Optimal. Leaf size=82 \[ -\frac {b \sec (c+d x)}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {a \tanh ^{-1}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.28, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3512, 731, 725, 206} \[ -\frac {b \sec (c+d x)}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {a \sec (c+d x) \tanh ^{-1}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right )}{d \left (a^2+b^2\right )^{3/2} \sqrt {\sec ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 731
Rule 3512
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac {\sec (c+d x) \operatorname {Subst}\left (\int \frac {1}{(a+x)^2 \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {b \sec (c+d x)}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {(a \sec (c+d x)) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b \left (a^2+b^2\right ) d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {b \sec (c+d x)}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {(a \sec (c+d x)) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\sec ^2(c+d x)}}\right )}{b \left (a^2+b^2\right ) d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {a \tanh ^{-1}\left (\frac {b \left (1-\frac {a \tan (c+d x)}{b}\right )}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{\left (a^2+b^2\right )^{3/2} d \sqrt {\sec ^2(c+d x)}}-\frac {b \sec (c+d x)}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 78, normalized size = 0.95 \[ \frac {\frac {2 a \tanh ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )-b}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \sec (c+d x)}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 215, normalized size = 2.62 \[ -\frac {2 \, a^{2} b + 2 \, b^{3} - {\left (a^{2} \cos \left (d x + c\right ) + a b \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right )}{2 \, {\left ({\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \cos \left (d x + c\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.94, size = 138, normalized size = 1.68 \[ -\frac {\frac {a \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a b\right )}}{{\left (a^{3} + a b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 118, normalized size = 1.44 \[ \frac {-\frac {2 \left (-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}+b^{2}\right ) a}-\frac {b}{a^{2}+b^{2}}\right )}{a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a}+\frac {2 a \arctanh \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.69, size = 182, normalized size = 2.22 \[ -\frac {\frac {a \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (a b + \frac {b^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}}{a^{4} + a^{2} b^{2} + \frac {2 \, {\left (a^{3} b + a b^{3}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {{\left (a^{4} + a^{2} b^{2}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.98, size = 136, normalized size = 1.66 \[ -\frac {\frac {2\,b}{a^2+b^2}+\frac {2\,b^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,\left (a^2+b^2\right )}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}+\frac {a\,\mathrm {atan}\left (\frac {a^2\,b\,1{}\mathrm {i}+b^3\,1{}\mathrm {i}-a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^2+b^2\right )\,1{}\mathrm {i}}{{\left (a^2+b^2\right )}^{3/2}}\right )\,2{}\mathrm {i}}{d\,{\left (a^2+b^2\right )}^{3/2}} \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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