Optimal. Leaf size=273 \[ \frac {2 (a B+A b-b C) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \sqrt {c+d \tan (e+f x)} (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac {(b+i a) (c-i d)^{3/2} (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(-b+i a) (c+i d)^{3/2} (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}-\frac {2 (-7 a C d-7 b B d+2 b c C) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f} \]
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Rubi [A] time = 0.88, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3637, 3630, 3528, 3539, 3537, 63, 208} \[ \frac {2 (a B+A b-b C) (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 \sqrt {c+d \tan (e+f x)} (a A d+a B c-a C d+A b c-b B d-b c C)}{f}-\frac {(b+i a) (c-i d)^{3/2} (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(-b+i a) (c+i d)^{3/2} (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}-\frac {2 (-7 a C d-7 b B d+2 b c C) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 3528
Rule 3537
Rule 3539
Rule 3630
Rule 3637
Rubi steps
\begin {align*} \int (a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} (2 b c C-7 a A d)-\frac {7}{2} (A b+a B-b C) d \tan (e+f x)+\frac {1}{2} (2 b c C-7 b B d-7 a C d) \tan ^2(e+f x)\right ) \, dx}{7 d}\\ &=-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int (c+d \tan (e+f x))^{3/2} \left (\frac {7}{2} (b B-a (A-C)) d-\frac {7}{2} (A b+a B-b C) d \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (\frac {7}{2} d (b B c+b (A-C) d-a (A c-c C-B d))-\frac {7}{2} d (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {2 \int \frac {\frac {7}{2} 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 )-\frac {7}{2} 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)}{\sqrt {c+d \tan (e+f x)}} \, dx}{7 d}\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {1}{2} \left ((a-i b) (A-i B-C) (c-i d)^2\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b) (A+i B-C) (c+i d)^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}+\frac {\left ((i a+b) (A-i B-C) (c-i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {\left ((i a-b) (A+i B-C) (c+i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}-\frac {\left ((a-i b) (A-i B-C) (c-i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {\left ((a+i b) (A+i B-C) (c+i d)^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac {(a-i b) (i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(i a-b) (A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 (A b c+a B c-b c C+a A d-b B d-a C d) \sqrt {c+d \tan (e+f x)}}{f}+\frac {2 (A b+a B-b C) (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {2 (2 b c C-7 b B d-7 a C d) (c+d \tan (e+f x))^{5/2}}{35 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{7 d f}\\ \end {align*}
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Mathematica [A] time = 4.44, size = 260, normalized size = 0.95 \[ \frac {\frac {35}{3} d (b+i a) (A-i B-C) \left (\sqrt {c+d \tan (e+f x)} (4 c+d \tan (e+f x)-3 i d)-3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )\right )+\frac {35}{3} d (b-i a) (A+i B-C) \left (\sqrt {c+d \tan (e+f x)} (4 c+d \tan (e+f x)+3 i d)-3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )\right )+\frac {2 (7 a C d+7 b B d-2 b c C) (c+d \tan (e+f x))^{5/2}}{d}+10 b C \tan (e+f x) (c+d \tan (e+f x))^{5/2}}{35 d f} \]
Antiderivative was successfully verified.
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fricas [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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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 = 0.55, size = 5149, normalized size = 18.86 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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