Optimal. Leaf size=309 \[ -\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)}-\frac {2 b^2 c \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.53, 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 = {4743, 4773, 3323, 2264, 2190, 2279, 2391} \[ -\frac {2 b^2 c \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 i b c \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{e (d+e x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3323
Rule 4743
Rule 4773
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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \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 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 i b^2 c\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 \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 ) \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 \sqrt {c^2 d^2-e^2}}-\frac {\left (2 b^2 c\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 \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 = 0.34, size = 231, normalized size = 0.75 \[ \frac {-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}+\frac {2 b c \left (-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )\right )-b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.70, size = 646, normalized size = 2.09 \[ -\frac {c \,a^{2}}{\left (c e x +d c \right ) e}-\frac {c \,b^{2} \arcsin \left (c x \right )^{2}}{e \left (c e x +d c \right )}+\frac {2 c \,b^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 c \,b^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \arcsin \left (c x \right ) \ln \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {2 i c \,b^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \dilog \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e -\sqrt {-c^{2} d^{2}+e^{2}}}{i d c -\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 i c \,b^{2} \sqrt {-c^{2} d^{2}+e^{2}}\, \dilog \left (\frac {i d c +\left (i c x +\sqrt {-c^{2} x^{2}+1}\right ) e +\sqrt {-c^{2} d^{2}+e^{2}}}{i d c +\sqrt {-c^{2} d^{2}+e^{2}}}\right )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {2 c a b \arcsin \left (c x \right )}{\left (c e x +d c \right ) e}-\frac {2 c a b \ln \left (\frac {-\frac {2 \left (c^{2} d^{2}-e^{2}\right )}{e^{2}}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {-\left (c x +\frac {d c}{e}\right )^{2}+\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{e^{2} \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________