Optimal. Leaf size=498 \[ \frac {e^2 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \sin \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac {d e \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}-\frac {d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.76, antiderivative size = 486, normalized size of antiderivative = 0.98, number of steps used = 26, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4667, 4621, 4723, 3303, 3299, 3302, 4631} \[ \frac {d e \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{2 b^2 c^3}-\frac {3 d e \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b^2 c^3}+\frac {e^2 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^5}-\frac {9 e^2 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^5}+\frac {5 e^2 \sin \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^5}-\frac {d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{2 b^2 c^3}+\frac {3 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b^2 c^3}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^5}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^5}-\frac {5 e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^5}+\frac {d^2 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac {d^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )} \]
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 {\left (d+e x^2\right )^2}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac {d^2}{\left (a+b \sin ^{-1}(c x)\right )^2}+\frac {2 d e x^2}{\left (a+b \sin ^{-1}(c x)\right )^2}+\frac {e^2 x^4}{\left (a+b \sin ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d^2 \int \frac {1}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx+(2 d e) \int \frac {x^2}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx+e^2 \int \frac {x^4}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx\\ &=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (c d^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b}+\frac {(2 d 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 {e^2 \operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{8 (a+b x)}+\frac {9 \sin (3 x)}{16 (a+b x)}-\frac {5 \sin (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^5}\\ &=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c}-\frac {(d e) \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c^3}+\frac {(3 d e) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{2 b c^3}-\frac {e^2 \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^5}-\frac {\left (5 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^5}+\frac {\left (9 e^2\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^5}\\ &=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (d^2 \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 (d 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 )}{2 b c^3}-\frac {\left (e^2 \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 )}{8 b c^5}+\frac {\left (3 d 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 )}{2 b c^3}+\frac {\left (9 e^2 \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 )}{16 b c^5}-\frac {\left (5 e^2 \cos \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^5}+\frac {\left (d^2 \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 (d 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 )}{2 b c^3}+\frac {\left (e^2 \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 )}{8 b c^5}-\frac {\left (3 d 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 )}{2 b c^3}-\frac {\left (9 e^2 \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 )}{16 b c^5}+\frac {\left (5 e^2 \sin \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^5}\\ &=-\frac {d^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1-c^2 x^2}}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {d^2 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{b^2 c}+\frac {d e \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{2 b^2 c^3}+\frac {e^2 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{8 b^2 c^5}-\frac {3 d e \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{2 b^2 c^3}-\frac {9 e^2 \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right ) \sin \left (\frac {5 a}{b}\right )}{16 b^2 c^5}-\frac {d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b^2 c}-\frac {d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{2 b^2 c^3}-\frac {e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b^2 c^5}+\frac {3 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{2 b^2 c^3}+\frac {9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b^2 c^5}-\frac {5 e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b^2 c^5}\\ \end {align*}
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Mathematica [A] time = 2.25, size = 359, normalized size = 0.72 \[ -\frac {16 c^4 d^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+3 e \sin \left (\frac {3 a}{b}\right ) \left (8 c^2 d+3 e\right ) \text {Ci}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+8 c^2 d e \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-24 c^2 d e \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )-2 \sin \left (\frac {a}{b}\right ) \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )+\frac {16 b c^4 d^2 \sqrt {1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac {32 b c^4 d e x^2 \sqrt {1-c^2 x^2}}{a+b \sin ^{-1}(c x)}+\frac {16 b c^4 e^2 x^4 \sqrt {1-c^2 x^2}}{a+b \sin ^{-1}(c x)}-5 e^2 \sin \left (\frac {5 a}{b}\right ) \text {Ci}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+2 e^2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )-9 e^2 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )+5 e^2 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\sin ^{-1}(c x)\right )\right )}{16 b^2 c^5} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}{b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.34, size = 2324, normalized size = 4.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 795, normalized size = 1.60 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {\left (d + e x^{2}\right )^{2}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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