Optimal. Leaf size=85 \[ -\frac {a (b B-a C) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac {(a B+b C) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {x (b B-a C)}{a^2+b^2} \]
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Rubi [A] time = 0.16, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1629, 635, 203, 260} \[ -\frac {a (b B-a C) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac {(a B+b C) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac {x (b B-a C)}{a^2+b^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 1629
Rubi steps
\begin {align*} \int \frac {B \tan (c+d x)+C \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x (B+C x)}{(a+b x) \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a (-b B+a C)}{\left (a^2+b^2\right ) (a+b x)}+\frac {b B-a C+(a B+b C) x}{\left (a^2+b^2\right ) \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a (b B-a C) \log (a+b \tan (c+d x))}{b \left (a^2+b^2\right ) d}+\frac {\operatorname {Subst}\left (\int \frac {b B-a C+(a B+b C) x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac {a (b B-a C) \log (a+b \tan (c+d x))}{b \left (a^2+b^2\right ) d}+\frac {(b B-a C) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {(a B+b C) \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {(b B-a C) x}{a^2+b^2}-\frac {(a B+b C) \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a (b B-a C) \log (a+b \tan (c+d x))}{b \left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 98, normalized size = 1.15 \[ \frac {b (a-i b) (B+i C) \log (-\tan (c+d x)+i)+b (a+i b) (B-i C) \log (\tan (c+d x)+i)+2 a (a C-b B) \log (a+b \tan (c+d x))}{2 b d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.31, size = 110, normalized size = 1.29 \[ -\frac {2 \, {\left (C a b - B b^{2}\right )} d x - {\left (C a^{2} - B a b\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left (C a^{2} + C b^{2}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{2} b + b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.52, size = 95, normalized size = 1.12 \[ -\frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, {\left (C a^{2} - B a b\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b + b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 159, normalized size = 1.87 \[ -\frac {a \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \left (a^{2}+b^{2}\right )}+\frac {a^{2} \ln \left (a +b \tan \left (d x +c \right )\right ) C}{d \left (a^{2}+b^{2}\right ) b}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a B}{2 d \left (a^{2}+b^{2}\right )}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C b}{2 d \left (a^{2}+b^{2}\right )}+\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b}{d \left (a^{2}+b^{2}\right )}-\frac {C \arctan \left (\tan \left (d x +c \right )\right ) a}{d \left (a^{2}+b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 94, normalized size = 1.11 \[ -\frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {2 \, {\left (C a^{2} - B a b\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b + b^{3}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.07, size = 100, normalized size = 1.18 \[ \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-b+a\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}-\frac {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (B\,b-C\,a\right )}{b\,d\,\left (a^2+b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 724, normalized size = 8.52 \[ \begin {cases} \frac {\tilde {\infty } x \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right )}{\tan {\relax (c )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {B d x \tan {\left (c + d x \right )}}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i B d x}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {B}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {i C d x \tan {\left (c + d x \right )}}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {C d x}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i C}{- 2 b d \tan {\left (c + d x \right )} + 2 i b d} & \text {for}\: a = - i b \\- \frac {B d x \tan {\left (c + d x \right )}}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i B d x}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {B}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {i C d x \tan {\left (c + d x \right )}}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {C d x}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i C}{- 2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = i b \\\frac {\frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - C x + \frac {C \tan {\left (c + d x \right )}}{d}}{a} & \text {for}\: b = 0 \\\frac {x \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right )}{a + b \tan {\relax (c )}} & \text {for}\: d = 0 \\- \frac {2 B a b \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b d + 2 b^{3} d} + \frac {B a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b d + 2 b^{3} d} + \frac {2 B b^{2} d x}{2 a^{2} b d + 2 b^{3} d} + \frac {2 C 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 C a b d x}{2 a^{2} b d + 2 b^{3} d} + \frac {C 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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