Optimal. Leaf size=425 \[ \frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b e x \left (c^2 d+6 e\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c \sqrt {-c^2 x^2-1} \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c^2 x \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {b c^3 x^2 \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rubi [A] time = 0.49, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {271, 264, 6302, 12, 580, 583, 531, 418, 492, 411} \[ \frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b c^3 x^2 \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}-\frac {b c \sqrt {-c^2 x^2-1} \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}-\frac {b e x \left (c^2 d+6 e\right ) \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^2 x \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 271
Rule 411
Rule 418
Rule 492
Rule 531
Rule 580
Rule 583
Rule 6302
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{3 d^2 x^4 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-d+2 e x^2\right )}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{3 d^2 \sqrt {-c^2 x^2}}\\ &=\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {-d \left (2 c^2 d+5 e\right )-e \left (c^2 d+6 e\right ) x^2}{x^2 \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \int \frac {-d e \left (c^2 d+6 e\right )-c^2 d e \left (2 c^2 d+5 e\right ) x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^3 \sqrt {-c^2 x^2}}\\ &=-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {\left (b c^3 e \left (2 c^2 d+5 e\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}-\frac {\left (b c e \left (c^2 d+6 e\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c^3 \left (2 c^2 d+5 e\right ) x^2 \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {b e \left (c^2 d+6 e\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {\left (b c^3 \left (2 c^2 d+5 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{9 d^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b c^3 \left (2 c^2 d+5 e\right ) x^2 \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {b c^2 \left (2 c^2 d+5 e\right ) x \sqrt {d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {b e \left (c^2 d+6 e\right ) x \sqrt {d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end {align*}
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Mathematica [C] time = 0.64, size = 239, normalized size = 0.56 \[ -\frac {\sqrt {d+e x^2} \left (3 a \left (d-2 e x^2\right )+b c x \sqrt {\frac {1}{c^2 x^2}+1} \left (2 c^2 d x^2-d+5 e x^2\right )+3 b \text {csch}^{-1}(c x) \left (d-2 e x^2\right )\right )}{9 d^2 x^3}-\frac {i b c x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {e x^2}{d}+1} \left (c^2 d \left (2 c^2 d+5 e\right ) E\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )-2 \left (c^4 d^2+2 c^2 d e-3 e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )\right )}{9 \sqrt {c^2} d^2 \sqrt {c^2 x^2+1} \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{e x^{6} + d x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}\, dx \]
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 \[ \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {e x^{2} + d} e}{d^{2} x} - \frac {\sqrt {e x^{2} + d}}{d x^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {{\left (2 \, e^{2} x^{4} + d e x^{2} - d^{2}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x^{2} + d} d^{2} x^{3}} + 3 \, \int \frac {2 \, c^{2} e^{2} x^{4} + c^{2} d e x^{2} - c^{2} d^{2}}{3 \, {\left ({\left (c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x^{2} + d} + {\left (c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} \sqrt {e x^{2} + d}\right )}}\,{d x} - 3 \, \int \frac {2 \, c^{2} e^{2} x^{6} + c^{2} d e x^{4} + {\left (3 \, d^{2} \log \relax (c) - d^{2}\right )} c^{2} x^{2} + 3 \, d^{2} \log \relax (c) + 3 \, {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \log \relax (x)}{3 \, {\left (c^{2} d^{2} x^{6} + d^{2} x^{4}\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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