Optimal. Leaf size=382 \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b f \left (a^2+b^2\right ) \sqrt{a+b \tan (e+f x)}}+\frac{d \left (3 a^2 C-2 a b B+2 A b^2+b^2 C\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 f \left (a^2+b^2\right )}-\frac{(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2}}-\frac{(c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2}}+\frac{\sqrt{d} (-3 a C d+2 b B d+3 b c C) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{5/2} f} \]
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Rubi [A] time = 5.73866, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {3645, 3647, 3655, 6725, 63, 217, 206, 93, 208} \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b f \left (a^2+b^2\right ) \sqrt{a+b \tan (e+f x)}}+\frac{d \left (3 a^2 C-2 a b B+2 A b^2+b^2 C\right ) \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 f \left (a^2+b^2\right )}-\frac{(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{3/2}}-\frac{(c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{3/2}}+\frac{\sqrt{d} (-3 a C d+2 b B d+3 b c C) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{5/2} f} \]
Antiderivative was successfully verified.
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Rule 3645
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^{3/2}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \int \frac{\sqrt{c+d \tan (e+f x)} \left (\frac{1}{2} ((b B-a C) (b c-3 a d)+A b (a c+3 b d))-\frac{1}{2} b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)+\frac{1}{2} \left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \tan ^2(e+f x)\right )}{\sqrt{a+b \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \int \frac{\frac{1}{4} \left (-\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d (b c+a d)+2 b c ((b B-a C) (b c-3 a d)+A b (a c+3 b d))\right )+\frac{1}{2} b^2 \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)+\frac{1}{4} \left (a^2+b^2\right ) d (3 b c C+2 b B d-3 a C d) \tan ^2(e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{4} \left (-\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d (b c+a d)+2 b c ((b B-a C) (b c-3 a d)+A b (a c+3 b d))\right )+\frac{1}{2} b^2 \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 ) x+\frac{1}{4} \left (a^2+b^2\right ) d (3 b c C+2 b B d-3 a C d) x^2}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^2 \left (a^2+b^2\right ) f}\\ &=\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{\left (a^2+b^2\right ) d (3 b c C+2 b B d-3 a C d)}{4 \sqrt{a+b x} \sqrt{c+d x}}+\frac{-b^2 \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 )+b^2 \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 ) x}{2 \sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{b^2 \left (a^2+b^2\right ) f}\\ &=\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{-b^2 \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 )+b^2 \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 ) x}{\sqrt{a+b x} \sqrt{c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^2 \left (a^2+b^2\right ) f}+\frac{(d (3 b c C+2 b B d-3 a C d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b^2 f}\\ &=\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\operatorname{Subst}\left (\int \left (\frac{-b^2 \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 )-i b^2 \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 )}{2 (i-x) \sqrt{a+b x} \sqrt{c+d x}}+\frac{b^2 \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 )-i b^2 \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 )}{2 (i+x) \sqrt{a+b x} \sqrt{c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^2 \left (a^2+b^2\right ) f}+\frac{(d (3 b c C+2 b B d-3 a C d)) \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 )}{b^3 f}\\ &=\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\left ((i A+B-i C) (c-i d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{(i+x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b) f}+\frac{(d (3 b c C+2 b B d-3 a C d)) \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 )}{b^3 f}-\frac{\left (b^2 \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 )+i b^2 \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 )\right ) \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^2 \left (a^2+b^2\right ) f}\\ &=\frac{\sqrt{d} (3 b c C+2 b B d-3 a C d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{5/2} f}+\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}+\frac{\left ((i A+B-i C) (c-i d)^2\right ) \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 )}{(a-i b) f}-\frac{\left (b^2 \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 )+i b^2 \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 )\right ) \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^2 \left (a^2+b^2\right ) f}\\ &=-\frac{(i A+B-i C) (c-i d)^{3/2} \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 )}{(a-i b)^{3/2} f}+\frac{(i A-B-i C) (c+i d)^{3/2} \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 )}{(a+i b)^{3/2} f}+\frac{\sqrt{d} (3 b c C+2 b B d-3 a C d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b \tan (e+f x)}}{\sqrt{b} \sqrt{c+d \tan (e+f x)}}\right )}{b^{5/2} f}+\frac{\left (2 A b^2-2 a b B+3 a^2 C+b^2 C\right ) d \sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}}{b^2 \left (a^2+b^2\right ) f}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d \tan (e+f x))^{3/2}}{b \left (a^2+b^2\right ) f \sqrt{a+b \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 39.4796, size = 1073629, normalized size = 2810.55 \[ \text{Result too large to show} \]
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}) \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( a+b\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{\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 )}{\left (a + b \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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