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 = 4.97305, 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 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
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*}
Mathematica [A] time = 7.70825, size = 619, normalized size = 1.62 \[ \frac{\frac{-\frac{\sqrt{b} \sqrt{c-\frac{a d}{b}} \left (a^2 C d^2-2 a b d (2 B d+c C)+b^2 \left (-8 d^2 (A-C)-4 B c d+c^2 C\right )\right ) \sqrt{\frac{b c+b d \tan (e+f x)}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c-\frac{a d}{b}}}\right )}{2 \sqrt{d} \sqrt{c+d \tan (e+f x)}}+\frac{2 b d \left (b (a A d+a B c-a C d+A b c-b B d-b c C)-\sqrt{-b^2} (-a (A c-B d-c C)+b d (A-C)+b B c)\right ) \tan ^{-1}\left (\frac{\sqrt{\frac{b d}{\sqrt{-b^2}}+c} \sqrt{a+b \tan (e+f x)}}{\sqrt{\sqrt{-b^2}-a} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{\sqrt{-b^2}-a} \sqrt{\frac{b d}{\sqrt{-b^2}}+c}}-\frac{2 b d \left (\sqrt{-b^2} (-a (A c-B d-c C)+b d (A-C)+b B c)+b (a A d+a B c-a C d+A b c-b B d-b c C)\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{\sqrt{-b^2} d+b c}{b}} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+\sqrt{-b^2}} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{a+\sqrt{-b^2}} \sqrt{-\frac{\sqrt{-b^2} d+b c}{b}}}}{b^2 f}+\frac{(a C d+4 b B d-b c C) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{2 b f}}{2 d}+\frac{C \sqrt{a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 d f} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\tan \left ( fx+e \right ) }\sqrt{c+d\tan \left ( fx+e \right ) } \left ( A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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