Optimal. Leaf size=79 \[ -\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}} \]
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Rubi [A] time = 0.11, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3700, 1247, 724, 206} \[ -\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 1247
Rule 3700
Rubi steps
\begin {align*} \int \frac {\tan (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{4 a-4 b+4 c-x^2} \, dx,x,\frac {2 a-b-(-b+2 c) \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}\\ &=-\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 79, normalized size = 1.00 \[ -\frac {\tanh ^{-1}\left (\frac {2 a+(b-2 c) \tan ^2(d+e x)-b}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 e \sqrt {a-b+c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 299, normalized size = 3.78 \[ \left [\frac {\log \left (\frac {{\left (b^{2} + 4 \, {\left (a - 2 \, b\right )} c + 8 \, c^{2}\right )} \tan \left (e x + d\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2} - 4 \, {\left (a - b\right )} c\right )} \tan \left (e x + d\right )^{2} - 4 \, \sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (e x + d\right )^{2} + 2 \, a - b\right )} \sqrt {a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan \left (e x + d\right )^{4} + 2 \, \tan \left (e x + d\right )^{2} + 1}\right )}{4 \, \sqrt {a - b + c} e}, -\frac {\sqrt {-a + b - c} \arctan \left (-\frac {\sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (e x + d\right )^{2} + 2 \, a - b\right )} \sqrt {-a + b - c}}{2 \, {\left ({\left ({\left (a - b\right )} c + c^{2}\right )} \tan \left (e x + d\right )^{4} + {\left (a b - b^{2} + b c\right )} \tan \left (e x + d\right )^{2} + a^{2} - a b + a c\right )}}\right )}{2 \, {\left (a - b + c\right )} e}\right ] \]
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 \[ \int \frac {\tan \left (e x + d\right )}{\sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 102, normalized size = 1.29 \[ -\frac {\ln \left (\frac {2 a -2 b +2 c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+2 \sqrt {a -b +c}\, \sqrt {\left (1+\tan ^{2}\left (e x +d \right )\right )^{2} c +\left (b -2 c \right ) \left (1+\tan ^{2}\left (e x +d \right )\right )+a -b +c}}{1+\tan ^{2}\left (e x +d \right )}\right )}{2 e \sqrt {a -b +c}} \]
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 \[ \int \frac {\tan \left (e x + d\right )}{\sqrt {c \tan \left (e x + d\right )^{4} + b \tan \left (e x + d\right )^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {tan}\left (d+e\,x\right )}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^4+b\,{\mathrm {tan}\left (d+e\,x\right )}^2+a}} \,d x \]
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 \[ \int \frac {\tan {\left (d + e x \right )}}{\sqrt {a + b \tan ^{2}{\left (d + e x \right )} + c \tan ^{4}{\left (d + e x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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