Optimal. Leaf size=389 \[ \frac {d \tan (e+f x) \left (A \left (2 a c d+b \left (c^2-d^2\right )\right )+a \left (B c^2-B d^2-2 c C d\right )-b \left (2 B c d+c^2 C-C d^2\right )\right )}{f}-\frac {\log (\cos (e+f x)) \left (A \left (3 a c^2 d-a d^3+b c^3-3 b c d^2\right )+a \left (B c^3-3 B c d^2-3 c^2 C d+C d^3\right )-b \left (3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )}{f}+x \left (a \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )-b \left (d (A-C) \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right )\right )+\frac {(a B+A b-b C) (c+d \tan (e+f x))^3}{3 f}+\frac {(c+d \tan (e+f x))^2 (a A d+a B c-a C d+A b c-b B d-b c C)}{2 f}-\frac {(-5 a C d-5 b B d+b c C) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f} \]
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Rubi [A] time = 0.71, antiderivative size = 387, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3637, 3630, 3528, 3525, 3475} \[ \frac {d \tan (e+f x) \left (2 a A c d+a B \left (c^2-d^2\right )-2 a c C d+A b \left (c^2-d^2\right )-b \left (2 B c d+c^2 C-C d^2\right )\right )}{f}-\frac {\log (\cos (e+f x)) \left (A \left (3 a c^2 d-a d^3+b c^3-3 b c d^2\right )+a \left (B c^3-3 B c d^2-3 c^2 C d+C d^3\right )-b \left (3 B c^2 d-B d^3+c^3 C-3 c C d^2\right )\right )}{f}-x \left (-a \left (A c^3-3 A c d^2-3 B c^2 d+B d^3-c^3 C+3 c C d^2\right )+b d (A-C) \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )\right )+\frac {(a B+A b-b C) (c+d \tan (e+f x))^3}{3 f}+\frac {(c+d \tan (e+f x))^2 (a A d+a B c-a C d+A b c-b B d-b c C)}{2 f}-\frac {(-5 a C d-5 b B d+b c C) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rule 3630
Rule 3637
Rubi steps
\begin {align*} \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {\int (c+d \tan (e+f x))^3 \left (b c C-5 a A d-5 (A b+a B-b C) d \tan (e+f x)+(b c C-5 b B d-5 a C d) \tan ^2(e+f x)\right ) \, dx}{5 d}\\ &=-\frac {(b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {\int (c+d \tan (e+f x))^3 (5 (b B-a (A-C)) d-5 (A b+a B-b C) d \tan (e+f x)) \, dx}{5 d}\\ &=\frac {(A b+a B-b C) (c+d \tan (e+f x))^3}{3 f}-\frac {(b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {\int (c+d \tan (e+f x))^2 (5 d (b B c+b (A-C) d-a (A c-c C-B d))-5 d (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)) \, dx}{5 d}\\ &=\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^2}{2 f}+\frac {(A b+a B-b C) (c+d \tan (e+f x))^3}{3 f}-\frac {(b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {\int (c+d \tan (e+f x)) \left (5 d \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )-5 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)\right ) \, dx}{5 d}\\ &=-\left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )\right ) x+\frac {d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)}{f}+\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^2}{2 f}+\frac {(A b+a B-b C) (c+d \tan (e+f x))^3}{3 f}-\frac {(b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\left (-A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )+b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )-a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right ) \int \tan (e+f x) \, dx\\ &=-\left (b (A-C) d \left (3 c^2-d^2\right )+b B \left (c^3-3 c d^2\right )-a \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right )\right ) x-\frac {\left (A \left (b c^3+3 a c^2 d-3 b c d^2-a d^3\right )-b \left (c^3 C+3 B c^2 d-3 c C d^2-B d^3\right )+a \left (B c^3-3 c^2 C d-3 B c d^2+C d^3\right )\right ) \log (\cos (e+f x))}{f}+\frac {d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)}{f}+\frac {(A b c+a B c-b c C+a A d-b B d-a C d) (c+d \tan (e+f x))^2}{2 f}+\frac {(A b+a B-b C) (c+d \tan (e+f x))^3}{3 f}-\frac {(b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^4}{20 d^2 f}+\frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}\\ \end {align*}
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Mathematica [C] time = 6.33, size = 297, normalized size = 0.76 \[ \frac {b C \tan (e+f x) (c+d \tan (e+f x))^4}{5 d f}-\frac {\frac {(-5 a C d-5 b B d+b c C) (c+d \tan (e+f x))^4}{4 d f}+\frac {5 \left ((a B+A b-b C) \left (-6 d^2 \left (6 c^2-d^2\right ) \tan (e+f x)-12 c d^3 \tan ^2(e+f x)-3 i (c-i d)^4 \log (\tan (e+f x)+i)+3 i (c+i d)^4 \log (-\tan (e+f x)+i)-2 d^4 \tan ^3(e+f x)\right )+3 (-a A d+a B c+a C d+A b c+b B d-b c C) \left (6 c d^2 \tan (e+f x)+(-d+i c)^3 \log (-\tan (e+f x)+i)-(d+i c)^3 \log (\tan (e+f x)+i)+d^3 \tan ^2(e+f x)\right )\right )}{6 f}}{5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.48, size = 386, normalized size = 0.99 \[ \frac {12 \, C b d^{3} \tan \left (f x + e\right )^{5} + 15 \, {\left (3 \, C b c d^{2} + {\left (C a + B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (3 \, C b c^{2} d + 3 \, {\left (C a + B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} \tan \left (f x + e\right )^{3} + 60 \, {\left ({\left ({\left (A - C\right )} a - B b\right )} c^{3} - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} f x + 30 \, {\left (C b c^{3} + 3 \, {\left (C a + B b\right )} c^{2} d + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} + {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} - 30 \, {\left ({\left (B a + {\left (A - C\right )} b\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{2} d - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} - {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 60 \, {\left ({\left (C a + B b\right )} c^{3} + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} - {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} \tan \left (f x + e\right )}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 994, normalized size = 2.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 387, normalized size = 0.99 \[ \frac {12 \, C b d^{3} \tan \left (f x + e\right )^{5} + 15 \, {\left (3 \, C b c d^{2} + {\left (C a + B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (3 \, C b c^{2} d + 3 \, {\left (C a + B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} \tan \left (f x + e\right )^{3} + 30 \, {\left (C b c^{3} + 3 \, {\left (C a + B b\right )} c^{2} d + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} + {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \tan \left (f x + e\right )^{2} + 60 \, {\left ({\left ({\left (A - C\right )} a - B b\right )} c^{3} - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d - 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} + {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} {\left (f x + e\right )} + 30 \, {\left ({\left (B a + {\left (A - C\right )} b\right )} c^{3} + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c^{2} d - 3 \, {\left (B a + {\left (A - C\right )} b\right )} c d^{2} - {\left ({\left (A - C\right )} a - B b\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 60 \, {\left ({\left (C a + B b\right )} c^{3} + 3 \, {\left (B a + {\left (A - C\right )} b\right )} c^{2} d + 3 \, {\left ({\left (A - C\right )} a - B b\right )} c d^{2} - {\left (B a + {\left (A - C\right )} b\right )} d^{3}\right )} \tan \left (f x + e\right )}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.04, size = 478, normalized size = 1.23 \[ x\,\left (A\,a\,c^3+A\,b\,d^3+B\,a\,d^3-B\,b\,c^3-C\,a\,c^3-C\,b\,d^3-3\,A\,a\,c\,d^2-3\,A\,b\,c^2\,d-3\,B\,a\,c^2\,d+3\,B\,b\,c\,d^2+3\,C\,a\,c\,d^2+3\,C\,b\,c^2\,d\right )+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {B\,b\,d^3}{4}+\frac {C\,a\,d^3}{4}+\frac {3\,C\,b\,c\,d^2}{4}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {A\,b\,d^3}{3}+\frac {B\,a\,d^3}{3}-\frac {C\,b\,d^3}{3}+B\,b\,c\,d^2+C\,a\,c\,d^2+C\,b\,c^2\,d\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {A\,a\,d^3}{2}-\frac {B\,b\,d^3}{2}-\frac {C\,a\,d^3}{2}+\frac {C\,b\,c^3}{2}+\frac {3\,A\,b\,c\,d^2}{2}+\frac {3\,B\,a\,c\,d^2}{2}+\frac {3\,B\,b\,c^2\,d}{2}+\frac {3\,C\,a\,c^2\,d}{2}-\frac {3\,C\,b\,c\,d^2}{2}\right )}{f}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )\,\left (\frac {A\,a\,d^3}{2}-\frac {A\,b\,c^3}{2}-\frac {B\,a\,c^3}{2}-\frac {B\,b\,d^3}{2}-\frac {C\,a\,d^3}{2}+\frac {C\,b\,c^3}{2}-\frac {3\,A\,a\,c^2\,d}{2}+\frac {3\,A\,b\,c\,d^2}{2}+\frac {3\,B\,a\,c\,d^2}{2}+\frac {3\,B\,b\,c^2\,d}{2}+\frac {3\,C\,a\,c^2\,d}{2}-\frac {3\,C\,b\,c\,d^2}{2}\right )}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (B\,b\,c^3-B\,a\,d^3-A\,b\,d^3+C\,a\,c^3+C\,b\,d^3+3\,A\,a\,c\,d^2+3\,A\,b\,c^2\,d+3\,B\,a\,c^2\,d-3\,B\,b\,c\,d^2-3\,C\,a\,c\,d^2-3\,C\,b\,c^2\,d\right )}{f}+\frac {C\,b\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.65, size = 1001, normalized size = 2.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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