Optimal. Leaf size=309 \[ -\frac {(a+b \text {ArcSin}(c x))^2}{e (d+e x)}-\frac {2 i b c (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 i b c (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.37, antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {4827, 4857,
3404, 2296, 2221, 2317, 2438} \begin {gather*} -\frac {2 i b c (a+b \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 \sqrt {c^2 d^2-e^2}}+\frac {2 i b c (a+b \text {ArcSin}(c x)) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{e (d+e x)}-\frac {2 b^2 c \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3404
Rule 4827
Rule 4857
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \int \frac {a+b \sin ^{-1}(c x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e}\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(2 b c) \text {Subst}\left (\int \frac {a+b x}{c d+e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}+\frac {(4 b c) \text {Subst}\left (\int \frac {e^{i x} (a+b x)}{i e+2 c d e^{i x}-i e e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e}\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac {(4 i b c) \text {Subst}\left (\int \frac {e^{i x} (a+b 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 )}{\sqrt {c^2 d^2-e^2}}+\frac {(4 i b c) \text {Subst}\left (\int \frac {e^{i x} (a+b 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 )}{\sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {\left (2 i b^2 c\right ) \text {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 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 i b^2 c\right ) \text {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 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {\left (2 b^2 c\right ) \text {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 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c\right ) \text {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 \sqrt {c^2 d^2-e^2}}\\ &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.23, size = 230, normalized size = 0.74 \begin {gather*} \frac {-\frac {(a+b \text {ArcSin}(c x))^2}{d+e x}+\frac {2 b c \left (-i (a+b \text {ArcSin}(c x)) \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 )-b \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 (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}}{e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 665 vs. \(2 (319 ) = 638\).
time = 0.97, size = 666, normalized size = 2.16
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +d c \right ) e}-\frac {b^{2} c^{2} \arcsin \left (c x \right )^{2}}{\left (c e x +d c \right ) e}+\frac {2 b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \dilog \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \dilog \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 a b \,c^{2} \arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {2 a b \,c^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\) | \(666\) |
default | \(\frac {-\frac {a^{2} c^{2}}{\left (c e x +d c \right ) e}-\frac {b^{2} c^{2} \arcsin \left (c x \right )^{2}}{\left (c e x +d c \right ) e}+\frac {2 b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \dilog \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i b^{2} c^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \dilog \left (\frac {i d c +e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 a b \,c^{2} \arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {2 a b \,c^{2} \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\) | \(666\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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}}{\left (d + e x\right )^{2}}\, 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}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________