Optimal. Leaf size=331 \[ -\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Chi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{24 b^4 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d} \]
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Rubi [A]
time = 0.48, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5859, 12,
5779, 5818, 5778, 3384, 3379, 3382, 5773, 5819} \begin {gather*} \frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}-\frac {3 e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {(c+d x)^2+1}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5773
Rule 5778
Rule 5779
Rule 5818
Rule 5819
Rule 5859
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \sinh ^{-1}(c+d x)\right )^4} \, dx &=\frac {\text {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \sinh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {x^2}{\left (a+b \sinh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}+\frac {\left (2 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}+\frac {e^2 \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}+\frac {e^2 \text {Subst}\left (\int \frac {1}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )} \, dx,x,c+d x\right )}{3 b^3 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 (a+b x)}+\frac {3 \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{2 b^3 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 b^3 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (3 e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {\left (9 e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (3 e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {\left (9 e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 b^3 d}\\ &=-\frac {e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )}{24 b^4 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{8 b^4 d}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 258, normalized size = 0.78 \begin {gather*} \frac {e^2 \left (-\frac {8 b^3 (c+d x)^2 \sqrt {1+(c+d x)^2}}{\left (a+b \sinh ^{-1}(c+d x)\right )^3}+\frac {4 b^2 \left (-2 (c+d x)-3 (c+d x)^3\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {4 b \sqrt {1+(c+d x)^2} \left (2+9 (c+d x)^2\right )}{a+b \sinh ^{-1}(c+d x)}-80 \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )+80 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+27 \left (3 \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )\right )}{24 b^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(708\) vs.
\(2(307)=614\).
time = 6.63, size = 709, normalized size = 2.14
method | result | size |
derivativedivides | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (9 b^{2} \arcsinh \left (d x +c \right )^{2}+18 a b \arcsinh \left (d x +c \right )-3 b^{2} \arcsinh \left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \arcsinh \left (d x +c \right )^{3}+3 a \,b^{2} \arcsinh \left (d x +c \right )^{2}+3 a^{2} b \arcsinh \left (d x +c \right )+a^{3}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \arcsinh \left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )-b^{2} \arcsinh \left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \arcsinh \left (d x +c \right )^{3}+3 a \,b^{2} \arcsinh \left (d x +c \right )^{2}+3 a^{2} b \arcsinh \left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{3} \left (a +b \arcsinh \left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{3} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(709\) |
default | \(\frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right )^{3}-\sqrt {1+\left (d x +c \right )^{2}}+3 d x +3 c \right ) e^{2} \left (9 b^{2} \arcsinh \left (d x +c \right )^{2}+18 a b \arcsinh \left (d x +c \right )-3 b^{2} \arcsinh \left (d x +c \right )+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \arcsinh \left (d x +c \right )^{3}+3 a \,b^{2} \arcsinh \left (d x +c \right )^{2}+3 a^{2} b \arcsinh \left (d x +c \right )+a^{3}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \arcsinh \left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}-\frac {\left (-\sqrt {1+\left (d x +c \right )^{2}}+d x +c \right ) e^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )-b^{2} \arcsinh \left (d x +c \right )+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \arcsinh \left (d x +c \right )^{3}+3 a \,b^{2} \arcsinh \left (d x +c \right )^{2}+3 a^{2} b \arcsinh \left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}+\frac {e^{2} \left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{48 b^{3} \left (a +b \arcsinh \left (d x +c \right )\right )}+\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}+3 d x +3 c +4 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+\sqrt {1+\left (d x +c \right )^{2}}\right )}{16 b^{3} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d}\) | \(709\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} e^{2} \left (\int \frac {c^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx\right ) \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 (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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