Optimal. Leaf size=306 \[ -\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d^2 e} \]
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Rubi [A] time = 0.19, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6288, 961, 266, 63, 208, 731, 725, 204} \[ -\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right )}{2 d^2 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d^2 e} \]
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 208
Rule 266
Rule 725
Rule 731
Rule 961
Rule 6288
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x (d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \left (\frac {1}{d^2 x \sqrt {1-c^2 x^2}}-\frac {e}{d (d+e x)^2 \sqrt {1-c^2 x^2}}-\frac {e}{d^2 (d+e x) \sqrt {1-c^2 x^2}}\right ) \, dx}{2 e}\\ &=-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 d^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{(d+e x)^2 \sqrt {1-c^2 x^2}} \, dx}{2 d}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{2 d^2 e}\\ &=\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 d^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{4 d^2 e}+\frac {\left (b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 d^2 \sqrt {c^2 d^2-e^2}}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{2 c^2 d^2 e}-\frac {\left (b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {1}{-c^2 d^2+e^2-x^2} \, dx,x,\frac {e+c^2 d x}{\sqrt {1-c^2 x^2}}\right )}{2 \left (c^2 d^2-e^2\right )}\\ &=\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{2 d \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {sech}^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tan ^{-1}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 d^2 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{2 d^2 e}\\ \end {align*}
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Mathematica [C] time = 0.70, size = 342, normalized size = 1.12 \[ \frac {1}{2} \left (-\frac {a}{e (d+e x)^2}-\frac {i b \left (2 c^2 d^2-e^2\right ) \log \left (\frac {4 d^2 e \sqrt {c^2 d^2-e^2} \left (c x \sqrt {\frac {1-c x}{c x+1}} \sqrt {c^2 d^2-e^2}+\sqrt {\frac {1-c x}{c x+1}} \sqrt {c^2 d^2-e^2}+i c^2 d x+i e\right )}{b \left (2 c^2 d^2-e^2\right ) (d+e x)}\right )}{d^2 (c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}+\frac {b \log \left (c x \sqrt {\frac {1-c x}{c x+1}}+\sqrt {\frac {1-c x}{c x+1}}+1\right )}{d^2 e}+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c e x+e)}{d (c d-e) (c d+e) (d+e x)}-\frac {b \text {sech}^{-1}(c x)}{e (d+e x)^2}-\frac {b \log (x)}{d^2 e}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 1212, normalized size = 3.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 1090, normalized size = 3.56 \[ -\frac {c^{2} a}{2 \left (c x e +c d \right )^{2} e}-\frac {c^{2} b \,\mathrm {arcsech}\left (c x \right )}{2 \left (c x e +c d \right )^{2} e}+\frac {c^{4} b \sqrt {-\frac {c x -1}{c x}}\, x^{2} \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2 \sqrt {-c^{2} x^{2}+1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )}+\frac {c^{4} b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, d \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2 e \sqrt {-c^{2} x^{2}+1}\, \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )}-\frac {c^{4} b \sqrt {-\frac {c x -1}{c x}}\, x^{2} \sqrt {\frac {c x +1}{c x}}\, \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 c^{2} d x +2 e}{c x e +c d}\right )}{\sqrt {-c^{2} x^{2}+1}\, \left (c d +e \right ) \left (c d -e \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c x e +c d \right )}-\frac {c^{4} b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, d \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 c^{2} d x +2 e}{c x e +c d}\right )}{e \sqrt {-c^{2} x^{2}+1}\, \left (c d +e \right ) \left (c d -e \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c x e +c d \right )}+\frac {c^{2} b e \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}}{2 d \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )}-\frac {c^{2} b \,e^{2} \sqrt {-\frac {c x -1}{c x}}\, x^{2} \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2 \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )}-\frac {c^{2} b e \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{2 \sqrt {-c^{2} x^{2}+1}\, d \left (c d +e \right ) \left (c d -e \right ) \left (c x e +c d \right )}+\frac {c^{2} b \,e^{2} \sqrt {-\frac {c x -1}{c x}}\, x^{2} \sqrt {\frac {c x +1}{c x}}\, \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 c^{2} d x +2 e}{c x e +c d}\right )}{2 \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c d +e \right ) \left (c d -e \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c x e +c d \right )}+\frac {c^{2} b e \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \ln \left (\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +2 c^{2} d x +2 e}{c x e +c d}\right )}{2 \sqrt {-c^{2} x^{2}+1}\, d \left (c d +e \right ) \left (c d -e \right ) \sqrt {-\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \left (c x e +c d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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