Optimal. Leaf size=249 \[ -\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{c d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d^2 \sqrt {\frac {e x^2}{d}+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 249, 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, 191, 6301, 12, 583, 524, 426, 424, 421, 419} \[ -\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d^2 x}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (c^2 d+2 e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{c d^2 \sqrt {d+e x^2}}+\frac {b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{d^2 \sqrt {\frac {e x^2}{d}+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 191
Rule 271
Rule 419
Rule 421
Rule 424
Rule 426
Rule 524
Rule 583
Rule 6301
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d-2 e x^2}{d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d-2 e x^2}{x^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{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}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d e-c^2 d e x^2}{\sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{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}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b c^2 \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}{d^2}-\frac {\left (b \left (c^2 d+2 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}{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}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {\left (b c^2 \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}{d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {\left (b \left (c^2 d+2 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}{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}}{d^2 x}-\frac {a+b \text {sech}^{-1}(c x)}{d x \sqrt {d+e x^2}}-\frac {2 e x \left (a+b \text {sech}^{-1}(c x)\right )}{d^2 \sqrt {d+e x^2}}+\frac {b c \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 )}{d^2 \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+2 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 )}{c d^2 \sqrt {d+e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 4.49, size = 501, normalized size = 2.01 \[ \frac {-\frac {a \left (d+2 e x^2\right )}{x}+\frac {b \sqrt {\frac {1-c x}{c x+1}} \left (-c^2 \left (d+e x^2\right )+\frac {(c x+1) \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{(c x+1) \left (c \sqrt {d}-i \sqrt {e}\right )}} \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{(c x+1) \left (c \sqrt {d}+i \sqrt {e}\right )}} \left (2 \sqrt {e} \left (c \sqrt {d}-2 i \sqrt {e}\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (d c^2+e\right ) (1-c x)}{\left (\sqrt {d} c+i \sqrt {e}\right )^2 (c x+1)}}\right )|\frac {\left (\sqrt {d} c+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )-i \left (c \sqrt {d}-i \sqrt {e}\right )^2 E\left (i \sinh ^{-1}\left (\sqrt {\frac {\left (d c^2+e\right ) (1-c x)}{\left (\sqrt {d} c+i \sqrt {e}\right )^2 (c x+1)}}\right )|\frac {\left (\sqrt {d} c+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{\sqrt {-\frac {(c x-1) \left (c \sqrt {d}-i \sqrt {e}\right )}{(c x+1) \left (c \sqrt {d}+i \sqrt {e}\right )}}}\right )}{c}+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (d+e x^2\right )}{x}-\frac {b \text {sech}^{-1}(c x) \left (d+2 e x^2\right )}{x}}{d^2 \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.51, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arcsech}\left (c x \right )}{x^{2} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -a {\left (\frac {2 \, e x}{\sqrt {e x^{2} + d} d^{2}} + \frac {1}{\sqrt {e x^{2} + d} d x}\right )} + b \int \frac {\log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2} \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________