Optimal. Leaf size=249 \[ \frac {e \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{4 b^2 c^3}-\frac {3 e \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b^2 c^3}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.42, antiderivative size = 241, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4667, 4621, 4723, 3303, 3299, 3302, 4631} \[ \frac {e \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b^2 c^3}-\frac {3 e \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2 c^3}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2 c^3}+\frac {d \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac {d \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 4621
Rule 4631
Rule 4667
Rule 4723
Rubi steps
\begin {align*} \int \frac {d+e x^2}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac {d}{\left (a+b \sin ^{-1}(c x)\right )^2}+\frac {e x^2}{\left (a+b \sin ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d \int \frac {1}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx+e \int \frac {x^2}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx\\ &=-\frac {d \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {(c d) \int \frac {x}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b}+\frac {e \operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{4 (a+b x)}+\frac {3 \sin (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {d \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {d \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c}-\frac {e \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac {(3 e) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {d \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (d \cos \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 )}{b c}-\frac {\left (e \cos \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^3}+\frac {\left (3 e \cos \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 )}{4 b c^3}+\frac {\left (d \sin \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 )}{b c}+\frac {\left (e \sin \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^3}-\frac {\left (3 e \sin \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 )}{4 b c^3}\\ &=-\frac {d \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {d \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {e \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c^3}-\frac {3 e \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c^3}-\frac {d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac {e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b^2 c^3}+\frac {3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2 c^3}\\ \end {align*}
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Mathematica [A] time = 1.04, size = 191, normalized size = 0.77 \[ -\frac {-\sin \left (\frac {a}{b}\right ) \left (4 c^2 d+e\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+4 c^2 d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+\frac {4 b c^2 d \sqrt {1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac {4 b c^2 e x^2 \sqrt {1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+3 e \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-3 e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )}{4 b^2 c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e x^{2} + d}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.82, size = 905, normalized size = 3.63 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 367, normalized size = 1.47 \[ \frac {-4 \arcsin \left (c x \right ) \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b \,c^{2} d +4 \arcsin \left (c x \right ) \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b \,c^{2} d -4 \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a \,c^{2} d +4 \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a \,c^{2} d +3 \arcsin \left (c x \right ) \Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b e -3 \arcsin \left (c x \right ) \Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b e -\arcsin \left (c x \right ) \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b e +\arcsin \left (c x \right ) \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b e -4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d +3 \Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a e -3 \Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a e -\Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a e +\Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a e +\cos \left (3 \arcsin \left (c x \right )\right ) b e -\sqrt {-c^{2} x^{2}+1}\, b e}{4 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {e\,x^2+d}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {d + e x^{2}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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