Optimal. Leaf size=249 \[ -\frac {2 b c^3 \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt {-1-c^2 x^2}}{11025 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {2 b c d \left (15 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {b c \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt {-1-c^2 x^2}}{11025 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3} \]
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Rubi [A]
time = 0.14, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {276, 6437, 12,
1279, 464, 277, 270} \begin {gather*} -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}+\frac {b c d^2 \sqrt {-c^2 x^2-1}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {2 b c d \sqrt {-c^2 x^2-1} \left (15 c^2 d-49 e\right )}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right )}{11025 x^2 \sqrt {-c^2 x^2}}-\frac {2 b c^3 \sqrt {-c^2 x^2-1} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right )}{11025 \sqrt {-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 276
Rule 277
Rule 464
Rule 1279
Rule 6437
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {(b c x) \int \frac {-15 d^2-42 d e x^2-35 e^2 x^4}{105 x^8 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {(b c x) \int \frac {-15 d^2-42 d e x^2-35 e^2 x^4}{x^8 \sqrt {-1-c^2 x^2}} \, dx}{105 \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {(b c x) \int \frac {6 d \left (15 c^2 d-49 e\right )-245 e^2 x^2}{x^6 \sqrt {-1-c^2 x^2}} \, dx}{735 \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {2 b c d \left (15 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {\left (b c \left (-360 c^4 d^2+1176 c^2 d e-1225 e^2\right ) x\right ) \int \frac {1}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{3675 \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {2 b c d \left (15 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {b c \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt {-1-c^2 x^2}}{11025 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}+\frac {\left (2 b c^3 \left (-360 c^4 d^2+1176 c^2 d e-1225 e^2\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1-c^2 x^2}} \, dx}{11025 \sqrt {-c^2 x^2}}\\ &=-\frac {2 b c^3 \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt {-1-c^2 x^2}}{11025 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {2 b c d \left (15 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {b c \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt {-1-c^2 x^2}}{11025 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 152, normalized size = 0.61 \begin {gather*} \frac {-105 a \left (15 d^2+42 d e x^2+35 e^2 x^4\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1225 e^2 x^4 \left (1-2 c^2 x^2\right )+294 d e x^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )-45 d^2 \left (-5+6 c^2 x^2-8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (15 d^2+42 d e x^2+35 e^2 x^4\right ) \text {csch}^{-1}(c x)}{11025 x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 223, normalized size = 0.90
method | result | size |
derivativedivides | \(c^{7} \left (\frac {a \left (-\frac {e^{2}}{3 c^{3} x^{3}}-\frac {2 d e}{5 c^{3} x^{5}}-\frac {d^{2}}{7 c^{3} x^{7}}\right )}{c^{4}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) e^{2}}{3 c^{3} x^{3}}-\frac {2 \,\mathrm {arccsch}\left (c x \right ) d e}{5 c^{3} x^{5}}-\frac {\mathrm {arccsch}\left (c x \right ) d^{2}}{7 c^{3} x^{7}}-\frac {\left (c^{2} x^{2}+1\right ) \left (720 c^{10} d^{2} x^{6}-2352 c^{8} d e \,x^{6}-360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}-1225 c^{4} e^{2} x^{4}-882 c^{4} d e \,x^{2}-225 c^{4} d^{2}\right )}{11025 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{4}}\right )\) | \(223\) |
default | \(c^{7} \left (\frac {a \left (-\frac {e^{2}}{3 c^{3} x^{3}}-\frac {2 d e}{5 c^{3} x^{5}}-\frac {d^{2}}{7 c^{3} x^{7}}\right )}{c^{4}}+\frac {b \left (-\frac {\mathrm {arccsch}\left (c x \right ) e^{2}}{3 c^{3} x^{3}}-\frac {2 \,\mathrm {arccsch}\left (c x \right ) d e}{5 c^{3} x^{5}}-\frac {\mathrm {arccsch}\left (c x \right ) d^{2}}{7 c^{3} x^{7}}-\frac {\left (c^{2} x^{2}+1\right ) \left (720 c^{10} d^{2} x^{6}-2352 c^{8} d e \,x^{6}-360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}-1225 c^{4} e^{2} x^{4}-882 c^{4} d e \,x^{2}-225 c^{4} d^{2}\right )}{11025 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{4}}\right )\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 232, normalized size = 0.93 \begin {gather*} \frac {1}{245} \, b d^{2} {\left (\frac {5 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, c^{8} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {35 \, \operatorname {arcsch}\left (c x\right )}{x^{7}}\right )} + \frac {2}{75} \, b d {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsch}\left (c x\right )}{x^{5}}\right )} e + \frac {1}{9} \, b {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} e^{2} - \frac {a e^{2}}{3 \, x^{3}} - \frac {2 \, a d e}{5 \, x^{5}} - \frac {a d^{2}}{7 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 347, normalized size = 1.39 \begin {gather*} -\frac {3675 \, a x^{4} \cosh \left (1\right )^{2} + 3675 \, a x^{4} \sinh \left (1\right )^{2} + 4410 \, a d x^{2} \cosh \left (1\right ) + 1575 \, a d^{2} + 105 \, {\left (35 \, b x^{4} \cosh \left (1\right )^{2} + 35 \, b x^{4} \sinh \left (1\right )^{2} + 42 \, b d x^{2} \cosh \left (1\right ) + 15 \, b d^{2} + 14 \, {\left (5 \, b x^{4} \cosh \left (1\right ) + 3 \, b d x^{2}\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 1470 \, {\left (5 \, a x^{4} \cosh \left (1\right ) + 3 \, a d x^{2}\right )} \sinh \left (1\right ) + {\left (720 \, b c^{7} d^{2} x^{7} - 360 \, b c^{5} d^{2} x^{5} + 270 \, b c^{3} d^{2} x^{3} - 225 \, b c d^{2} x + 1225 \, {\left (2 \, b c^{3} x^{7} - b c x^{5}\right )} \cosh \left (1\right )^{2} + 1225 \, {\left (2 \, b c^{3} x^{7} - b c x^{5}\right )} \sinh \left (1\right )^{2} - 294 \, {\left (8 \, b c^{5} d x^{7} - 4 \, b c^{3} d x^{5} + 3 \, b c d x^{3}\right )} \cosh \left (1\right ) - 98 \, {\left (24 \, b c^{5} d x^{7} - 12 \, b c^{3} d x^{5} + 9 \, b c d x^{3} - 25 \, {\left (2 \, b c^{3} x^{7} - b c x^{5}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{11025 \, x^{7}} \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 {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{8}}\, dx \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 (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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