Optimal. Leaf size=25 \[ \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {\pi }} \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783}
\begin {gather*} \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 \sqrt {\pi } b c} \end {gather*}
Antiderivative was successfully verified.
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Rule 5783
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx &=\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {\pi }}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {\pi }} \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. \(52\) vs.
\(2(21)=42\).
time = 0.67, size = 53, normalized size = 2.12
method | result | size |
default | \(\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b \arcsinh \left (c x \right )^{2}}{2 c \sqrt {\pi }}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 28, normalized size = 1.12 \begin {gather*} \frac {b \operatorname {arsinh}\left (c x\right )^{2}}{2 \, \sqrt {\pi } c} + \frac {a \operatorname {arsinh}\left (c x\right )}{\sqrt {\pi } 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (19) = 38\).
time = 1.50, size = 85, normalized size = 3.40 \begin {gather*} \begin {cases} a \left (\begin {cases} \frac {\sqrt {- \frac {1}{c^{2}}} \operatorname {asin}{\left (x \sqrt {- c^{2}} \right )}}{\sqrt {\pi }} & \text {for}\: \pi c^{2} < 0 \\\frac {\sqrt {\frac {1}{c^{2}}} \operatorname {asinh}{\left (x \sqrt {c^{2}} \right )}}{\sqrt {\pi }} & \text {for}\: \pi c^{2} > 0 \end {cases}\right ) & \text {for}\: b = 0 \\\frac {a x}{\sqrt {\pi }} & \text {for}\: c = 0 \\\frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{2 \sqrt {\pi } b c} & \text {otherwise} \end {cases} \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 = -0.04 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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