Optimal. Leaf size=179 \[ \frac {e \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{4 b c^3}-\frac {e \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b c^3}+\frac {e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{4 b c^3}-\frac {e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{4 b c^3}+\frac {d \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c} \]
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Rubi [A] time = 0.34, antiderivative size = 175, normalized size of antiderivative = 0.98, 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, 4623, 3303, 3299, 3302, 4635, 4406} \[ \frac {e \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b c^3}-\frac {e \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b c^3}+\frac {e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b c^3}-\frac {e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b c^3}+\frac {d \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rule 4623
Rule 4635
Rule 4667
Rubi steps
\begin {align*} \int \frac {d+e x^2}{a+b \sin ^{-1}(c x)} \, dx &=\int \left (\frac {d}{a+b \sin ^{-1}(c x)}+\frac {e x^2}{a+b \sin ^{-1}(c x)}\right ) \, dx\\ &=d \int \frac {1}{a+b \sin ^{-1}(c x)} \, dx+e \int \frac {x^2}{a+b \sin ^{-1}(c x)} \, dx\\ &=\frac {d \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}+\frac {e \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^3}\\ &=\frac {e \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 (a+b x)}-\frac {\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3}+\frac {\left (d \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}+\frac {\left (d \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}\\ &=\frac {d \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac {e \operatorname {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}-\frac {e \operatorname {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3}\\ &=\frac {d \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac {\left (e \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 c^3}-\frac {\left (e \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 )}{4 c^3}+\frac {\left (e \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 c^3}-\frac {\left (e \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 )}{4 c^3}\\ &=\frac {e \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b c^3}-\frac {e \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b c^3}+\frac {d \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}+\frac {e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b c^3}-\frac {e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b c^3}+\frac {d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b c}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 125, normalized size = 0.70 \[ \frac {\cos \left (\frac {a}{b}\right ) \left (4 c^2 d+e\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+4 c^2 d \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-e \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+e \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-e \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )}{4 b c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e x^{2} + d}{b \arcsin \left (c x\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.98, size = 235, normalized size = 1.31 \[ \frac {d \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} - \frac {\cos \left (\frac {a}{b}\right )^{3} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) e}{b c^{3}} - \frac {\cos \left (\frac {a}{b}\right )^{2} e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {d \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{b c} + \frac {3 \, \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) e}{4 \, b c^{3}} + \frac {\cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) e}{4 \, b c^{3}} + \frac {e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {e \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{4 \, b c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 142, normalized size = 0.79 \[ -\frac {-4 \Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) c^{2} d -4 \Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) c^{2} d +\Si \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) e +\Ci \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) e -\Si \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) e -\Ci \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) e}{4 c^{3} b} \]
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 \[ \int \frac {e x^{2} + d}{b \arcsin \left (c x\right ) + a}\,{d x} \]
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 \[ \int \frac {e\,x^2+d}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \]
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 \[ \int \frac {d + e x^{2}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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