Optimal. Leaf size=204 \[ -\frac {b^2 x}{3 \pi ^{5/2} \sqrt {1+c^2 x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^{5/2} \left (1+c^2 x^2\right )}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^{5/2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 b \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^{5/2}}-\frac {2 b^2 \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^{5/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5788, 5787,
5797, 3799, 2221, 2317, 2438, 5798, 197} \begin {gather*} \frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} c \left (c^2 x^2+1\right )}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^{5/2} c}-\frac {4 b \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^{5/2} c}-\frac {b^2 x}{3 \pi ^{5/2} \sqrt {c^2 x^2+1}}-\frac {2 b^2 \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 \pi ^{5/2} c} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5787
Rule 5788
Rule 5797
Rule 5798
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{3 \pi }-\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^2} \, dx}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (4 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}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {\left (8 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 )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 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 )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (4 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 )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 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 )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (2 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 )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b^2 x}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \left (a+b \sinh ^{-1}(c x)\right )}{3 c \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {4 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 )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {2 b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 293, normalized size = 1.44 \begin {gather*} \frac {3 a^2 c x-b^2 c x+2 a^2 c^3 x^3-b^2 c^3 x^3+a b \sqrt {1+c^2 x^2}-b^2 \left (-3 c x-2 c^3 x^3+2 \sqrt {1+c^2 x^2}+2 c^2 x^2 \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)^2-b \sinh ^{-1}(c x) \left (-6 a c x-4 a c^3 x^3-b \sqrt {1+c^2 x^2}+4 b \left (1+c^2 x^2\right )^{3/2} \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )\right )-2 a b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )-2 a b c^2 x^2 \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )+2 b^2 \left (1+c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )}{3 c \pi ^{5/2} \left (1+c^2 x^2\right )^{3/2}} \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. \(1728\) vs.
\(2(192)=384\).
time = 6.10, size = 1729, normalized size = 8.48
method | result | size |
default | \(\text {Expression too large to display}\) | \(1729\) |
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^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 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^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 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^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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