Optimal. Leaf size=156 \[ -\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^3 d}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d} \]
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Rubi [A]
time = 0.22, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5859, 12,
5779, 5818, 5780, 5556, 3384, 3379, 3382, 5783} \begin {gather*} -\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{b^3 d}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e \sqrt {(c+d x)^2+1} (c+d x)}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5779
Rule 5780
Rule 5783
Rule 5818
Rule 5859
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \sinh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}+\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}+\frac {e \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {(2 e) \text {Subst}\left (\int \frac {x}{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {(2 e) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {(2 e) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\left (e \cosh \left (\frac {2 a}{b}\right )\right ) \text {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^2 d}-\frac {\left (e \sinh \left (\frac {2 a}{b}\right )\right ) \text {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^2 d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{2 b d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac {e}{2 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e (c+d x)^2}{b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac {e \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right ) \sinh \left (\frac {2 a}{b}\right )}{b^3 d}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{b^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 120, normalized size = 0.77 \begin {gather*} \frac {e \left (-\frac {b^2 (c+d x) \sqrt {1+(c+d x)^2}}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}+\frac {b \left (-1-2 (c+d x)^2\right )}{a+b \sinh ^{-1}(c+d x)}-2 \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )}{2 b^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.26, size = 239, normalized size = 1.53
method | result | size |
derivativedivides | \(\frac {-\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e \left (2 b \arcsinh \left (d x +c \right )+2 a -b \right )}{8 b^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )+a^{2}\right )}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \arcsinh \left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{4 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \arcsinh \left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{3}}}{d}\) | \(239\) |
default | \(\frac {-\frac {\left (-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 \left (d x +c \right )^{2}+1\right ) e \left (2 b \arcsinh \left (d x +c \right )+2 a -b \right )}{8 b^{2} \left (b^{2} \arcsinh \left (d x +c \right )^{2}+2 a b \arcsinh \left (d x +c \right )+a^{2}\right )}+\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \expIntegral \left (1, 2 \arcsinh \left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{3}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{8 b \left (a +b \arcsinh \left (d x +c \right )\right )^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{4 b^{2} \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \expIntegral \left (1, -2 \arcsinh \left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{3}}}{d}\) | \(239\) |
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*} e \left (\int \frac {c}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{3} + 3 a^{2} b \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {asinh}^{3}{\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 = -0.01 \begin {gather*} \int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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