Optimal. Leaf size=160 \[ -\frac{\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{6 b^4 d}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{6 b^4 d}-\frac{c+d x}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{\sqrt{(c+d x)^2+1}}{6 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac{\sqrt{(c+d x)^2+1}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3} \]
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Rubi [A] time = 0.267411, antiderivative size = 156, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5863, 5655, 5774, 5779, 3303, 3298, 3301} \[ -\frac{\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{6 b^4 d}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{6 b^4 d}-\frac{c+d x}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{\sqrt{(c+d x)^2+1}}{6 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac{\sqrt{(c+d x)^2+1}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 5863
Rule 5655
Rule 5774
Rule 5779
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sinh ^{-1}(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \sinh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}+\frac{\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{\sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac{c+d x}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{6 b^2 d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac{c+d x}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{\sqrt{1+(c+d x)^2}}{6 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )} \, dx,x,c+d x\right )}{6 b^3 d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac{c+d x}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{\sqrt{1+(c+d x)^2}}{6 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac{c+d x}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{\sqrt{1+(c+d x)^2}}{6 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac{\cosh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6 b^3 d}-\frac{\sinh \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac{\sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^3}-\frac{c+d x}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^2}-\frac{\sqrt{1+(c+d x)^2}}{6 b^3 d \left (a+b \sinh ^{-1}(c+d x)\right )}-\frac{\text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right ) \sinh \left (\frac{a}{b}\right )}{6 b^4 d}+\frac{\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{6 b^4 d}\\ \end{align*}
Mathematica [A] time = 0.436369, size = 130, normalized size = 0.81 \[ -\frac{\frac{2 b^3 \sqrt{(c+d x)^2+1}}{\left (a+b \sinh ^{-1}(c+d x)\right )^3}+\frac{b^2 (c+d x)}{\left (a+b \sinh ^{-1}(c+d x)\right )^2}+\sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )-\cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )+\frac{b \sqrt{(c+d x)^2+1}}{a+b \sinh ^{-1}(c+d x)}}{6 b^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 272, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{2} \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}+2\,ab{\it Arcsinh} \left ( dx+c \right ) -{\it Arcsinh} \left ( dx+c \right ){b}^{2}+{a}^{2}-ab+2\,{b}^{2}}{12\,{b}^{3} \left ({b}^{3} \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{3}+3\, \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}a{b}^{2}+3\,{a}^{2}b{\it Arcsinh} \left ( dx+c \right ) +{a}^{3} \right ) } \left ( -\sqrt{1+ \left ( dx+c \right ) ^{2}}+dx+c \right ) }+{\frac{1}{12\,{b}^{4}}{{\rm e}^{{\frac{a}{b}}}}{\it Ei} \left ( 1,{\it Arcsinh} \left ( dx+c \right ) +{\frac{a}{b}} \right ) }-{\frac{1}{6\,b \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{3}} \left ( dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }-{\frac{1}{12\,{b}^{2} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}} \left ( dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }-{\frac{1}{12\,{b}^{3} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) } \left ( dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }-{\frac{1}{12\,{b}^{4}}{{\rm e}^{-{\frac{a}{b}}}}{\it Ei} \left ( 1,-{\it Arcsinh} \left ( dx+c \right ) -{\frac{a}{b}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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