Optimal. Leaf size=40 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b+c} \tan (c+d x)}{\sqrt {a+c}}\right )}{d \sqrt {a+c} \sqrt {b+c}} \]
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Rubi [A] time = 0.61, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b+c} \tan (c+d x)}{\sqrt {a+c}}\right )}{d \sqrt {a+c} \sqrt {b+c}} \]
Antiderivative was successfully verified.
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Rule 205
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+c+(b+c) x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b+c} \tan (c+d x)}{\sqrt {a+c}}\right )}{\sqrt {a+c} \sqrt {b+c} d}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 40, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b+c} \tan (c+d x)}{\sqrt {a+c}}\right )}{d \sqrt {a+c} \sqrt {b+c}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 300, normalized size = 7.50 \[ \left [-\frac {\sqrt {-a b - {\left (a + b\right )} c - c^{2}} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2} + 8 \, {\left (a + b\right )} c + 8 \, c^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a b + b^{2} + {\left (3 \, a + 5 \, b\right )} c + 4 \, c^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b + 2 \, c\right )} \cos \left (d x + c\right )^{3} - {\left (b + c\right )} \cos \left (d x + c\right )\right )} \sqrt {-a b - {\left (a + b\right )} c - c^{2}} \sin \left (d x + c\right ) + b^{2} + 2 \, b c + c^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (a b - b^{2} + {\left (a - b\right )} c\right )} \cos \left (d x + c\right )^{2} + b^{2} + 2 \, b c + c^{2}}\right )}{4 \, {\left (a b + {\left (a + b\right )} c + c^{2}\right )} d}, -\frac {\arctan \left (\frac {{\left (a + b + 2 \, c\right )} \cos \left (d x + c\right )^{2} - b - c}{2 \, \sqrt {a b + {\left (a + b\right )} c + c^{2}} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right )}{2 \, \sqrt {a b + {\left (a + b\right )} c + c^{2}} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 76, normalized size = 1.90 \[ \frac {\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, b + 2 \, c\right ) + \arctan \left (\frac {b \tan \left (d x + c\right ) + c \tan \left (d x + c\right )}{\sqrt {a b + a c + b c + c^{2}}}\right )}{\sqrt {a b + a c + b c + c^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 34, normalized size = 0.85 \[ \frac {\arctan \left (\frac {\left (b +c \right ) \tan \left (d x +c \right )}{\sqrt {\left (a +c \right ) \left (b +c \right )}}\right )}{d \sqrt {\left (a +c \right ) \left (b +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 43, normalized size = 1.08 \[ \frac {\arctan \left (\frac {{\left (b + c\right )} \tan \left (d x + c\right )}{\sqrt {a b + {\left (a + b\right )} c + c^{2}}}\right )}{\sqrt {a b + {\left (a + b\right )} c + c^{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 45, normalized size = 1.12 \[ \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (b+c\right )}{\sqrt {a\,b+a\,c+b\,c+c^2}}\right )}{d\,\sqrt {a\,b+a\,c+b\,c+c^2}} \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )} + c \sec ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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