Optimal. Leaf size=347 \[ -\frac {i (a+b \text {ArcSin}(c x))^3}{3 b e}+\frac {(a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 b^2 \text {PolyLog}\left (3,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 b^2 \text {PolyLog}\left (3,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.35, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4825, 4615,
2221, 2611, 2320, 6724} \begin {gather*} -\frac {2 i b (a+b \text {ArcSin}(c x)) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b (a+b \text {ArcSin}(c x)) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {(a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e}-\frac {i (a+b \text {ArcSin}(c x))^3}{3 b e}+\frac {2 b^2 \text {Li}_3\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 b^2 \text {Li}_3\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 4615
Rule 4825
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d+e x} \, dx &=\text {Subst}\left (\int \frac {(a+b x)^2 \cos (x)}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b e}+\text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{c d-\sqrt {c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )+\text {Subst}\left (\int \frac {e^{i x} (a+b x)^2}{c d+\sqrt {c^2 d^2-e^2}-i e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-\frac {i e e^{i x}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e}-\frac {(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-\frac {i e e^{i x}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\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}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\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}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {i e e^{i x}}{c d-\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (\frac {i e e^{i x}}{c d+\sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\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}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\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}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i e x}{c d-\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i e x}{c d+\sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e}\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^3}{3 b e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\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}-\frac {2 i b \left (a+b \sin ^{-1}(c x)\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}+\frac {2 b^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {2 b^2 \text {Li}_3\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 330, normalized size = 0.95 \begin {gather*} \frac {-\frac {i (a+b \text {ArcSin}(c x))^3}{b}+3 (a+b \text {ArcSin}(c x))^2 \log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+3 (a+b \text {ArcSin}(c x))^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+6 b \left (-i (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+b \text {PolyLog}\left (3,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )\right )+6 b \left (-i (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+b \text {PolyLog}\left (3,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{3 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (c x \right )\right )^{2}}{e x +d}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )^{2}}{d + e x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________