Optimal. Leaf size=520 \[ \frac {2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}-\frac {2 b^2 h x}{e} \]
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Rubi [A] time = 1.64, antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 18, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {683, 4757, 6742, 261, 725, 204, 4799, 1654, 12, 4797, 4677, 8, 4773, 3323, 2264, 2190, 2279, 2391} \[ -\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \sin ^{-1}(c x) \left (e^2 f-d^2 h\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}-\frac {2 b^2 h x}{e} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 204
Rule 261
Rule 683
Rule 725
Rule 1654
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3323
Rule 4677
Rule 4757
Rule 4773
Rule 4797
Rule 4799
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (e f+2 d h x+e h x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}-(2 b c) \int \frac {\left (\frac {h x}{e}-\frac {f-\frac {d^2 h}{e^2}}{d+e x}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}-(2 b c) \int \left (\frac {a \left (-e^2 f+d^2 h+d e h x+e^2 h x^2\right )}{e^2 (d+e x) \sqrt {1-c^2 x^2}}+\frac {b \left (-e^2 f+d^2 h+d e h x+e^2 h x^2\right ) \sin ^{-1}(c x)}{e^2 (d+e x) \sqrt {1-c^2 x^2}}\right ) \, dx\\ &=\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}-\frac {(2 a b c) \int \frac {-e^2 f+d^2 h+d e h x+e^2 h x^2}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2}-\frac {\left (2 b^2 c\right ) \int \frac {\left (-e^2 f+d^2 h+d e h x+e^2 h x^2\right ) \sin ^{-1}(c x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2}\\ &=\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {(2 a b) \int \frac {c^2 e^2 \left (e^2 f-d^2 h\right )}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{c e^4}-\frac {\left (2 b^2 c\right ) \int \left (\frac {e h x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}}+\frac {\left (-e^2 f+d^2 h\right ) \sin ^{-1}(c x)}{(d+e x) \sqrt {1-c^2 x^2}}\right ) \, dx}{e^2}\\ &=\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}-\frac {\left (2 b^2 c h\right ) \int \frac {x \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{e}+\frac {\left (2 a b c \left (e^2 f-d^2 h\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2}-\frac {\left (2 b^2 c \left (-e^2 f+d^2 h\right )\right ) \int \frac {\sin ^{-1}(c x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^2}\\ &=\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}-\frac {\left (2 b^2 h\right ) \int 1 \, dx}{e}-\frac {\left (2 a b c \left (e^2 f-d^2 h\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{e^2}-\frac {\left (2 b^2 c \left (-e^2 f+d^2 h\right )\right ) \operatorname {Subst}\left (\int \frac {x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}\\ &=-\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 b^2 c \left (-e^2 f+d^2 h\right )\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^2}\\ &=-\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (4 i b^2 c \left (e^2 f-d^2 h\right )\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c d-2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {\left (4 i b^2 c \left (e^2 f-d^2 h\right )\right ) \operatorname {Subst}\left (\int \frac {e^{i x} x}{2 c d+2 \sqrt {c^2 d^2-e^2}-2 i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e \sqrt {c^2 d^2-e^2}}\\ &=-\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 i b^2 c \left (e^2 f-d^2 h\right )\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 i b^2 c \left (e^2 f-d^2 h\right )\right ) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e e^{i x}}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c \left (e^2 f-d^2 h\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c \left (e^2 f-d^2 h\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {2 b^2 h x}{e}+\frac {2 a b h \sqrt {1-c^2 x^2}}{c e}+\frac {2 b^2 h \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{c e}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {2 a b c \left (e^2 f-d^2 h\right ) \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 c \left (e^2 f-d^2 h\right ) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \left (e^2 f-d^2 h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2 \sqrt {c^2 d^2-e^2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 307, normalized size = 0.59 \[ \frac {2 b c \left (e^2 f-d^2 h\right ) \left (-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )\right )-b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 b h \left (b x-\frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )}{e}-\frac {\left (f-\frac {d^2 h}{e^2}\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {h x \left (a+b \sin ^{-1}(c x)\right )^2}{e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} e h x^{2} + 2 \, a^{2} d h x + a^{2} e f + {\left (b^{2} e h x^{2} + 2 \, b^{2} d h x + b^{2} e f\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b e h x^{2} + 2 \, a b d h x + a b e f\right )} \arcsin \left (c x\right )}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e h x^{2} + 2 \, d h x + e f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.15, size = 1399, normalized size = 2.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (e\,h\,x^2+2\,d\,h\,x+e\,f\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (2 d h x + e f + e h x^{2}\right )}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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