Integrand size = 23, antiderivative size = 701 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {2 b c^2 \left (c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 \left (2 c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
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Time = 0.99 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {277, 197, 5346, 12, 6874, 432, 430, 491, 597, 538, 438, 437, 435, 507} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {2 b c^2 x \sqrt {1-c^2 x^2} \left (c^2 d-e\right ) \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b c^2 x \sqrt {1-c^2 x^2} \left (2 c^2 d-e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}+\frac {2 b c \sqrt {c^2 x^2-1} \left (c^2 d-e\right ) \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}} \]
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Rule 12
Rule 197
Rule 277
Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 491
Rule 507
Rule 538
Rule 597
Rule 5346
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{3 d^3 x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-d^2+4 d e x^2+8 e^2 x^4}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \left (\frac {8 e^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {d^2}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {4 d e}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\right ) \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}+\frac {(b c x) \int \frac {1}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d \sqrt {c^2 x^2}}-\frac {(4 b c e x) \int \frac {1}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-2 \left (c^2 d-e\right )-c^2 e 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 {(4 b c e x) \int \frac {c^2 e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {(b c x) \int \frac {-c^2 d e+2 c^2 \left (c^2 d-e\right ) e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2}}+\frac {\left (4 b c^3 e^2 x\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (2 b c^3 \left (c^2 d-e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2}}+\frac {\left (b c^3 \left (2 c^2 d-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 {\left (4 b c^3 e x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2}}-\frac {\left (4 b c^3 e x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{3 d^2 \sqrt {c^2 x^2}} \\ & = \frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (2 b c^3 \left (c^2 d-e\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (4 b c^3 e x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (b c^3 \left (2 c^2 d-e\right ) x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{9 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (4 b c^3 e x \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (2 b c^3 \left (c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (4 b c^3 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (b c^3 \left (2 c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (4 b c^3 e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}}} \, dx}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {2 b c \left (c^2 d-e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{3 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x^2 \sqrt {c^2 x^2}}-\frac {a+b \sec ^{-1}(c x)}{3 d x^3 \sqrt {d+e x^2}}+\frac {4 e \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x \sqrt {d+e x^2}}+\frac {8 e^2 x \left (a+b \sec ^{-1}(c x)\right )}{3 d^3 \sqrt {d+e x^2}}-\frac {2 b c^2 \left (c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {b c^2 \left (2 c^2 d-e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {4 b c^2 e x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{3 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.45 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.40 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+2 c^2 d x^2-14 e x^2\right ) \left (d+e x^2\right )-3 a \left (d^2-4 d e x^2-8 e^2 x^4\right )-3 b \left (d^2-4 d e x^2-8 e^2 x^4\right ) \sec ^{-1}(c x)}{9 d^3 x^3 \sqrt {d+e x^2}}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (2 c^2 d \left (c^2 d-7 e\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+\left (-2 c^4 d^2+13 c^2 d e+24 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{9 \sqrt {-c^2} d^3 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{4} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
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none
Time = 0.12 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.47 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\frac {{\left (24 \, a c d e^{2} x^{4} + 12 \, a c d^{2} e x^{2} - 3 \, a c d^{3} + 3 \, {\left (8 \, b c d e^{2} x^{4} + 4 \, b c d^{2} e x^{2} - b c d^{3}\right )} \operatorname {arcsec}\left (c x\right ) + {\left (b c d^{3} + 2 \, {\left (b c^{3} d^{2} e - 7 \, b c d e^{2}\right )} x^{4} + {\left (2 \, b c^{3} d^{3} - 13 \, b c d^{2} e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} + {\left (2 \, {\left ({\left (b c^{6} d^{2} e - 7 \, b c^{4} d e^{2}\right )} x^{5} + {\left (b c^{6} d^{3} - 7 \, b c^{4} d^{2} e\right )} x^{3}\right )} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left ({\left (2 \, b c^{6} d^{2} e - {\left (14 \, b c^{4} - b c^{2}\right )} d e^{2} - 24 \, b e^{3}\right )} x^{5} + {\left (2 \, b c^{6} d^{3} - {\left (14 \, b c^{4} - b c^{2}\right )} d^{2} e - 24 \, b d e^{2}\right )} x^{3}\right )} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{9 \, {\left (c d^{4} e x^{5} + c d^{5} x^{3}\right )}} \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^4 \left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^4\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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