Optimal. Leaf size=76 \[ -\frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a b d}-\frac {x}{a}+\frac {\tanh ^{-1}(\sin (c+d x))}{b d} \]
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Rubi [A] time = 0.18, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3894, 4051, 3770, 3919, 3831, 2659, 208} \[ -\frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a b d}-\frac {x}{a}+\frac {\tanh ^{-1}(\sin (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 2659
Rule 3770
Rule 3831
Rule 3894
Rule 3919
Rule 4051
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x)}{a+b \sec (c+d x)} \, dx &=\int \frac {-1+\sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx\\ &=\frac {\int \sec (c+d x) \, dx}{b}+\frac {\int \frac {-b-a \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b}\\ &=-\frac {x}{a}+\frac {\tanh ^{-1}(\sin (c+d x))}{b d}-\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx\\ &=-\frac {x}{a}+\frac {\tanh ^{-1}(\sin (c+d x))}{b d}-\frac {\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{b}\\ &=-\frac {x}{a}+\frac {\tanh ^{-1}(\sin (c+d x))}{b d}-\frac {\left (2 \left (\frac {a}{b}-\frac {b}{a}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}\\ &=-\frac {x}{a}+\frac {\tanh ^{-1}(\sin (c+d x))}{b d}-\frac {2 \sqrt {a-b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a b d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 115, normalized size = 1.51 \[ -\frac {-2 \sqrt {a^2-b^2} \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+b c+b d x}{a b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 253, normalized size = 3.33 \[ \left [-\frac {2 \, b d x - a \log \left (\sin \left (d x + c\right ) + 1\right ) + a \log \left (-\sin \left (d x + c\right ) + 1\right ) - \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right )}{2 \, a b d}, -\frac {2 \, b d x - a \log \left (\sin \left (d x + c\right ) + 1\right ) + a \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right )}{2 \, a b d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.47, size = 140, normalized size = 1.84 \[ -\frac {\frac {d x + c}{a} - \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b} + \frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )} {\left (a^{2} - b^{2}\right )}}{\sqrt {-a^{2} + b^{2}} a b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 153, normalized size = 2.01 \[ -\frac {2 a \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d b \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 b \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d a \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d b}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d b}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 121, normalized size = 1.59 \[ \frac {2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b\,d}-\frac {2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a\,d}-\frac {2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {a^2-b^2}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a+b\right )}\right )\,\sqrt {a^2-b^2}}{a\,b\,d} \]
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 \[ \int \frac {\tan ^{2}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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