Optimal. Leaf size=52 \[ \frac {\tan (c+d x)}{b d}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d} \]
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Rubi [A] time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3675, 388, 205} \[ \frac {\tan (c+d x)}{b d}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 388
Rule 3675
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{a+b \tan ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{a+b x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan (c+d x)}{b d}-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (c+d x)\right )}{b d}\\ &=-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}+\frac {\tan (c+d x)}{b d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 52, normalized size = 1.00 \[ \frac {\tan (c+d x)}{b d}-\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 267, normalized size = 5.13 \[ \left [\frac {\sqrt {-a b} {\left (a - b\right )} \cos \left (d x + c\right ) \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cos \left (d x + c\right )^{3} - b \cos \left (d x + c\right )\right )} \sqrt {-a b} \sin \left (d x + c\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 4 \, a b \sin \left (d x + c\right )}{4 \, a b^{2} d \cos \left (d x + c\right )}, \frac {\sqrt {a b} {\left (a - b\right )} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (d x + c\right )^{2} - b\right )} \sqrt {a b}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \cos \left (d x + c\right ) + 2 \, a b \sin \left (d x + c\right )}{2 \, a b^{2} d \cos \left (d x + c\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 62, normalized size = 1.19 \[ -\frac {\frac {{\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (d x + c\right )}{\sqrt {a b}}\right )\right )} {\left (a - b\right )}}{\sqrt {a b} b} - \frac {\tan \left (d x + c\right )}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 66, normalized size = 1.27 \[ \frac {\tan \left (d x +c \right )}{b d}-\frac {\arctan \left (\frac {\tan \left (d x +c \right ) b}{\sqrt {a b}}\right ) a}{d b \sqrt {a b}}+\frac {\arctan \left (\frac {\tan \left (d x +c \right ) b}{\sqrt {a b}}\right )}{d \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 45, normalized size = 0.87 \[ -\frac {\frac {{\left (a - b\right )} \arctan \left (\frac {b \tan \left (d x + c\right )}{\sqrt {a b}}\right )}{\sqrt {a b} b} - \frac {\tan \left (d x + c\right )}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.40, size = 44, normalized size = 0.85 \[ \frac {\mathrm {tan}\left (c+d\,x\right )}{b\,d}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {a}}\right )\,\left (a-b\right )}{\sqrt {a}\,b^{3/2}\,d} \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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