Optimal. Leaf size=299 \[ -\frac{2 \left (A d^2-B c d+c^2 C\right ) \sqrt{a+b \tan (e+f x)}}{d f \left (c^2+d^2\right ) \sqrt{c+d \tan (e+f x)}}-\frac{\sqrt{a-i b} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac{\sqrt{a+i b} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}}+\frac{2 \sqrt{b} C \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f} \]
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Rubi [A] time = 3.33311, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {3645, 3655, 6725, 63, 217, 206, 93, 208} \[ -\frac{2 \left (A d^2-B c d+c^2 C\right ) \sqrt{a+b \tan (e+f x)}}{d f \left (c^2+d^2\right ) \sqrt{c+d \tan (e+f x)}}-\frac{\sqrt{a-i b} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c-i d)^{3/2}}-\frac{\sqrt{a+i b} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (c+i d)^{3/2}}+\frac{2 \sqrt{b} C \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 3645
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{2 \int \frac{\frac{1}{2} (A d (a c+b d)+(b c-a d) (c C-B d))+\frac{1}{2} d ((A-C) (b c-a d)+B (a c+b d)) \tan (e+f x)+\frac{1}{2} b C \left (c^2+d^2\right ) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} (A d (a c+b d)+(b c-a d) (c C-B d))+\frac{1}{2} d ((A-C) (b c-a d)+B (a c+b d)) x+\frac{1}{2} b C \left (c^2+d^2\right ) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{b C \left (c^2+d^2\right )}{2 \sqrt{a+b x} \sqrt{c+d x}}+\frac{d (a (A c-c C+B d)-b (B c-(A-C) d))+d (A b c+a B c-b c C-a A d+b B d+a C d) x}{2 \sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{(b C) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{d f}+\frac{\operatorname{Subst}\left (\int \frac{d (a (A c-c C+B d)-b (B c-(A-C) d))+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 )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{(2 C) \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 )}{d f}+\frac{\operatorname{Subst}\left (\int \left (\frac{-d (A b c+a B c-b c C-a A d+b B d+a C d)+i d (a (A c-c C+B d)-b (B c-(A-C) d))}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{d (A b c+a B c-b c C-a A d+b B d+a C d)+i d (a (A c-c C+B d)-b (B c-(A-C) d))}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{d \left (c^2+d^2\right ) f}\\ &=-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{((i a+b) (A-i B-C)) \operatorname{Subst}\left (\int \frac{1}{(i+x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c-i d) f}+\frac{((i a-b) (A+i B-C)) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (c+i d) f}+\frac{(2 C) \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 )}{d f}\\ &=\frac{2 \sqrt{b} C \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}+\frac{((i a+b) (A-i B-C)) \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 )}{(c-i d) f}+\frac{((i a-b) (A+i B-C)) \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 )}{(c+i d) f}\\ &=-\frac{\sqrt{a-i b} (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 )}{(c-i d)^{3/2} f}-\frac{\sqrt{a+i b} (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 )}{(c+i d)^{3/2} f}+\frac{2 \sqrt{b} C \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac{2 \left (c^2 C-B c d+A d^2\right ) \sqrt{a+b \tan (e+f x)}}{d \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 35.4382, size = 621084, normalized size = 2077.2 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{(A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2})\sqrt{a+b\tan \left ( fx+e \right ) } \left ( c+d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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 \frac{\sqrt{a + b \tan{\left (e + f x \right )}} \left (A + B \tan{\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (c + d \tan{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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