Optimal. Leaf size=59 \[ \frac {\left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^3 d}-\frac {a \tan (c+d x)}{b^2 d}+\frac {\tan ^2(c+d x)}{2 b d} \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3506, 697} \[ \frac {\left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^3 d}-\frac {a \tan (c+d x)}{b^2 d}+\frac {\tan ^2(c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+\frac {x^2}{b^2}}{a+x} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b^2}+\frac {x}{b^2}+\frac {a^2+b^2}{b^2 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\left (a^2+b^2\right ) \log (a+b \tan (c+d x))}{b^3 d}-\frac {a \tan (c+d x)}{b^2 d}+\frac {\tan ^2(c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 52, normalized size = 0.88 \[ \frac {\left (a^2+b^2\right ) \log (a+b \tan (c+d x))-a b \tan (c+d x)+\frac {1}{2} b^2 \tan ^2(c+d x)}{b^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 117, normalized size = 1.98 \[ \frac {{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\cos \left (d x + c\right )^{2}\right ) - 2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b^{2}}{2 \, b^{3} d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.77, size = 54, normalized size = 0.92 \[ \frac {\frac {b \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{b^{2}} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 72, normalized size = 1.22 \[ \frac {\tan ^{2}\left (d x +c \right )}{2 b d}-\frac {a \tan \left (d x +c \right )}{b^{2} d}+\frac {\ln \left (a +b \tan \left (d x +c \right )\right ) a^{2}}{d \,b^{3}}+\frac {\ln \left (a +b \tan \left (d x +c \right )\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 53, normalized size = 0.90 \[ \frac {\frac {b \tan \left (d x + c\right )^{2} - 2 \, a \tan \left (d x + c\right )}{b^{2}} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 57, normalized size = 0.97 \[ \frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b\,d}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2+b^2\right )}{b^3\,d}-\frac {a\,\mathrm {tan}\left (c+d\,x\right )}{b^2\,d} \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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