Optimal. Leaf size=483 \[ -\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 (a+b \text {ArcSin}(c x))^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}} \]
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Rubi [A]
time = 0.58, antiderivative size = 483, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 11, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {4789, 4745,
4765, 3800, 2221, 2317, 2438, 4769, 4504, 4268, 270} \begin {gather*} -\frac {4 c^2 (a+b \text {ArcSin}(c x))^2}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4268
Rule 4504
Rule 4745
Rule 4765
Rule 4769
Rule 4789
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{3/2}} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}+\frac {1}{3} \left (4 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {1}{3} \left (8 c^4\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c^5 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (32 i b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (8 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {\left (16 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {\left (8 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1-c^2 x^2\right )}{3 d x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 d x^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d-c^2 d x^2}}-\frac {4 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {8 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \sqrt {d-c^2 d x^2}}-\frac {20 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}+\frac {16 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {5 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 462, normalized size = 0.96 \begin {gather*} \frac {-a^2-4 a^2 c^2 x^2-b^2 c^2 x^2+8 a^2 c^4 x^4+b^2 c^4 x^4-a b c x \sqrt {1-c^2 x^2}-2 a b \text {ArcSin}(c x)-8 a b c^2 x^2 \text {ArcSin}(c x)+16 a b c^4 x^4 \text {ArcSin}(c x)-b^2 c x \sqrt {1-c^2 x^2} \text {ArcSin}(c x)-b^2 \text {ArcSin}(c x)^2-4 b^2 c^2 x^2 \text {ArcSin}(c x)^2+8 b^2 c^4 x^4 \text {ArcSin}(c x)^2-8 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2+10 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+6 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )+10 a b c^3 x^3 \sqrt {1-c^2 x^2} \log (c x)+3 a b c^3 x^3 \sqrt {1-c^2 x^2} \log \left (1-c^2 x^2\right )-3 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-5 i b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 d x^3 \sqrt {d-c^2 d x^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. 2843 vs. \(2 (470 ) = 940\).
time = 0.58, size = 2844, normalized size = 5.89
method | result | size |
default | \(\text {Expression too large to display}\) | \(2844\) |
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 )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, 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 (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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