Optimal. Leaf size=51 \[ \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \log \left (1+c^2 x^2\right )}{2 c \pi ^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5787, 266}
\begin {gather*} \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 5787
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{\pi \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi +c^2 \pi x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c \pi \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 66, normalized size = 1.29 \begin {gather*} \frac {2 a c x+2 b c x \sinh ^{-1}(c x)-b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 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. \(131\) vs.
\(2(45)=90\).
time = 1.60, size = 132, normalized size = 2.59
method | result | size |
default | \(\frac {a x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2 b \arcsinh \left (c x \right )}{c \,\pi ^{\frac {3}{2}}}-\frac {b \arcsinh \left (c x \right ) c \,x^{2}}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}+\frac {b \arcsinh \left (c x \right ) x}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}-\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} c \left (c^{2} x^{2}+1\right )}-\frac {b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c \,\pi ^{\frac {3}{2}}}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 58, normalized size = 1.14 \begin {gather*} \frac {b x \operatorname {arsinh}\left (c x\right )}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {a x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} - \frac {b \log \left (x^{2} + \frac {1}{c^{2}}\right )}{2 \, \pi ^{\frac {3}{2}} c} \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}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {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 = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\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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