Optimal. Leaf size=250 \[ -\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}+\frac {a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
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Rubi [A]
time = 0.37, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3713, 3686,
3716, 3707, 3698, 31, 3556} \begin {gather*} \frac {a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^3 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 (\cos (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}+\frac {a \left (a^5 C+3 a^3 b^2 C+a^2 b^3 B+6 a b^4 C-3 b^5 B\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 3556
Rule 3686
Rule 3698
Rule 3707
Rule 3713
Rule 3716
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx &=\int \frac {\tan ^3(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\\ &=\frac {a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan (c+d x) \left (-2 a (b B-a C)+2 b (b B-a C) \tan (c+d x)+2 \left (a^2+b^2\right ) C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {-2 a \left (2 b^3 B-a^3 C-3 a b^2 C\right )-2 b^2 \left (a^2 B-b^2 B+2 a b C\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 C \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^2 \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) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \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 \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \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 (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^3 d}\\ &=-\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 (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}+\frac {a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.23, size = 462, normalized size = 1.85 \begin {gather*} \frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (a^3 b^2 \left (a^2+b^2\right ) (b B-a C)-2 a b \left (a^2+b^2\right ) \left (-3 b^3 B+a^3 C+4 a b^2 C\right ) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+2 b^3 \left (-3 a^2 b B+b^3 B+a^3 C-3 a b^2 C\right ) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2+2 i a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2-2 i a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) \text {ArcTan}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2-2 \left (a^2+b^2\right )^3 C \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right ) (a \cos (c+d x)+b \sin (c+d x))^2\right ) (B+C \tan (c+d x))}{2 b^3 \left (a^2+b^2\right )^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 242, normalized size = 0.97 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 366, normalized size = 1.46 \begin {gather*} \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 (C a^{6} + 3 \, C a^{4} b^{2} + B a^{3} b^{3} + 6 \, C a^{2} b^{4} - 3 \, B a b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \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 {3 \, C a^{6} - B a^{5} b + 7 \, C a^{4} b^{2} - 5 \, B a^{3} b^{3} + 2 \, {\left (2 \, C a^{5} b - B a^{4} b^{2} + 4 \, C a^{3} b^{3} - 3 \, B a^{2} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} + {\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs.
\(2 (243) = 486\).
time = 5.07, size = 666, normalized size = 2.66 \begin {gather*} \frac {C a^{6} b^{2} + B a^{5} b^{3} + 7 \, C a^{4} b^{4} - 5 \, B a^{3} b^{5} + 2 \, {\left (C a^{5} b^{3} - 3 \, B a^{4} b^{4} - 3 \, C a^{3} b^{5} + B a^{2} b^{6}\right )} d x - {\left (3 \, C a^{6} b^{2} - B a^{5} b^{3} + 9 \, C a^{4} b^{4} - 7 \, B a^{3} b^{5} - 2 \, {\left (C a^{3} b^{5} - 3 \, B a^{2} b^{6} - 3 \, C a b^{7} + B b^{8}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (C a^{8} + 3 \, C a^{6} b^{2} + B a^{5} b^{3} + 6 \, C a^{4} b^{4} - 3 \, B a^{3} b^{5} + {\left (C a^{6} b^{2} + 3 \, C a^{4} b^{4} + B a^{3} b^{5} + 6 \, C a^{2} b^{6} - 3 \, B a b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b + 3 \, C a^{5} b^{3} + B a^{4} b^{4} + 6 \, C a^{3} b^{5} - 3 \, B a^{2} b^{6}\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 ) - {\left (C a^{8} + 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} + C a^{2} b^{6} + {\left (C a^{6} b^{2} + 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} + C b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (C a^{7} b + 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} + C a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (C a^{7} b + 3 \, C a^{5} b^{3} - 3 \, B a^{4} b^{4} - 4 \, C a^{3} b^{5} + 3 \, B a^{2} b^{6} - 2 \, {\left (C a^{4} b^{4} - 3 \, B a^{3} b^{5} - 3 \, C a^{2} b^{6} + B a b^{7}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right ) + {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}} \end {gather*}
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 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 458, normalized size = 1.83 \begin {gather*} \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 (C a^{6} + 3 \, C a^{4} b^{2} + B a^{3} b^{3} + 6 \, C a^{2} b^{4} - 3 \, B a b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac {3 \, C a^{6} b \tan \left (d x + c\right )^{2} + 9 \, C a^{4} b^{3} \tan \left (d x + c\right )^{2} + 3 \, B a^{3} b^{4} \tan \left (d x + c\right )^{2} + 18 \, C a^{2} b^{5} \tan \left (d x + c\right )^{2} - 9 \, B a b^{6} \tan \left (d x + c\right )^{2} + 2 \, C a^{7} \tan \left (d x + c\right ) + 2 \, B a^{6} b \tan \left (d x + c\right ) + 6 \, C a^{5} b^{2} \tan \left (d x + c\right ) + 14 \, B a^{4} b^{3} \tan \left (d x + c\right ) + 28 \, C a^{3} b^{4} \tan \left (d x + c\right ) - 12 \, B a^{2} b^{5} \tan \left (d x + c\right ) + B a^{7} - C a^{6} b + 9 \, B a^{5} b^{2} + 11 \, C a^{4} b^{3} - 4 \, B a^{3} b^{4}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.32, size = 307, normalized size = 1.23 \begin {gather*} \frac {\frac {3\,C\,a^6-B\,a^5\,b+7\,C\,a^4\,b^2-5\,B\,a^3\,b^3}{2\,b^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {a^2\,\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,C\,a^3+B\,a^2\,b-4\,C\,a\,b^2+3\,B\,b^3\right )}{b^2\,\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 (\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 )}+\frac {a\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (C\,a^5+3\,C\,a^3\,b^2+B\,a^2\,b^3+6\,C\,a\,b^4-3\,B\,b^5\right )}{b^3\,d\,{\left (a^2+b^2\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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