Optimal. Leaf size=66 \[ \frac {a^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b d \left (a^2+b^2\right )}-\frac {a x}{a^2+b^2}-\frac {\log (\cos (c+d x))}{b d} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3541, 3475, 3484, 3530} \[ \frac {a^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b d \left (a^2+b^2\right )}+\frac {a^3 x}{b^2 \left (a^2+b^2\right )}-\frac {a x}{b^2}-\frac {\log (\cos (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3484
Rule 3530
Rule 3541
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx &=-\frac {a x}{b^2}+\frac {a^2 \int \frac {1}{a+b \tan (c+d x)} \, dx}{b^2}+\frac {\int \tan (c+d x) \, dx}{b}\\ &=-\frac {a x}{b^2}+\frac {a^3 x}{b^2 \left (a^2+b^2\right )}-\frac {\log (\cos (c+d x))}{b d}+\frac {a^2 \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {a x}{b^2}+\frac {a^3 x}{b^2 \left (a^2+b^2\right )}-\frac {\log (\cos (c+d x))}{b d}+\frac {a^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 78, normalized size = 1.18 \[ \frac {2 a^2 \log (a+b \tan (c+d x))+b (b+i a) \log (-\tan (c+d x)+i)+b (b-i a) \log (\tan (c+d x)+i)}{2 b d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 89, normalized size = 1.35 \[ -\frac {2 \, a b d x - a^{2} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (a^{2} + b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{2} b + b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 73, normalized size = 1.11 \[ \frac {\frac {2 \, a^{2} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b + b^{3}} - \frac {2 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 80, normalized size = 1.21 \[ \frac {a^{2} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right ) b}+\frac {b \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {a \arctan \left (\tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 72, normalized size = 1.09 \[ \frac {\frac {2 \, a^{2} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b + b^{3}} - \frac {2 \, {\left (d x + c\right )} a}{a^{2} + b^{2}} + \frac {b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.05, size = 78, normalized size = 1.18 \[ \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{b\,d\,\left (a^2+b^2\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.05, size = 405, normalized size = 6.14 \[ \begin {cases} \tilde {\infty } x \tan {\relax (c )} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {d x \tan {\left (c + d x \right )}}{2 i b d \tan {\left (c + d x \right )} + 2 b d} + \frac {i d x}{2 i b d \tan {\left (c + d x \right )} + 2 b d} + \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 i b d \tan {\left (c + d x \right )} + 2 b d} + \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 i b d \tan {\left (c + d x \right )} + 2 b d} + \frac {1}{2 i b d \tan {\left (c + d x \right )} + 2 b d} & \text {for}\: a = - i b \\- \frac {d x \tan {\left (c + d x \right )}}{- 2 i b d \tan {\left (c + d x \right )} + 2 b d} - \frac {i d x}{- 2 i b d \tan {\left (c + d x \right )} + 2 b d} - \frac {i \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{- 2 i b d \tan {\left (c + d x \right )} + 2 b d} + \frac {\log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{- 2 i b d \tan {\left (c + d x \right )} + 2 b d} + \frac {1}{- 2 i b d \tan {\left (c + d x \right )} + 2 b d} & \text {for}\: a = i b \\\frac {- x + \frac {\tan {\left (c + d x \right )}}{d}}{a} & \text {for}\: b = 0 \\\frac {x \tan ^{2}{\relax (c )}}{a + b \tan {\relax (c )}} & \text {for}\: d = 0 \\\frac {2 a^{2} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b d + 2 b^{3} d} - \frac {2 a b d x}{2 a^{2} b d + 2 b^{3} d} + \frac {b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b d + 2 b^{3} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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