Optimal. Leaf size=436 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}-\frac {\sqrt [4]{c} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \Pi \left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \]
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Rubi [A]
time = 0.23, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1230, 1117,
1720} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}}-\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} e \left (\sqrt {a}-\sqrt {c}\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}} \Pi \left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}};2 \text {ArcTan}\left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{4 \sqrt [4]{a} \sqrt [4]{c} e \left (\sqrt {a}-\sqrt {c}\right ) \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 1230
Rule 1720
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\sqrt {a} \text {Subst}\left (\int \frac {1+\frac {\sqrt {c} x^2}{\sqrt {a}}}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{\left (\sqrt {a}-\sqrt {c}\right ) e}-\frac {\sqrt {c} \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{\left (\sqrt {a}-\sqrt {c}\right ) e}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a-b+c} \tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}-\frac {\sqrt [4]{c} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}+\frac {\left (\sqrt {a}+\sqrt {c}\right ) \Pi \left (-\frac {\left (\sqrt {a}-\sqrt {c}\right )^2}{4 \sqrt {a} \sqrt {c}};2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \tan (d+e x)}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right ) \sqrt {\frac {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}{\left (\sqrt {a}+\sqrt {c} \tan ^2(d+e x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {c}\right ) \sqrt [4]{c} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.88, size = 235, normalized size = 0.54 \begin {gather*} -\frac {i \Pi \left (\frac {b+\sqrt {b^2-4 a c}}{2 c};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \tan (d+e x)\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c \tan ^2(d+e x)}{b+\sqrt {b^2-4 a c}}} \sqrt {1-\frac {2 c \tan ^2(d+e x)}{-b+\sqrt {b^2-4 a c}}}}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} e \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 231, normalized size = 0.53
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \sqrt {1+\frac {b \left (\tan ^{2}\left (e x +d \right )\right )}{2 a}-\frac {\left (\tan ^{2}\left (e x +d \right )\right ) \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \left (\tan ^{2}\left (e x +d \right )\right )}{2 a}+\frac {\left (\tan ^{2}\left (e x +d \right )\right ) \sqrt {-4 a c +b^{2}}}{2 a}}\, \EllipticPi \left (\frac {\tan \left (e x +d \right ) \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a}{-b +\sqrt {-4 a c +b^{2}}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{e \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}}\) | \(231\) |
default | \(\frac {\sqrt {2}\, \sqrt {1+\frac {b \left (\tan ^{2}\left (e x +d \right )\right )}{2 a}-\frac {\left (\tan ^{2}\left (e x +d \right )\right ) \sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \left (\tan ^{2}\left (e x +d \right )\right )}{2 a}+\frac {\left (\tan ^{2}\left (e x +d \right )\right ) \sqrt {-4 a c +b^{2}}}{2 a}}\, \EllipticPi \left (\frac {\tan \left (e x +d \right ) \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, -\frac {2 a}{-b +\sqrt {-4 a c +b^{2}}}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{e \sqrt {-\frac {b}{a}+\frac {\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}}\) | \(231\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{\sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}}}\, 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 {1}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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