Optimal. Leaf size=179 \[ \frac {a (b B-a C)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a^2 B+2 a b C-b^2 B}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right )}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.25, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3628, 3529, 3531, 3530} \[ \frac {a (b B-a C)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a^2 B+2 a b C-b^2 B}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3530
Rule 3531
Rule 3628
Rubi steps
\begin {align*} \int \frac {B \tan (c+d x)+C \tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {a (b B-a C)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {b B-a C+(a B+b C) \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{a^2+b^2}\\ &=\frac {a (b B-a C)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2 B-b^2 B+2 a b C}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a b B-a^2 C+b^2 C+\left (a^2 B-b^2 B+2 a b C\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {a (b B-a C)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2 B-b^2 B+2 a b C}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a (b B-a C)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a^2 B-b^2 B+2 a b C}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 4.07, size = 188, normalized size = 1.05 \[ \frac {\frac {a (b B-a C)}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 \left (a^2 B+2 a b C-b^2 B\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {2 \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac {(B+i C) \log (-\tan (c+d x)+i)}{(a+i b)^3}+\frac {(B-i C) \log (\tan (c+d x)+i)}{(a-i b)^3}}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 488, normalized size = 2.73 \[ -\frac {3 \, C a^{4} b - 5 \, B a^{3} b^{2} - 3 \, C a^{2} b^{3} + B a b^{4} + 2 \, {\left (C a^{5} - 3 \, B a^{4} b - 3 \, C a^{3} b^{2} + B a^{2} b^{3}\right )} d x - {\left (C a^{4} b - 3 \, B a^{3} b^{2} - 5 \, C a^{2} b^{3} + 3 \, B a b^{4} - 2 \, {\left (C a^{3} b^{2} - 3 \, B a^{2} b^{3} - 3 \, C a b^{4} + B b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (B a^{5} + 3 \, C a^{4} b - 3 \, B a^{3} b^{2} - C a^{2} b^{3} + {\left (B a^{3} b^{2} + 3 \, C a^{2} b^{3} - 3 \, B a b^{4} - C b^{5}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{4} b + 3 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3} - C a b^{4}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (C a^{5} - 2 \, B a^{4} b - 3 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3} + 2 \, C a b^{4} - B b^{5} - 2 \, {\left (C a^{4} b - 3 \, B a^{3} b^{2} - 3 \, C a^{2} b^{3} + B a b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.65, size = 410, normalized size = 2.29 \[ -\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (B a^{3} b + 3 \, C a^{2} b^{2} - 3 \, B a b^{3} - C b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {3 \, B a^{3} b^{3} \tan \left (d x + c\right )^{2} + 9 \, C a^{2} b^{4} \tan \left (d x + c\right )^{2} - 9 \, B a b^{5} \tan \left (d x + c\right )^{2} - 3 \, C b^{6} \tan \left (d x + c\right )^{2} + 8 \, B a^{4} b^{2} \tan \left (d x + c\right ) + 22 \, C a^{3} b^{3} \tan \left (d x + c\right ) - 18 \, B a^{2} b^{4} \tan \left (d x + c\right ) - 2 \, C a b^{5} \tan \left (d x + c\right ) - 2 \, B b^{6} \tan \left (d x + c\right ) - C a^{6} + 6 \, B a^{5} b + 11 \, C a^{4} b^{2} - 7 \, B a^{3} b^{3} - B a b^{5}}{{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 488, normalized size = 2.73 \[ \frac {a B}{2 d \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {a^{2} C}{2 d \left (a^{2}+b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{2} B}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{2} B}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 C a b}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{3} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 a \,b^{2} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 a^{2} b \ln \left (a +b \tan \left (d x +c \right )\right ) C}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {\ln \left (a +b \tan \left (d x +c \right )\right ) b^{3} C}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} B}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B a \,b^{2}}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C \,a^{2} b}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{3} C}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 B \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {C \arctan \left (\tan \left (d x +c \right )\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 C \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 330, normalized size = 1.84 \[ -\frac {\frac {2 \, {\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {C a^{4} - 3 \, B a^{3} b - 3 \, C a^{2} b^{2} + B a b^{3} - 2 \, {\left (B a^{2} b^{2} + 2 \, C a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.28, size = 282, normalized size = 1.58 \[ \frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^2\,b+2\,C\,a\,b^2-B\,b^3\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {C\,a^4-3\,B\,a^3\,b-3\,C\,a^2\,b^2+B\,a\,b^3}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {B\,a+3\,C\,b}{{\left (a^2+b^2\right )}^2}-\frac {4\,b^2\,\left (B\,a+C\,b\right )}{{\left (a^2+b^2\right )}^3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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