Optimal. Leaf size=394 \[ \frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{4 b c^4}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^4}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{4 b c^4}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^4}-\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {3 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {3 d^2 e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c^2}+\frac {3 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c^2}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c} \]
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Rubi [A] time = 1.17, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5805, 6742, 3303, 3298, 3301, 5448, 12} \[ -\frac {3 d^2 e \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c^2}-\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{4 b c^4}-\frac {e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^4}+\frac {3 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c^2}+\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {3 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{4 b c^4}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^4}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5805
Rule 6742
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{a+b \sinh ^{-1}(c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh (x) (c d+e \sinh (x))^3}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c^4}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {c^3 d^3 \cosh (x)}{a+b x}+\frac {3 c^2 d^2 e \cosh (x) \sinh (x)}{a+b x}+\frac {3 c d e^2 \cosh (x) \sinh ^2(x)}{a+b x}+\frac {e^3 \cosh (x) \sinh ^3(x)}{a+b x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4}\\ &=\frac {d^3 \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c}+\frac {\left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2}+\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}+\frac {e^3 \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{c^4}\\ &=\frac {\left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 (a+b x)} \, dx,x,\sinh ^{-1}(c x)\right )}{c^2}+\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 (a+b x)}+\frac {\cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3}+\frac {e^3 \operatorname {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^4}+\frac {\left (d^3 \cosh \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 x)\right )}{c}-\frac {\left (d^3 \sinh \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 x)\right )}{c}\\ &=\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}+\frac {\left (3 d^2 e\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 c^2}-\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3}+\frac {\left (3 d e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^3}+\frac {e^3 \operatorname {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^4}-\frac {e^3 \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^4}\\ &=\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}-\frac {\left (3 d e^2 \cosh \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 x)\right )}{4 c^3}+\frac {\left (3 d^2 e \cosh \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 x)\right )}{2 c^2}-\frac {\left (e^3 \cosh \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 x)\right )}{4 c^4}+\frac {\left (3 d e^2 \cosh \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 x)\right )}{4 c^3}+\frac {\left (e^3 \cosh \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 x)\right )}{8 c^4}+\frac {\left (3 d e^2 \sinh \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 x)\right )}{4 c^3}-\frac {\left (3 d^2 e \sinh \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 x)\right )}{2 c^2}+\frac {\left (e^3 \sinh \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 x)\right )}{4 c^4}-\frac {\left (3 d e^2 \sinh \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 x)\right )}{4 c^3}-\frac {\left (e^3 \sinh \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 x)\right )}{8 c^4}\\ &=\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}-\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b c^3}-\frac {3 d^2 e \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{2 b c^2}+\frac {e^3 \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{4 b c^4}-\frac {e^3 \text {Chi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b c^4}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b c}+\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {3 d^2 e \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{2 b c^2}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )}{4 b c^4}-\frac {3 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{4 b c^3}+\frac {e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \sinh ^{-1}(c x)\right )}{8 b c^4}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 305, normalized size = 0.77 \[ \frac {e^3 \left (2 \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-2 \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )\right )}{8 b c^4}+\frac {3 d e^2 \left (-\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )\right )}{4 b c^3}-\frac {3 d^2 e \left (\sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )-\cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c x)\right )\right )}{2 b c^2}+\frac {d^3 \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{b c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{b \operatorname {arsinh}\left (c x\right ) + a}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 394, normalized size = 1.00 \[ \frac {-\frac {e^{3} {\mathrm e}^{-\frac {4 a}{b}} \Ei \left (1, -4 \arcsinh \left (c x \right )-\frac {4 a}{b}\right )}{16 c^{3} b}+\frac {e^{3} {\mathrm e}^{\frac {4 a}{b}} \Ei \left (1, 4 \arcsinh \left (c x \right )+\frac {4 a}{b}\right )}{16 c^{3} b}+\frac {3 e \,{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right ) d^{2}}{4 c b}-\frac {e^{3} {\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (c x \right )+\frac {2 a}{b}\right )}{8 c^{3} b}-\frac {3 e \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right ) d^{2}}{4 c b}+\frac {e^{3} {\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (c x \right )-\frac {2 a}{b}\right )}{8 c^{3} b}-\frac {3 d \,e^{2} {\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right )}{8 c^{2} b}-\frac {{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 d \,{\mathrm e}^{\frac {a}{b}} \Ei \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 d \,{\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) e^{2}}{8 c^{2} b}-\frac {3 d \,e^{2} {\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, 3 \arcsinh \left (c x \right )+\frac {3 a}{b}\right )}{8 c^{2} b}}{c} \]
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 \[ \int \frac {{\left (e x + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^3}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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 \[ \int \frac {\left (d + e x\right )^{3}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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