Optimal. Leaf size=204 \[ \frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} c \left (c^2 x^2+1\right )}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^{5/2} c}-\frac {4 b \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} c}-\frac {b^2 x}{3 \pi ^{5/2} \sqrt {c^2 x^2+1}}-\frac {2 b^2 \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 \pi ^{5/2} c} \]
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Rubi [A] time = 0.28, antiderivative size = 292, normalized size of antiderivative = 1.43, number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5690, 5687, 5714, 3718, 2190, 2279, 2391, 5717, 191} \[ -\frac {2 b^2 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 \pi ^2 c \sqrt {\pi c^2 x^2+\pi }}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^2 c \sqrt {c^2 x^2+1} \sqrt {\pi c^2 x^2+\pi }}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 c \sqrt {\pi c^2 x^2+\pi }}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {4 b \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^2 c \sqrt {\pi c^2 x^2+\pi }}-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }} \]
Antiderivative was successfully verified.
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Rule 191
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5687
Rule 5690
Rule 5714
Rule 5717
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{3 \pi }-\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^2} \, dx}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (4 b c \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 293, normalized size = 1.44 \[ \frac {2 a^2 c^3 x^3+3 a^2 c x+a b \sqrt {c^2 x^2+1}-2 a b c^2 x^2 \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )-2 a b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )-b \sinh ^{-1}(c x) \left (-4 a c^3 x^3-6 a c x-b \sqrt {c^2 x^2+1}+4 b \left (c^2 x^2+1\right )^{3/2} \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )\right )-b^2 c^3 x^3+2 b^2 \left (c^2 x^2+1\right )^{3/2} \text {Li}_2\left (-e^{-2 \sinh ^{-1}(c x)}\right )-b^2 \left (-2 c^3 x^3+2 c^2 x^2 \sqrt {c^2 x^2+1}+2 \sqrt {c^2 x^2+1}-3 c x\right ) \sinh ^{-1}(c x)^2-b^2 c x}{3 \pi ^{5/2} c \left (c^2 x^2+1\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\pi + \pi c^{2} x^{2}} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{\pi ^{3} c^{6} x^{6} + 3 \, \pi ^{3} c^{4} x^{4} + 3 \, \pi ^{3} c^{2} x^{2} + \pi ^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 1730, normalized size = 8.48 \[ \text {result too large to display} \]
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 \[ \frac {1}{3} \, a b c {\left (\frac {1}{\pi ^{\frac {5}{2}} c^{4} x^{2} + \pi ^{\frac {5}{2}} c^{2}} - \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{\pi ^{\frac {5}{2}} c^{2}}\right )} + \frac {2}{3} \, a b {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}}}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a^{2} {\left (\frac {x}{\pi {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, x}{\pi ^{2} \sqrt {\pi + \pi c^{2} x^{2}}}\right )} + b^{2} \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]
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 \[ \frac {\int \frac {a^{2}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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