Integrand size = 23, antiderivative size = 425 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=-\frac {b c^3 \left (2 c^2 d+5 e\right ) x^2 \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}-\frac {b c \left (2 c^2 d+5 e\right ) \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}+\frac {b c^2 \left (2 c^2 d+5 e\right ) x \sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}}-\frac {b e \left (c^2 d+6 e\right ) x \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2} \sqrt {\frac {d+e x^2}{d \left (1+c^2 x^2\right )}}} \]
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Time = 0.34 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {277, 270, 6437, 12, 594, 597, 545, 429, 506, 422} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\frac {2 e \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 d x^3}-\frac {b e x \left (c^2 d+6 e\right ) \sqrt {d+e x^2} \operatorname {EllipticF}\left (\arctan (c x),1-\frac {e}{c^2 d}\right )}{9 d^3 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac {b c^2 x \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2} E\left (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1} \sqrt {\frac {d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac {b c \sqrt {-c^2 x^2-1} \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2}}+\frac {b c \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {-c^2 x^2}}-\frac {b c^3 x^2 \left (2 c^2 d+5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}} \]
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Rule 12
Rule 270
Rule 277
Rule 422
Rule 429
Rule 506
Rule 545
Rule 594
Rule 597
Rule 6437
Rubi steps \begin{align*} \text {integral}& = -\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} \operatorname {EllipticF}\left (\arctan (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 (\arctan (c x)\left |1-\frac {e}{c^2 d}\right .\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} \operatorname {EllipticF}\left (\arctan (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*}
Result contains complex when optimal does not.
Time = 8.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.56 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=-\frac {\sqrt {d+e x^2} \left (3 a \left (d-2 e x^2\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-d+2 c^2 d x^2+5 e x^2\right )+3 b \left (d-2 e x^2\right ) \text {csch}^{-1}(c x)\right )}{9 d^2 x^3}-\frac {i b c \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (2 c^2 d+5 e\right ) E\left (i \text {arcsinh}\left (\sqrt {c^2} x\right )|\frac {e}{c^2 d}\right )-2 \left (c^4 d^2+2 c^2 d e-3 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {c^2} x\right ),\frac {e}{c^2 d}\right )\right )}{9 \sqrt {c^2} d^2 \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}d x\]
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none
Time = 0.10 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.64 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\frac {{\left (2 \, b c^{6} d^{2} + 5 \, b c^{4} d e\right )} \sqrt {-c^{2}} \sqrt {d} x^{3} E(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) - {\left (2 \, b c^{6} d^{2} + {\left (5 \, b c^{4} + b c^{2}\right )} d e + 6 \, b e^{2}\right )} \sqrt {-c^{2}} \sqrt {d} x^{3} F(\arcsin \left (\sqrt {-c^{2}} x\right )\,|\,\frac {e}{c^{2} d}) + 3 \, {\left (2 \, b c^{2} d e x^{2} - b c^{2} d^{2}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, a c^{2} d e x^{2} - 3 \, a c^{2} d^{2} + {\left (b c^{3} d^{2} x - {\left (2 \, b c^{5} d^{2} + 5 \, b c^{3} d e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{9 \, c^{2} d^{3} x^{3}} \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \]
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