Optimal. Leaf size=108 \[ -\frac {1}{4} b c^3 \pi ^{3/2} x^2+\frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {3 c \pi ^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b}+b c \pi ^{3/2} \log (x) \]
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Rubi [A]
time = 0.11, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5807, 5785,
5783, 30, 14} \begin {gather*} \frac {3}{2} \pi c^2 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {3 \pi ^{3/2} c \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b}-\frac {1}{4} \pi ^{3/2} b c^3 x^2+\pi ^{3/2} b c \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 5783
Rule 5785
Rule 5807
Rubi steps
\begin {align*} \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\left (3 c^2 \pi \right ) \int \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {1+c^2 x^2}{x} \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {\left (b c \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \left (\frac {1}{x}+c^2 x\right ) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (3 c^2 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c^3 \pi \sqrt {\pi +c^2 \pi x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c^3 \pi x^2 \sqrt {\pi +c^2 \pi x^2}}{4 \sqrt {1+c^2 x^2}}+\frac {3}{2} c^2 \pi x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x}+\frac {3 c \pi \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b \sqrt {1+c^2 x^2}}+\frac {b c \pi \sqrt {\pi +c^2 \pi x^2} \log (x)}{\sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 122, normalized size = 1.13 \begin {gather*} \frac {\pi ^{3/2} \left (-8 a \sqrt {1+c^2 x^2}+4 a c^2 x^2 \sqrt {1+c^2 x^2}+6 b c x \sinh ^{-1}(c x)^2-b c x \cosh \left (2 \sinh ^{-1}(c x)\right )+8 b c x \log (c x)+2 \sinh ^{-1}(c x) \left (6 a c x-4 b \sqrt {1+c^2 x^2}+b c x \sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )}{8 x} \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. \(221\) vs.
\(2(92)=184\).
time = 4.48, size = 222, normalized size = 2.06
method | result | size |
default | \(-\frac {a \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{\pi x}+a \,c^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}+\frac {3 a \,c^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2}+\frac {3 a \,c^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 \sqrt {\pi \,c^{2}}}+\frac {3 b c \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right )^{2}}{4}+\frac {b \arcsinh \left (c x \right ) \pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}\, x \,c^{2}}{2}-\frac {b \,c^{3} \pi ^{\frac {3}{2}} x^{2}}{4}-b c \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right )-\frac {b \,\pi ^{\frac {3}{2}} c}{8}-\frac {b \,\pi ^{\frac {3}{2}} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{x}+b c \,\pi ^{\frac {3}{2}} \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )\) | \(222\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \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*} \pi ^{\frac {3}{2}} \left (\int a c^{2} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{2}}\, dx + \int b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{2}}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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