Optimal. Leaf size=362 \[ \frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {b c^2 \left (2 c^2 d-5 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}} \]
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Rubi [A]
time = 0.32, antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {277, 270,
5346, 12, 594, 597, 538, 438, 437, 435, 432, 430} \begin {gather*} \frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 b x \sqrt {1-c^2 x^2} \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} F\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}-\frac {b c^2 x \sqrt {1-c^2 x^2} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2} E\left (\text {ArcSin}(c x)\left |-\frac {e}{c^2 d}\right .\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {c^2 x^2-1} \sqrt {\frac {e x^2}{d}+1}}+\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d-5 e\right ) \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 277
Rule 430
Rule 432
Rule 435
Rule 437
Rule 438
Rule 538
Rule 594
Rule 597
Rule 5346
Rubi steps
\begin {align*} \int \frac {a+b \sec ^{-1}(c x)}{x^4 \sqrt {d+e x^2}} \, dx &=-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \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}{\sqrt {c^2 x^2}}\\ &=-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {(b c x) \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 \sqrt {c^2 x^2}}\\ &=\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \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 \sqrt {c^2 x^2}}\\ &=\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}+\frac {(b c x) \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 \sqrt {c^2 x^2}}\\ &=\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\left (b c^3 \left (2 c^2 d-5 e\right ) x\right ) \int \frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}} \, dx}{9 d^2 \sqrt {c^2 x^2}}+\frac {\left (2 b c \left (c^2 d-3 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}{9 d^2 \sqrt {c^2 x^2}}\\ &=\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\left (b c^3 \left (2 c^2 d-5 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^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2}}+\frac {\left (2 b c \left (c^2 d-3 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}{9 d^2 \sqrt {c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {\left (b c^3 \left (2 c^2 d-5 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^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {\left (2 b c \left (c^2 d-3 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}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}\\ &=\frac {b c \left (2 c^2 d-5 e\right ) \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d^2 \sqrt {c^2 x^2}}+\frac {b c \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{9 d x^2 \sqrt {c^2 x^2}}-\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d x^3}+\frac {2 e \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{3 d^2 x}-\frac {b c^2 \left (2 c^2 d-5 e\right ) x \sqrt {1-c^2 x^2} \sqrt {d+e x^2} E\left (\sin ^{-1}(c x)|-\frac {e}{c^2 d}\right )}{9 d^2 \sqrt {c^2 x^2} \sqrt {-1+c^2 x^2} \sqrt {1+\frac {e x^2}{d}}}+\frac {2 b \left (c^2 d-3 e\right ) \left (c^2 d+e\right ) x \sqrt {1-c^2 x^2} \sqrt {1+\frac {e x^2}{d}} F\left (\sin ^{-1}(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}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.52, size = 249, normalized size = 0.69 \begin {gather*} \frac {\sqrt {d+e x^2} \left (b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+2 c^2 d x^2-5 e x^2\right )-3 a \left (d-2 e x^2\right )-3 b \left (d-2 e x^2\right ) \sec ^{-1}(c x)\right )}{9 d^2 x^3}-\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 (2 c^2 d-5 e\right ) E\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )+2 \left (-c^4 d^2+2 c^2 d e+3 e^2\right ) F\left (i \sinh ^{-1}\left (\sqrt {-c^2} x\right )|-\frac {e}{c^2 d}\right )\right )}{9 \sqrt {-c^2} d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.38, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arcsec}\left (c x \right )}{x^{4} \sqrt {e \,x^{2}+d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {a + b \operatorname {asec}{\left (c x \right )}}{x^{4} \sqrt {d + e x^{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{x^4\,\sqrt {e\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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