Optimal. Leaf size=346 \[ \frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{9 d^2 x}-\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 c d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {\frac {e x^2}{d}+1}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.48, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {271, 264, 6301, 12, 580, 583, 524, 426, 424, 421, 419} \[ \frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{9 d^2 x}-\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 c d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {\frac {e x^2}{d}+1}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 264
Rule 271
Rule 419
Rule 421
Rule 424
Rule 426
Rule 524
Rule 580
Rule 583
Rule 6301
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^3}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d \left (2 c^2 d-5 e\right )-\left (c^2 d-6 e\right ) e x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^3}+\frac {b \left (2 c^2 d-5 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {d \left (c^2 d-6 e\right ) e-c^2 d \left (2 c^2 d-5 e\right ) e x^2}{\sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^3}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^3}+\frac {b \left (2 c^2 d-5 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^2 \left (2 c^2 d-5 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^2}-\frac {\left (2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{9 d^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^3}+\frac {b \left (2 c^2 d-5 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}+\frac {\left (b c^2 \left (2 c^2 d-5 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2}\right ) \int \frac {\sqrt {1+\frac {e x^2}{d}}}{\sqrt {1-c^2 x^2}} \, dx}{9 d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c 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 {d+e x^2}}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d x^3}+\frac {b \left (2 c^2 d-5 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{9 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 d^2 x}+\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 c d^2 \sqrt {d+e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 5.03, size = 612, normalized size = 1.77 \[ \frac {-\frac {3 a \left (d-2 e x^2\right ) \left (d+e x^2\right )}{x^3}+\frac {b \sqrt {\frac {1-c x}{c x+1}} \left (2 c^2 d-5 e\right ) \left (d+e x^2\right )}{x}-\frac {b \sqrt {\frac {1-c x}{c x+1}} \left (\sqrt {e} x+i \sqrt {d}\right ) \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{(c x+1) \left (c \sqrt {d}+i \sqrt {e}\right )}} \left (2 \sqrt {e} \left (2 i c^2 d-c \sqrt {d} \sqrt {e}-6 i e\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (d c^2+e\right ) (1-c x)}{\left (\sqrt {d} c+i \sqrt {e}\right )^2 (c x+1)}}\right )|\frac {\left (\sqrt {d} c+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )+\left (2 c^3 d^{3/2}-2 i c^2 d \sqrt {e}-5 c \sqrt {d} e+5 i e^{3/2}\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (d c^2+e\right ) (1-c x)}{\left (\sqrt {d} c+i \sqrt {e}\right )^2 (c x+1)}}\right )|\frac {\left (\sqrt {d} c+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{\sqrt {-\frac {(c x-1) \left (c \sqrt {d}-i \sqrt {e}\right )}{(c x+1) \left (c \sqrt {d}+i \sqrt {e}\right )}} \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{(c x+1) \left (c \sqrt {d}-i \sqrt {e}\right )}}}+\frac {b c d \sqrt {\frac {1-c x}{c x+1}} \left (d+e x^2\right )}{x^2}+\frac {b d \sqrt {\frac {1-c x}{c x+1}} \left (d+e x^2\right )}{x^3}-\frac {3 b \text {sech}^{-1}(c x) \left (d-2 e x^2\right ) \left (d+e x^2\right )}{x^3}}{9 d^2 \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{e x^{6} + d x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arcsech}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {e x^{2} + d} e}{d^{2} x} - \frac {\sqrt {e x^{2} + d}}{d x^{3}}\right )} + \frac {1}{3} \, b {\left (\frac {{\left (2 \, e^{2} x^{5} + d e x^{3} - d^{2} x\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}{\sqrt {e x^{2} + d} d^{2} x^{4}} - 3 \, \int \frac {3 \, c^{2} d^{2} x^{2} \log \relax (c) - 3 \, d^{2} \log \relax (c) + {\left (2 \, c^{2} e^{2} x^{6} + c^{2} d e x^{4} + {\left (3 \, d^{2} \log \relax (c) - d^{2}\right )} c^{2} x^{2} - 3 \, d^{2} \log \relax (c) + 6 \, {\left (c^{2} d^{2} x^{2} - d^{2}\right )} \log \left (\sqrt {x}\right )\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} + 6 \, {\left (c^{2} d^{2} x^{2} - d^{2}\right )} \log \left (\sqrt {x}\right )}{3 \, {\left ({\left (c^{2} d^{2} x^{2} - d^{2}\right )} x^{4} + {\left (c^{2} d^{2} x^{2} - d^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right ) + 4 \, \log \relax (x)\right )}\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________