Optimal. Leaf size=192 \[ -\frac {b x^2}{4 c^5 \pi ^{5/2}}-\frac {b}{6 c^7 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 \pi ^3}-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 \pi ^{5/2}}-\frac {7 b \log \left (1+c^2 x^2\right )}{6 c^7 \pi ^{5/2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5810, 5812,
5783, 30, 272, 45} \begin {gather*} -\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 \pi ^{5/2} b c^7}-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}+\frac {5 x \sqrt {\pi c^2 x^2+\pi } \left (a+b \sinh ^{-1}(c x)\right )}{2 \pi ^3 c^6}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 \pi ^2 c^4 \sqrt {\pi c^2 x^2+\pi }}-\frac {b x^2}{4 \pi ^{5/2} c^5}-\frac {b}{6 \pi ^{5/2} c^7 \left (c^2 x^2+1\right )}-\frac {7 b \log \left (c^2 x^2+1\right )}{6 \pi ^{5/2} c^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 45
Rule 272
Rule 5783
Rule 5810
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^6 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx &=-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {5 \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{3 c^2 \pi }+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^5}{\left (1+c^2 x^2\right )^2} \, dx}{3 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{c^4 \pi ^2}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{3 c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 \pi ^3}-\frac {5 \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{2 c^6 \pi ^2}-\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 c^5 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{6 c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4}+\frac {1}{c^4 \left (1+c^2 x\right )^2}-\frac {2}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b}{6 c^7 \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}-\frac {13 b x^2 \sqrt {1+c^2 x^2}}{12 c^5 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 \pi ^3}-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 \pi ^{5/2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{3 c^7 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {\left (5 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 c^3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ &=-\frac {b}{6 c^7 \pi ^2 \sqrt {1+c^2 x^2} \sqrt {\pi +c^2 \pi x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^5 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {5 x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {5 x \sqrt {\pi +c^2 \pi x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c^6 \pi ^3}-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )^2}{4 b c^7 \pi ^{5/2}}-\frac {7 b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{6 c^7 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 202, normalized size = 1.05 \begin {gather*} \frac {60 a c x+80 a c^3 x^3+12 a c^5 x^5-7 b \sqrt {1+c^2 x^2}-9 b c^2 x^2 \sqrt {1+c^2 x^2}-6 b c^4 x^4 \sqrt {1+c^2 x^2}+4 \left (-15 a \left (1+c^2 x^2\right )^{3/2}+b c x \left (15+20 c^2 x^2+3 c^4 x^4\right )\right ) \sinh ^{-1}(c x)-30 b \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)^2-28 b \left (1+c^2 x^2\right )^{3/2} \log \left (1+c^2 x^2\right )}{24 c^7 \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. \(969\) vs.
\(2(164)=328\).
time = 8.03, size = 970, normalized size = 5.05
method | result | size |
default | \(\frac {385 b \arcsinh \left (c x \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}-\frac {2338 b \arcsinh \left (c x \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {49 b}{6 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c^{7}}-\frac {1463 b \arcsinh \left (c x \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}-\frac {147 b c \arcsinh \left (c x \right ) x^{8}}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {553 b \arcsinh \left (c x \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c}+\frac {1009 b \arcsinh \left (c x \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{4}}+\frac {98 b \arcsinh \left (c x \right ) x}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{6}}-\frac {b}{8 \pi ^{\frac {5}{2}} c^{7}}-\frac {49 b c \,x^{8}}{6 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {98 b \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {49 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {98 b \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {49 b \,x^{6}}{6 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right ) c}+\frac {14 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right ) c^{3}}+\frac {6 b \,x^{2}}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right ) c^{5}}+\frac {b \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x}{2 \pi ^{\frac {5}{2}} c^{6}}-\frac {343 b \arcsinh \left (c x \right )}{3 \pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{2} c^{7}}+\frac {147 b \arcsinh \left (c x \right ) x^{7}}{\pi ^{\frac {5}{2}} \left (63 c^{4} x^{4}+111 c^{2} x^{2}+49\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {5 a x}{2 c^{6} \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {5 a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{6} \pi ^{2} \sqrt {\pi \,c^{2}}}+\frac {5 a \,x^{3}}{6 c^{4} \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {a \,x^{5}}{2 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {b \,x^{2}}{4 c^{5} \pi ^{\frac {5}{2}}}-\frac {5 b \arcsinh \left (c x \right )^{2}}{4 c^{7} \pi ^{\frac {5}{2}}}+\frac {14 b \arcsinh \left (c x \right )}{3 c^{7} \pi ^{\frac {5}{2}}}-\frac {7 b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{7} \pi ^{\frac {5}{2}}}\) | \(970\) |
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 x^{6}}{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 x^{6} \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(-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 {x^6\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\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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