Optimal. Leaf size=331 \[ \frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}-\frac {3 e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {(c+d x)^2+1}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3} \]
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Rubi [A] time = 0.67, antiderivative size = 327, normalized size of antiderivative = 0.99, 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 = {5865, 12, 5667, 5774, 5665, 3303, 3298, 3301, 5655, 5779} \[ \frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{8 b^4 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c+d x)\right )}{8 b^4 d}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {3 e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{2 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {(c+d x)^2+1}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5655
Rule 5665
Rule 5667
Rule 5774
Rule 5779
Rule 5865
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \sinh ^{-1}(c+d x)\right )^4} \, dx &=\frac {\operatorname {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 \operatorname {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 ) \operatorname {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 \operatorname {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 \operatorname {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 ) \operatorname {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 \operatorname {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 ) \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.84, size = 258, normalized size = 0.78 \[ \frac {e^2 \left (-\frac {8 b^3 (c+d x)^2 \sqrt {(c+d x)^2+1}}{\left (a+b \sinh ^{-1}(c+d x)\right )^3}+\frac {4 b^2 \left (-3 (c+d x)^3-2 (c+d x)\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}+27 \left (3 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )-3 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )-80 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+80 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-\frac {4 b \sqrt {(c+d x)^2+1} \left (9 (c+d x)^2+2\right )}{a+b \sinh ^{-1}(c+d x)}\right )}{24 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}{b^{4} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arsinh}\left (d x + c\right ) + a^{4}}, x\right ) \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 709, normalized size = 2.14 \[ \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 \arcsinh \left (d x +c \right ) b^{2}+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}} \Ei \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 )-\arcsinh \left (d x +c \right ) b^{2}+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}} \Ei \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}} \Ei \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}} \Ei \left (1, -3 \arcsinh \left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \]
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 \[ e^{2} \left (\int \frac {c^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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