Optimal. Leaf size=93 \[ -\frac {a+b \sinh ^{-1}(c x)}{\pi x \sqrt {\pi +c^2 \pi x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b c \log (x)}{\pi ^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{2 \pi ^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {277, 197, 5804,
12, 457, 78} \begin {gather*} -\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi \sqrt {\pi c^2 x^2+\pi }}-\frac {a+b \sinh ^{-1}(c x)}{\pi x \sqrt {\pi c^2 x^2+\pi }}+\frac {b c \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2}}+\frac {b c \log (x)}{\pi ^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 78
Rule 197
Rule 277
Rule 457
Rule 5804
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \int \frac {-1-2 c^2 x^2}{x \left (1+c^2 x^2\right )} \, dx}{\pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {-1-2 c^2 x}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 \pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}-\frac {(b c) \text {Subst}\left (\int \left (-\frac {1}{x}-\frac {c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{2 \pi ^{3/2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}-\frac {2 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} \sqrt {1+c^2 x^2}}+\frac {b c \log (x)}{\pi ^{3/2}}+\frac {b c \log \left (1+c^2 x^2\right )}{2 \pi ^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 69, normalized size = 0.74 \begin {gather*} -\frac {\left (1+2 c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\pi ^{3/2} x \sqrt {1+c^2 x^2}}+\frac {b \left (c \log (x)+\frac {1}{2} c \log \left (1+c^2 x^2\right )\right )}{\pi ^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(180\) vs.
\(2(85)=170\).
time = 2.49, size = 181, normalized size = 1.95
method | result | size |
default | \(a \left (-\frac {1}{\pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {2 c^{2} x}{\pi \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )-\frac {4 b c \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}}}+\frac {2 b \arcsinh \left (c x \right ) x^{2} c^{3}}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}-\frac {2 b \arcsinh \left (c x \right ) x \,c^{2}}{\pi ^{\frac {3}{2}} \sqrt {c^{2} x^{2}+1}}+\frac {2 b \arcsinh \left (c x \right ) c}{\pi ^{\frac {3}{2}} \left (c^{2} x^{2}+1\right )}-\frac {b \arcsinh \left (c x \right )}{\pi ^{\frac {3}{2}} x \sqrt {c^{2} x^{2}+1}}+\frac {b c \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{4}-1\right )}{\pi ^{\frac {3}{2}}}\) | \(181\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 119, normalized size = 1.28 \begin {gather*} \frac {1}{2} \, b c {\left (\frac {\log \left (c^{2} x^{2} + 1\right )}{\pi ^{\frac {3}{2}}} + \frac {2 \, \log \left (x\right )}{\pi ^{\frac {3}{2}}}\right )} - {\left (\frac {2 \, c^{2} x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x}\right )} b \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, c^{2} x}{\pi \sqrt {\pi + \pi c^{2} x^{2}}} + \frac {1}{\pi \sqrt {\pi + \pi c^{2} x^{2}} x}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________