Optimal. Leaf size=512 \[ \frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {11 i b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 i b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}} \]
[Out]
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Rubi [A]
time = 0.60, antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5810, 5798,
5772, 267, 5812, 5789, 4265, 2317, 2438, 272, 45} \begin {gather*} -\frac {22 b \sqrt {c^2 x^2+1} \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {8 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {16 a b x \sqrt {c^2 x^2+1}}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}+\frac {11 b x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {c^2 d x^2+d}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {11 i b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {11 i b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}+\frac {2 b^2 \left (c^2 x^2+1\right )}{c^6 d^2 \sqrt {c^2 d x^2+d}}+\frac {b^2}{3 c^6 d^2 \sqrt {c^2 d x^2+d}}-\frac {16 b^2 x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {c^2 d x^2+d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 267
Rule 272
Rule 2317
Rule 2438
Rule 4265
Rule 5772
Rule 5789
Rule 5798
Rule 5810
Rule 5812
Rubi steps
\begin {align*} \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {4 \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{3 c^4 d^2}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b \sqrt {1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^{3/2}}+\frac {1}{c^2 \sqrt {1+c^2 x}}\right ) \, dx,x,x^2\right )}{6 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {10 b^2 \left (1+c^2 x^2\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}+\frac {11 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {11 i b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 i b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.16, size = 333, normalized size = 0.65 \begin {gather*} \frac {\sqrt {d+c^2 d x^2} \left (a^2 \left (8+12 c^2 x^2+3 c^4 x^4\right )+a b \left (2 \left (8+12 c^2 x^2+3 c^4 x^4\right ) \sinh ^{-1}(c x)-\sqrt {1+c^2 x^2} \left (c x \left (5+6 c^2 x^2\right )+22 \left (1+c^2 x^2\right ) \text {ArcTan}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )\right )\right )+b^2 \left (c x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-6 c x \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x)-\sinh ^{-1}(c x)^2+3 \left (1+c^2 x^2\right )^2 \left (2+\sinh ^{-1}(c x)^2\right )+\left (1+c^2 x^2\right ) \left (1+6 \sinh ^{-1}(c x)^2\right )+11 i \left (1+c^2 x^2\right )^{3/2} \sinh ^{-1}(c x) \left (\log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-\log \left (1+i e^{-\sinh ^{-1}(c x)}\right )\right )+11 i \left (1+c^2 x^2\right )^{3/2} \left (\text {PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )\right )\right )\right )}{3 c^6 d^3 \left (1+c^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1041 vs. \(2 (475 ) = 950\).
time = 4.11, size = 1042, normalized size = 2.04
method | result | size |
default | \(a^{2} \left (\frac {x^{4}}{c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {4 \left (-\frac {x^{2}}{c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{c^{2}}\right )+\frac {11 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{3}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{c^{6} d^{3} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{2}}{c^{4} d^{3} \left (c^{2} x^{2}+1\right )}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{2}}{d^{3} \left (c^{2} x^{2}+1\right )^{2} c^{4}}-\frac {11 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{3}}+\frac {11 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{3}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{c^{5} d^{3} \sqrt {c^{2} x^{2}+1}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{3 d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5}}-\frac {11 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{3}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{c^{4} d^{3} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{c^{6} d^{3} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{3 d^{3} \left (c^{2} x^{2}+1\right )^{2} c^{6}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{3 d^{3} \left (c^{2} x^{2}+1\right )^{2} c^{4}}+\frac {5 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{3 d^{3} \left (c^{2} x^{2}+1\right )^{2} c^{6}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2}}{c^{4} d^{3} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{c^{5} d^{3} \sqrt {c^{2} x^{2}+1}}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{c^{6} d^{3} \left (c^{2} x^{2}+1\right )}+\frac {4 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2}}{d^{3} \left (c^{2} x^{2}+1\right )^{2} c^{4}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{3 d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5}}+\frac {10 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{3 d^{3} \left (c^{2} x^{2}+1\right )^{2} c^{6}}-\frac {11 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{3}}+\frac {11 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{6} d^{3}}\) | \(1042\) |
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*} \int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \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 = -0.00 \begin {gather*} \int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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