^{2} \int \frac {x^{4}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x} + 3 \, \int \frac {x^{2}}{b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a}\,{d x}\right )} {\left (b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c\right )}}{b c}}{b^{2} c \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c} \]
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 \[ \int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x^{3} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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