Optimal. Leaf size=340 \[ -\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {8 e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^3 \sqrt {(c+d x)^2+1} (c+d x)}{b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3} \]
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Rubi [A] time = 0.70, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5865, 12, 5667, 5774, 5665, 3303, 3298, 3301} \[ -\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {8 e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^3 \sqrt {(c+d x)^2+1} (c+d x)}{b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5665
Rule 5667
Rule 5774
Rule 5865
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^3}{\left (a+b \sinh ^{-1}(c+d x)\right )^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \sinh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b \sinh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}+\frac {e^3 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{b d}+\frac {\left (4 e^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}+\frac {e^3 \operatorname {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (8 e^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^3 (c+d x) \sqrt {1+(c+d x)^2}}{b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {8 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e^3 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^3 d}+\frac {\left (8 e^3\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 (a+b x)}+\frac {\cosh (4 x)}{2 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^3 (c+d x) \sqrt {1+(c+d x)^2}}{b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {8 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {\left (4 e^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (4 e^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^3 d}-\frac {\left (e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^3 d}\\ &=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^3 (c+d x) \sqrt {1+(c+d x)^2}}{b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {8 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{b^4 d}-\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{b^4 d}-\frac {\left (4 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (4 e^3 \cosh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (4 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (4 e^3 \sinh \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e^3 (c+d x) \sqrt {1+(c+d x)^2}}{b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {8 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{3 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c+d x)\right )}{3 b^4 d}\\ \end {align*}
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Mathematica [A] time = 1.11, size = 318, normalized size = 0.94 \[ \frac {e^3 \left (-\frac {2 b^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{\left (a+b \sinh ^{-1}(c+d x)\right )^3}+\frac {b^2 \left (-4 (c+d x)^4-3 (c+d x)^2\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}+30 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )-\log \left (a+b \sinh ^{-1}(c+d x)\right )\right )+8 \left (-4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )+4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )+3 \log \left (a+b \sinh ^{-1}(c+d x)\right )\right )-\frac {2 b \sqrt {(c+d x)^2+1} \left (8 (c+d x)^3+3 (c+d x)\right )}{a+b \sinh ^{-1}(c+d x)}+6 \log \left (a+b \sinh ^{-1}(c+d x)\right )\right )}{6 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}{b^{4} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arsinh}\left (d x + c\right ) + a^{4}}, x\right ) \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.35, size = 800, normalized size = 2.35 \[ \frac {\frac {\left (8 \left (d x +c \right )^{4}-8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+8 \left (d x +c \right )^{2}-4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right ) e^{3} \left (8 b^{2} \arcsinh \left (d x +c \right )^{2}+16 a b \arcsinh \left (d x +c \right )-2 \arcsinh \left (d x +c \right ) b^{2}+8 a^{2}-2 a b +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 {2 e^{3} {\mathrm e}^{\frac {4 a}{b}} \Ei \left (1, 4 \arcsinh \left (d x +c \right )+\frac {4 a}{b}\right )}{3 b^{4}}-\frac {\left (2 \left (d x +c \right )^{2}-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right ) e^{3} \left (2 b^{2} \arcsinh \left (d x +c \right )^{2}+4 a b \arcsinh \left (d x +c \right )-\arcsinh \left (d x +c \right ) b^{2}+2 a^{2}-a b +b^{2}\right )}{24 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^{3} {\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (d x +c \right )+\frac {2 a}{b}\right )}{6 b^{4}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{24 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}+\frac {e^{3} \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{12 b^{3} \left (a +b \arcsinh \left (d x +c \right )\right )}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (d x +c \right )-\frac {2 a}{b}\right )}{6 b^{4}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{48 b \left (a +b \arcsinh \left (d x +c \right )\right )^{3}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{24 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {e^{3} \left (8 \left (d x +c \right )^{4}+8 \left (d x +c \right )^{2}+8 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+4 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right )}{6 b^{3} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {2 e^{3} {\mathrm e}^{-\frac {4 a}{b}} \Ei \left (1, -4 \arcsinh \left (d x +c \right )-\frac {4 a}{b}\right )}{3 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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4} \,d x \]
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 \[ e^{3} \left (\int \frac {c^{3}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{4} + 4 a^{3} b \operatorname {asinh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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