Integrand size = 23, antiderivative size = 1006 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=\frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {8 b c^2 e^2 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 )}{15 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 \left (2 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 )}{45 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 (8 c^4 d^2+3 c^2 d e-2 e^2\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 )}{75 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 (8 c^2 d-e\right ) \left (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 )}{75 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {8 b c^2 e \left (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 )}{45 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e^2 \left (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 )}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
[Out]
Time = 1.26 (sec) , antiderivative size = 1006, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {277, 270, 5346, 12, 6874, 486, 597, 538, 438, 437, 435, 432, 430, 21, 434} \[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=-\frac {8 b e^2 x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right ) c^2}{15 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {4 b e \left (2 d c^2+e\right ) x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right ) c^2}{45 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}-\frac {b \left (8 d^2 c^4+3 d e c^2-2 e^2\right ) x \sqrt {1-c^2 x^2} \sqrt {e x^2+d} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right ) c^2}{75 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b \left (8 c^2 d-e\right ) \left (d c^2+e\right ) 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 ) c^2}{75 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}-\frac {8 b e \left (d c^2+e\right ) 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 ) c^2}{45 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}}+\frac {8 b e^2 \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b e \left (2 d c^2+e\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{45 d^3 \sqrt {c^2 x^2}}+\frac {b \left (8 d^2 c^4+3 d e c^2-2 e^2\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{75 d^3 \sqrt {c^2 x^2}}-\frac {4 b e \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b \left (4 d c^2+e\right ) \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{75 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b \sqrt {c^2 x^2-1} \sqrt {e x^2+d} c}{25 d x^4 \sqrt {c^2 x^2}}-\frac {8 e^2 \sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}+\frac {4 e \sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {\sqrt {e x^2+d} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {8 b e^2 \left (d c^2+e\right ) 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 )}{15 d^3 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {e x^2+d}} \]
[In]
[Out]
Rule 12
Rule 21
Rule 270
Rule 277
Rule 430
Rule 432
Rule 434
Rule 435
Rule 437
Rule 438
Rule 486
Rule 538
Rule 597
Rule 5346
Rule 6874
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )}{15 d^3 x^6 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {\sqrt {d+e x^2} \left (-3 d^2+4 d e x^2-8 e^2 x^4\right )}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \left (-\frac {3 d^2 \sqrt {d+e x^2}}{x^6 \sqrt {-1+c^2 x^2}}+\frac {4 d e \sqrt {d+e x^2}}{x^4 \sqrt {-1+c^2 x^2}}-\frac {8 e^2 \sqrt {d+e x^2}}{x^2 \sqrt {-1+c^2 x^2}}\right ) \, dx}{15 d^3 \sqrt {c^2 x^2}} \\ & = -\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}+\frac {(b c x) \int \frac {\sqrt {d+e x^2}}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{5 d \sqrt {c^2 x^2}}-\frac {(4 b c e x) \int \frac {\sqrt {d+e x^2}}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{15 d^2 \sqrt {c^2 x^2}}+\frac {\left (8 b c e^2 x\right ) \int \frac {\sqrt {d+e x^2}}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}} \\ & = \frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {-4 c^2 d-e-3 c^2 e x^2}{x^4 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{25 d \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {-2 c^2 d-e-c^2 e x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{45 d^2 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^2 x\right ) \int \frac {-e+c^2 e x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}} \\ & = \frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {-8 c^4 d^2-3 c^2 d e+2 e^2-c^2 e \left (4 c^2 d+e\right ) x^2}{x^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}}+\frac {(4 b c e x) \int \frac {-c^2 d e+c^2 e \left (2 c^2 d+e\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{45 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c e^3 x\right ) \int \frac {\sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}} \\ & = \frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {(b c x) \int \frac {-c^2 d e \left (4 c^2 d+e\right )+c^2 e \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x^2}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e^2 x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{45 d^2 \sqrt {c^2 x^2}}+\frac {\left (8 b c e^2 \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2}}+\frac {\left (4 b c^3 e \left (2 c^2 d+e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{45 d^3 \sqrt {c^2 x^2}} \\ & = \frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}+\frac {\left (b c^3 \left (8 c^2 d-e\right ) \left (c^2 d+e\right ) x\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \, dx}{75 d^2 \sqrt {c^2 x^2}}-\frac {\left (b c^3 \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{75 d^3 \sqrt {c^2 x^2}}-\frac {\left (8 b c^3 e^2 x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (4 b c^3 e \left (2 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}{45 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (8 b c^3 e \left (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}{45 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}+\frac {\left (8 b c e^2 \left (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}{15 d^3 \sqrt {c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {\left (b c^3 \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) x \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{75 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}-\frac {\left (8 b c^3 e^2 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}{15 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 \left (2 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}{45 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 (8 c^2 d-e\right ) \left (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}{75 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (8 b c^3 e \left (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}{45 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {\left (8 b c e^2 \left (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}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {8 b c^2 e^2 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 )}{15 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 \left (2 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 )}{45 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}-\frac {8 b c^2 e \left (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 )}{45 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e^2 \left (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 )}{15 d^3 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {\left (b c^3 \left (8 c^4 d^2+3 c^2 d e-2 e^2\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}{75 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 (8 c^2 d-e\right ) \left (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}{75 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \\ & = \frac {8 b c e^2 \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{15 d^3 \sqrt {c^2 x^2}}-\frac {4 b c e \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^3 \sqrt {c^2 x^2}}+\frac {b c \left (8 c^4 d^2+3 c^2 d e-2 e^2\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^3 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{25 d x^4 \sqrt {c^2 x^2}}-\frac {4 b c e \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{45 d^2 x^2 \sqrt {c^2 x^2}}+\frac {b c \left (4 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{75 d^2 x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{5 d x^5}+\frac {4 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^2 x^3}-\frac {8 e^2 \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{15 d^3 x}-\frac {8 b c^2 e^2 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 )}{15 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 \left (2 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 )}{45 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 (8 c^4 d^2+3 c^2 d e-2 e^2\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 )}{75 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 (8 c^2 d-e\right ) \left (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 )}{75 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}-\frac {8 b c^2 e \left (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 )}{45 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}+\frac {8 b e^2 \left (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 )}{15 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 = 8.64 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.33 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=\frac {\sqrt {d+e x^2} \left (-15 a \left (3 d^2-4 d e x^2+8 e^2 x^4\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (94 e^2 x^4-d e x^2 \left (17+31 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d^2-4 d e x^2+8 e^2 x^4\right ) \sec ^{-1}(c x)\right )}{225 d^3 x^5}-\frac {i b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1+\frac {e x^2}{d}} \left (c^2 d \left (24 c^4 d^2-31 c^2 d e+94 e^2\right ) E\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )-\left (24 c^6 d^3-19 c^4 d^2 e+77 c^2 d e^2+120 e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-c^2} x\right ),-\frac {e}{c^2 d}\right )\right )}{225 \sqrt {-c^2} d^3 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \]
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\[\int \frac {a +b \,\operatorname {arcsec}\left (c x \right )}{x^{6} \sqrt {e \,x^{2}+d}}d x\]
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none
Time = 0.12 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.29 \[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=-\frac {{\left (120 \, a c d e^{2} x^{4} - 60 \, a c d^{2} e x^{2} + 45 \, a c d^{3} + 15 \, {\left (8 \, b c d e^{2} x^{4} - 4 \, b c d^{2} e x^{2} + 3 \, b c d^{3}\right )} \operatorname {arcsec}\left (c x\right ) - {\left (9 \, b c d^{3} + {\left (24 \, b c^{5} d^{3} - 31 \, b c^{3} d^{2} e + 94 \, b c d e^{2}\right )} x^{4} + {\left (12 \, b c^{3} d^{3} - 17 \, b c d^{2} e\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {e x^{2} + d} - {\left ({\left (24 \, b c^{8} d^{3} - 31 \, b c^{6} d^{2} e + 94 \, b c^{4} d e^{2}\right )} x^{5} E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (24 \, b c^{8} d^{3} - {\left (31 \, b c^{6} - 12 \, b c^{4}\right )} d^{2} e + {\left (94 \, b c^{4} - 17 \, b c^{2}\right )} d e^{2} + 120 \, b e^{3}\right )} x^{5} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {-d}}{225 \, c d^{4} x^{5}} \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {asec}{\left (c x \right )}}{x^{6} \sqrt {d + e x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{x^6 \sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^6\,\sqrt {e\,x^2+d}} \,d x \]
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