Optimal. Leaf size=111 \[ \frac {\left (a^2 B-b^2 B+2 a b C\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (2 a b B-a^2 C+b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b B-a C}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.15, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3713, 3610,
3612, 3611} \begin {gather*} -\frac {b B-a C}{d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\left (a^2 (-C)+2 a b B+b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {x \left (a^2 B+2 a b C-b^2 B\right )}{\left (a^2+b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rule 3713
Rubi steps
\begin {align*} \int \frac {\cot (c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx &=\int \frac {B+C \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx\\ &=-\frac {b B-a C}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a B+b C-(b B-a C) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {\left (a^2 B-b^2 B+2 a b C\right ) x}{\left (a^2+b^2\right )^2}-\frac {b B-a C}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (2 a b B-a^2 C+b^2 C\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {\left (a^2 B-b^2 B+2 a b C\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (2 a b B-a^2 C+b^2 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b B-a C}{\left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.52, size = 190, normalized size = 1.71 \begin {gather*} \frac {\frac {C ((-i a-b) \log (i-\tan (c+d x))+i (a+i b) \log (i+\tan (c+d x))+2 b \log (a+b \tan (c+d x)))}{a^2+b^2}-(b B-a C) \left (\frac {i \log (i-\tan (c+d x))}{(a+i b)^2}-\frac {i \log (i+\tan (c+d x))}{(a-i b)^2}+\frac {2 b \left (-2 a \log (a+b \tan (c+d x))+\frac {a^2+b^2}{a+b \tan (c+d x)}\right )}{\left (a^2+b^2\right )^2}\right )}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 141, normalized size = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-2 B a b +C \,a^{2}-b^{2} C \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{2} B -b^{2} B +2 C a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {B b -C a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (2 B a b -C \,a^{2}+b^{2} C \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(141\) |
default | \(\frac {\frac {\frac {\left (-2 B a b +C \,a^{2}-b^{2} C \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (a^{2} B -b^{2} B +2 C a b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {B b -C a}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\left (2 B a b -C \,a^{2}+b^{2} C \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(141\) |
norman | \(\frac {\frac {a \left (a^{2} B -b^{2} B +2 C a b \right ) x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {b \left (a^{2} B -b^{2} B +2 C a b \right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\left (B b -C a \right ) b \tan \left (d x +c \right )}{a d \left (a^{2}+b^{2}\right )}}{a +b \tan \left (d x +c \right )}+\frac {\left (2 B a b -C \,a^{2}+b^{2} C \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (2 B a b -C \,a^{2}+b^{2} C \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(226\) |
risch | \(-\frac {x B}{2 i b a -a^{2}+b^{2}}+\frac {i x C}{2 i b a -a^{2}+b^{2}}-\frac {4 i B a b x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i a^{2} C x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i C \,b^{2} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 i B a b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 i a^{2} C c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i C \,b^{2} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 i b^{2} B}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {2 i b C a}{\left (-i a +b \right ) d \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B a b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{2} C}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(482\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 177, normalized size = 1.59 \begin {gather*} \frac {\frac {2 \, {\left (B a^{2} + 2 \, C a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (C a^{2} - 2 \, B a b - C b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (C a^{2} - 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (C a - B b\right )}}{a^{3} + a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \left (d x + c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.13, size = 222, normalized size = 2.00 \begin {gather*} \frac {2 \, C a b^{2} - 2 \, B b^{3} + 2 \, {\left (B a^{3} + 2 \, C a^{2} b - B a b^{2}\right )} d x - {\left (C a^{3} - 2 \, B a^{2} b - C a b^{2} + {\left (C a^{2} b - 2 \, B a b^{2} - C b^{3}\right )} \tan \left (d x + c\right )\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 ) - 2 \, {\left (C a^{2} b - B a b^{2} - {\left (B a^{2} b + 2 \, C a b^{2} - B b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.23, size = 2895, normalized size = 26.08 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 234 vs.
\(2 (111) = 222\).
time = 0.95, size = 234, normalized size = 2.11 \begin {gather*} \frac {\frac {2 \, {\left (B a^{2} + 2 \, C a b - B b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (C a^{2} - 2 \, B a b - C b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (C a^{2} b - 2 \, B a b^{2} - C b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {2 \, {\left (C a^{2} b \tan \left (d x + c\right ) - 2 \, B a b^{2} \tan \left (d x + c\right ) - C b^{3} \tan \left (d x + c\right ) + 2 \, C a^{3} - 3 \, B a^{2} b - B b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.09, size = 153, normalized size = 1.38 \begin {gather*} \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-C\,a^2+2\,B\,a\,b+C\,b^2\right )}{d\,{\left (a^2+b^2\right )}^2}-\frac {B\,b-C\,a}{d\,\left (a^2+b^2\right )\,\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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