Optimal. Leaf size=344 \[ \frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \text {ArcSin}(c x)^2}{2 e^2}+\frac {g x (a+b \text {ArcSin}(c x))}{e}+\frac {b (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b (e f-d g) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {b (e f-d g) \text {ArcSin}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) (a+b \text {ArcSin}(c x)) \log (d+e x)}{e^2}-\frac {i b (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2} \]
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Rubi [A]
time = 0.48, antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 12, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {45, 4837, 12,
6874, 267, 222, 2451, 4825, 4615, 2221, 2317, 2438} \begin {gather*} \frac {(e f-d g) \log (d+e x) (a+b \text {ArcSin}(c x))}{e^2}+\frac {g x (a+b \text {ArcSin}(c x))}{e}-\frac {i b (e f-d g) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}-\frac {i b (e f-d g) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b \text {ArcSin}(c x) (e f-d g) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^2}+\frac {b \text {ArcSin}(c x) (e f-d g) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^2}-\frac {i b \text {ArcSin}(c x)^2 (e f-d g)}{2 e^2}-\frac {b \text {ArcSin}(c x) (e f-d g) \log (d+e x)}{e^2}+\frac {b g \sqrt {1-c^2 x^2}}{c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 222
Rule 267
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 4615
Rule 4825
Rule 4837
Rule 6874
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \sin ^{-1}(c x)\right )}{d+e x} \, dx &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-(b c) \int \frac {e g x+(e f-d g) \log (d+e x)}{e^2 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b c) \int \frac {e g x+(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^2}\\ &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b c) \int \left (\frac {e g x}{\sqrt {1-c^2 x^2}}+\frac {(e f-d g) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e^2}\\ &=\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b c g) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{e}-\frac {(b c (e f-d g)) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^2}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(b c (e f-d g)) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(b c (e f-d g)) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(b c (e f-d g)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}+\frac {(b c (e f-d g)) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {b (e f-d g) \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}+\frac {b (e f-d g) \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}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {(b (e f-d g)) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2}-\frac {(b (e f-d g)) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^2}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {b (e f-d g) \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}+\frac {b (e f-d g) \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}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}+\frac {(i b (e f-d g)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2}+\frac {(i b (e f-d g)) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^2}\\ &=\frac {b g \sqrt {1-c^2 x^2}}{c e}-\frac {i b (e f-d g) \sin ^{-1}(c x)^2}{2 e^2}+\frac {g x \left (a+b \sin ^{-1}(c x)\right )}{e}+\frac {b (e f-d g) \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}+\frac {b (e f-d g) \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}-\frac {b (e f-d g) \sin ^{-1}(c x) \log (d+e x)}{e^2}+\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^2}-\frac {i b (e f-d g) \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}-\frac {i b (e f-d g) \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}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 602, normalized size = 1.75 \begin {gather*} \frac {8 a c e g x+8 a c (e f-d g) \log (d+e x)+b g \left (8 e \sqrt {1-c^2 x^2}+8 c e x \text {ArcSin}(c x)-c d \left (i (\pi -2 \text {ArcSin}(c x))^2-32 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {(c d-e) \cot \left (\frac {1}{4} (\pi +2 \text {ArcSin}(c x))\right )}{\sqrt {c^2 d^2-e^2}}\right )-4 \left (\pi +4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1-\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )-4 \left (\pi -4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {c d}{e}}}{\sqrt {2}}\right )-2 \text {ArcSin}(c x)\right ) \log \left (1+\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+4 (\pi -2 \text {ArcSin}(c x)) \log (c (d+e x))+8 \text {ArcSin}(c x) \log (c (d+e x))+8 i \left (\text {PolyLog}\left (2,\frac {i \left (-c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )+\text {PolyLog}\left (2,-\frac {i \left (c d+\sqrt {c^2 d^2-e^2}\right ) e^{-i \text {ArcSin}(c x)}}{e}\right )\right )\right )\right )-4 i b c e f \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+\log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )\right )+2 \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{8 c e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1592 vs. \(2 (357 ) = 714\).
time = 1.32, size = 1593, normalized size = 4.63
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1593\) |
default | \(\text {Expression too large to display}\) | \(1593\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right ) \left (f + g x\right )}{d + e x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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