Optimal. Leaf size=142 \[ -\frac {\tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a} e}+\frac {\tanh ^{-1}\left (\frac {2 a-b+(b-2 c) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e} \]
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Rubi [A]
time = 0.15, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3781, 1265,
974, 738, 212} \begin {gather*} \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}}-\frac {\tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 974
Rule 1265
Rule 3781
Rubi steps
\begin {align*} \int \frac {\cot (d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x (1+x) \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{(-1-x) \sqrt {a+b x+c x^2}}+\frac {1}{x \sqrt {a+b x+c x^2}}\right ) \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=\frac {\text {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 {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,\tan ^2(d+e x)\right )}{2 e}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \tan ^2(d+e x)}{\sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{e}-\frac {\text {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 \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{2 \sqrt {a} 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 = 0.93, size = 136, normalized size = 0.96 \begin {gather*} -\frac {\frac {\tanh ^{-1}\left (\frac {2 a+b \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \tan ^2(d+e x)+c \tan ^4(d+e x)}}\right )}{\sqrt {a}}+\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 )}{\sqrt {a-b+c}}}{2 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.46, size = 0, normalized size = 0.00 \[\int \frac {\cot \left (e x +d \right )}{\sqrt {a +b \left (\tan ^{2}\left (e x +d \right )\right )+c \left (\tan ^{4}\left (e x +d \right )\right )}}\, dx\]
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 [A]
time = 7.76, size = 1057, normalized size = 7.44 \begin {gather*} \left [\frac {{\left (\sqrt {a - b + c} a \log \left (\frac {{\left (b^{2} + 4 \, {\left (a - 2 \, b\right )} c + 8 \, c^{2}\right )} \tan \left (x e + d\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2} - 4 \, {\left (a - b\right )} c\right )} \tan \left (x e + d\right )^{2} + 4 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2} + 2 \, a - b\right )} \sqrt {a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan \left (x e + d\right )^{4} + 2 \, \tan \left (x e + d\right )^{2} + 1}\right ) + {\left (a - b + c\right )} \sqrt {a} \log \left (\frac {{\left (b^{2} + 4 \, a c\right )} \tan \left (x e + d\right )^{4} + 8 \, a b \tan \left (x e + d\right )^{2} - 4 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{\tan \left (x e + d\right )^{4}}\right )\right )} e^{\left (-1\right )}}{4 \, {\left (a^{2} - a b + a c\right )}}, \frac {{\left (2 \, \sqrt {-a} {\left (a - b + c\right )} \arctan \left (\frac {\sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c \tan \left (x e + d\right )^{4} + a b \tan \left (x e + d\right )^{2} + a^{2}\right )}}\right ) + \sqrt {a - b + c} a \log \left (\frac {{\left (b^{2} + 4 \, {\left (a - 2 \, b\right )} c + 8 \, c^{2}\right )} \tan \left (x e + d\right )^{4} + 2 \, {\left (4 \, a b - 3 \, b^{2} - 4 \, {\left (a - b\right )} c\right )} \tan \left (x e + d\right )^{2} + 4 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2} + 2 \, a - b\right )} \sqrt {a - b + c} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, a c}{\tan \left (x e + d\right )^{4} + 2 \, \tan \left (x e + d\right )^{2} + 1}\right )\right )} e^{\left (-1\right )}}{4 \, {\left (a^{2} - a b + a c\right )}}, \frac {{\left (2 \, a \sqrt {-a + b - c} \arctan \left (-\frac {\sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2} + 2 \, a - b\right )} \sqrt {-a + b - c}}{2 \, {\left ({\left ({\left (a - b\right )} c + c^{2}\right )} \tan \left (x e + d\right )^{4} + {\left (a b - b^{2} + b c\right )} \tan \left (x e + d\right )^{2} + a^{2} - a b + a c\right )}}\right ) + {\left (a - b + c\right )} \sqrt {a} \log \left (\frac {{\left (b^{2} + 4 \, a c\right )} \tan \left (x e + d\right )^{4} + 8 \, a b \tan \left (x e + d\right )^{2} - 4 \, \sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{\tan \left (x e + d\right )^{4}}\right )\right )} e^{\left (-1\right )}}{4 \, {\left (a^{2} - a b + a c\right )}}, \frac {{\left (\sqrt {-a} {\left (a - b + c\right )} \arctan \left (\frac {\sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left (b \tan \left (x e + d\right )^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c \tan \left (x e + d\right )^{4} + a b \tan \left (x e + d\right )^{2} + a^{2}\right )}}\right ) + a \sqrt {-a + b - c} \arctan \left (-\frac {\sqrt {c \tan \left (x e + d\right )^{4} + b \tan \left (x e + d\right )^{2} + a} {\left ({\left (b - 2 \, c\right )} \tan \left (x e + d\right )^{2} + 2 \, a - b\right )} \sqrt {-a + b - c}}{2 \, {\left ({\left ({\left (a - b\right )} c + c^{2}\right )} \tan \left (x e + d\right )^{4} + {\left (a b - b^{2} + b c\right )} \tan \left (x e + d\right )^{2} + a^{2} - a b + a c\right )}}\right )\right )} e^{\left (-1\right )}}{2 \, {\left (a^{2} - a b + a c\right )}}\right ] \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 {\cot {\left (d + e x \right )}}{\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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {cot}\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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