Optimal. Leaf size=166 \[ -\frac{6 b^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 b^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 b^3 \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 b^3 \text{PolyLog}\left (3,e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}-\frac{6 b \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.255148, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {5865, 12, 5661, 5760, 4182, 2531, 2282, 6589} \[ -\frac{6 b^2 \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 b^2 \text{PolyLog}\left (2,e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e^2}+\frac{6 b^3 \text{PolyLog}\left (3,-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 b^3 \text{PolyLog}\left (3,e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}-\frac{6 b \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5865
Rule 12
Rule 5661
Rule 5760
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^3}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^3}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^2}{x \sqrt{1+x^2}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}+\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \text{csch}(x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}-\frac{6 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}-\frac{6 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}-\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e^2}\\ &=-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}-\frac{6 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{\left (6 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}\\ &=-\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e^2 (c+d x)}-\frac{6 b \left (a+b \sinh ^{-1}(c+d x)\right )^2 \tanh ^{-1}\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Li}_2\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 b^2 \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Li}_2\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}+\frac{6 b^3 \text{Li}_3\left (-e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}-\frac{6 b^3 \text{Li}_3\left (e^{\sinh ^{-1}(c+d x)}\right )}{d e^2}\\ \end{align*}
Mathematica [A] time = 0.779462, size = 315, normalized size = 1.9 \[ \frac{3 a b^2 \left (2 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c+d x)}\right )-2 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c+d x)}\right )+\sinh ^{-1}(c+d x) \left (-\frac{\sinh ^{-1}(c+d x)}{c+d x}+2 \log \left (1-e^{-\sinh ^{-1}(c+d x)}\right )-2 \log \left (e^{-\sinh ^{-1}(c+d x)}+1\right )\right )\right )+b^3 \left (6 \sinh ^{-1}(c+d x) \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c+d x)}\right )-6 \sinh ^{-1}(c+d x) \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c+d x)}\right )+6 \text{PolyLog}\left (3,-e^{-\sinh ^{-1}(c+d x)}\right )-6 \text{PolyLog}\left (3,e^{-\sinh ^{-1}(c+d x)}\right )-\frac{\sinh ^{-1}(c+d x)^3}{c+d x}+3 \sinh ^{-1}(c+d x)^2 \log \left (1-e^{-\sinh ^{-1}(c+d x)}\right )-3 \sinh ^{-1}(c+d x)^2 \log \left (e^{-\sinh ^{-1}(c+d x)}+1\right )\right )-3 a^2 b \log \left (\sqrt{c^2+2 c d x+d^2 x^2+1}+1\right )+3 a^2 b \log (c+d x)-\frac{3 a^2 b \sinh ^{-1}(c+d x)}{c+d x}-\frac{a^3}{c+d x}}{d e^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.046, size = 481, normalized size = 2.9 \begin{align*} -{\frac{{a}^{3}}{d{e}^{2} \left ( dx+c \right ) }}-{\frac{{b}^{3} \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{3}}{d{e}^{2} \left ( dx+c \right ) }}-3\,{\frac{{b}^{3} \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}\ln \left ( 1+dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}-6\,{\frac{{b}^{3}{\it Arcsinh} \left ( dx+c \right ){\it polylog} \left ( 2,-dx-c-\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}+6\,{\frac{{b}^{3}{\it polylog} \left ( 3,-dx-c-\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}+3\,{\frac{{b}^{3} \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}\ln \left ( 1-dx-c-\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}+6\,{\frac{{b}^{3}{\it Arcsinh} \left ( dx+c \right ){\it polylog} \left ( 2,dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}-6\,{\frac{{b}^{3}{\it polylog} \left ( 3,dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}-3\,{\frac{a{b}^{2} \left ({\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}}{d{e}^{2} \left ( dx+c \right ) }}-6\,{\frac{a{b}^{2}{\it Arcsinh} \left ( dx+c \right ) \ln \left ( 1+dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}-6\,{\frac{a{b}^{2}{\it polylog} \left ( 2,-dx-c-\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}+6\,{\frac{a{b}^{2}{\it Arcsinh} \left ( dx+c \right ) \ln \left ( 1-dx-c-\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}+6\,{\frac{a{b}^{2}{\it polylog} \left ( 2,dx+c+\sqrt{1+ \left ( dx+c \right ) ^{2}} \right ) }{d{e}^{2}}}-3\,{\frac{{a}^{2}b{\it Arcsinh} \left ( dx+c \right ) }{d{e}^{2} \left ( dx+c \right ) }}-3\,{\frac{{a}^{2}b{\it Artanh} \left ({\frac{1}{\sqrt{1+ \left ( dx+c \right ) ^{2}}}} \right ) }{d{e}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \operatorname{arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arsinh}\left (d x + c\right ) + a^{3}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{3}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{b^{3} \operatorname{asinh}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{3 a b^{2} \operatorname{asinh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac{3 a^{2} b \operatorname{asinh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]