Optimal. Leaf size=111 \[ \frac {1}{2} x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {c^2 x^2+1}}-\frac {b c x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5682, 5675, 30} \[ \frac {1}{2} x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {c^2 x^2+1}}-\frac {b c x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5675
Rule 5682
Rubi steps
\begin {align*} \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{2} x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {d+c^2 d x^2} \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {1}{2} x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 120, normalized size = 1.08 \[ \frac {1}{8} \left (4 a x \sqrt {c^2 d x^2+d}+\frac {4 a \sqrt {d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )}{c}+\frac {b \sqrt {c^2 d x^2+d} \left (2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )-\cosh \left (2 \sinh ^{-1}(c x)\right )\right )}{c \sqrt {c^2 x^2+1}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 222, normalized size = 2.00 \[ \frac {a x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {a d \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{4 \sqrt {c^{2} x^{2}+1}\, c}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c^{2} \arcsinh \left (c x \right ) x^{3}}{2 c^{2} x^{2}+2}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, c \,x^{2}}{4 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{2 c^{2} x^{2}+2}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}}{8 c \sqrt {c^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \]
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 \[ \int \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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