Optimal. Leaf size=104 \[ \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{c \pi ^{3/2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{c \pi ^{3/2}}-\frac {b^2 \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{c \pi ^{3/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {5787, 5797,
3799, 2221, 2317, 2438} \begin {gather*} \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi c^2 x^2+\pi }}+\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\pi ^{3/2} c}-\frac {2 b \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} c}-\frac {b^2 \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{\pi ^{3/2} c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5787
Rule 5797
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (2 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}{\pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 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 )}{c \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 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 )}{c \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{c \pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 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 )}{c \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{c \pi \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 153, normalized size = 1.47 \begin {gather*} \frac {-b^2 \left (-c x+\sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)^2+2 b \sinh ^{-1}(c x) \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )\right )+a \left (a c x-b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )\right )+b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )}{c \pi ^{3/2} \sqrt {1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs.
\(2(114)=228\).
time = 2.39, size = 306, normalized size = 2.94
method | result | size |
default | \(\frac {a^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {b^{2} \arcsinh \left (c x \right )^{2} c \,x^{2}}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \arcsinh \left (c x \right )^{2} x}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}+\frac {2 b^{2} \arcsinh \left (c x \right )^{2}}{c \,\pi ^{\frac {3}{2}}}-\frac {2 b^{2} \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}-\frac {b^{2} \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}+\frac {4 a b \arcsinh \left (c x \right )}{c \,\pi ^{\frac {3}{2}}}-\frac {2 a b \arcsinh \left (c x \right ) c \,x^{2}}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \arcsinh \left (c x \right ) x}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {2 a b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {2 a b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\) | \(306\) |
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*} \frac {\int \frac {a^{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^{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^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \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 {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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