Optimal. Leaf size=381 \[ -\frac {\left (a^2 C d^2-2 a b d (2 B d+c C)+b^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{3/2} d^{3/2} f}-\frac {\sqrt {a-i b} \sqrt {c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\sqrt {a+i b} \sqrt {c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(-a C d-4 b B d+b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f} \]
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Rubi [A] time = 4.97, antiderivative size = 383, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 8, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3647, 3655, 6725, 63, 217, 206, 93, 208} \[ -\frac {\left (a^2 C d^2-2 a b d (2 B d+c C)+b^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{3/2} d^{3/2} f}-\frac {\sqrt {a-i b} \sqrt {c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {a+i b} \sqrt {c+i d} (i A-B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(-a C d-4 b B d+b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 206
Rule 208
Rule 217
Rule 3647
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (\frac {1}{2} (-b c C+a (4 A-3 C) d)+2 (A b+a B-b C) d \tan (e+f x)-\frac {1}{2} (b c C-4 b B d-a C d) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{2 d}\\ &=-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {\int \frac {\frac {1}{4} \left (-a^2 C d^2+2 a b d (4 A c-3 c C-2 B d)-b^2 c (c C+4 B d)\right )+2 b d (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)+\frac {1}{4} \left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-4 b B d-a C d)\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{2 b d}\\ &=-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} \left (-a^2 C d^2+2 a b d (4 A c-3 c C-2 B d)-b^2 c (c C+4 B d)\right )+2 b d (A b c+a B c-b c C+a A d-b B d-a C d) x+\frac {1}{4} \left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-4 b B d-a C d)\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{2 b d f}\\ &=-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {\operatorname {Subst}\left (\int \left (\frac {-a^2 C d^2+2 a b d (c C+2 B d)-b^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )}{4 \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 (-b d (b B c+b (A-C) d-a (A c-c C-B d))+b d (A b c+a B c-b c C+a A d-b B d-a C d) x)}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{2 b d f}\\ &=-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {\operatorname {Subst}\left (\int \frac {-b d (b B c+b (A-C) d-a (A c-c C-B d))+b d (A b c+a B c-b c C+a A d-b B d-a C d) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b d f}-\frac {\left (a^2 C d^2-2 a b d (c C+2 B d)+b^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{8 b d f}\\ &=-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {\operatorname {Subst}\left (\int \left (\frac {-b d (A b c+a B c-b c C+a A d-b B d-a C d)-i b d (b B c+b (A-C) d-a (A c-c C-B d))}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {b d (A b c+a B c-b c C+a A d-b B d-a C d)-i b d (b B c+b (A-C) d-a (A c-c C-B d))}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b d f}-\frac {\left (a^2 C d^2-2 a b d (c C+2 B d)+b^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{4 b^2 d f}\\ &=-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {((i a+b) (A-i B-C) (c-i d)) \operatorname {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}-\frac {\left (a^2 C d^2-2 a b d (c C+2 B d)+b^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{4 b^2 d f}-\frac {(b d (A b c+a B c-b c C+a A d-b B d-a C d)+i b d (b B c+b (A-C) d-a (A c-c C-B d))) \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b d f}\\ &=-\frac {\left (a^2 C d^2-2 a b d (c C+2 B d)+b^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{3/2} d^{3/2} f}-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}+\frac {((i a+b) (A-i B-C) (c-i d)) \operatorname {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(b d (A b c+a B c-b c C+a A d-b B d-a C d)+i b d (b B c+b (A-C) d-a (A c-c C-B d))) \operatorname {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{b d f}\\ &=-\frac {\sqrt {a-i b} (i A+B-i C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {a+i b} (i A-B-i C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\left (a^2 C d^2-2 a b d (c C+2 B d)+b^2 \left (c^2 C-4 B c d-8 (A-C) d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{3/2} d^{3/2} f}-\frac {(b c C-4 b B d-a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f}\\ \end {align*}
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Mathematica [A] time = 7.79, size = 619, normalized size = 1.62 \[ \frac {\frac {-\frac {\sqrt {b} \sqrt {c-\frac {a d}{b}} \left (a^2 C d^2-2 a b d (2 B d+c C)+b^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right )}{2 \sqrt {d} \sqrt {c+d \tan (e+f x)}}+\frac {2 b d \left (b (a A d+a B c-a C d+A b c-b B d-b c C)-\sqrt {-b^2} (-a (A c-B d-c C)+b d (A-C)+b B c)\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}-c} \sqrt {a+b \tan (e+f x)}}{\sqrt {\sqrt {-b^2}-a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {\sqrt {-b^2}-a} \sqrt {\frac {\sqrt {-b^2} d}{b}-c}}-\frac {2 b d \left (\sqrt {-b^2} (-a (A c-B d-c C)+b d (A-C)+b B c)+b (a A d+a B c-a C d+A b c-b B d-b c C)\right ) \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {-b^2} d}{b}+c} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {\frac {\sqrt {-b^2} d}{b}+c}}}{b^2 f}+\frac {(a C d+4 b B d-b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{2 b f}}{2 d}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f} \]
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 [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a +b \tan \left (f x +e \right )}\, \sqrt {c +d \tan \left (f x +e \right )}\, \left (A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} \sqrt {b \tan \left (f x + e\right ) + a} \sqrt {d \tan \left (f x + e\right ) + c}\,{d x} \]
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 \[ \int \sqrt {a + b \tan {\left (e + f x \right )}} \sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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