Optimal. Leaf size=447 \[ -\frac{d \left (A \left (2 a^2 d^2+b^2 \left (c^2+3 d^2\right )\right )+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{\sqrt{b} \left (-a^2 b^2 (d (7 A-C)+2 B c)+5 a^3 b B d-3 a^4 C d+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{5/2}}-\frac{(i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (a-i b)^2 (c-i d)^{3/2}}-\frac{(B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (a+i b)^2 (c+i d)^{3/2}} \]
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Rubi [A] time = 2.88148, antiderivative size = 446, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.149, Rules used = {3649, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac{d \left (2 a^2 A d^2+a^2 \left (-2 B c d+3 c^2 C+C d^2\right )-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+2 b^2 c (c C-B d)\right )}{f \left (a^2+b^2\right ) \left (c^2+d^2\right ) (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{f \left (a^2+b^2\right ) (b c-a d) (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{\sqrt{b} \left (-a^2 b^2 (d (7 A-C)+2 B c)+5 a^3 b B d-3 a^4 C d+a b^3 (4 A c+B d-4 c C)+b^4 (2 B c-3 A d)\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{f \left (a^2+b^2\right )^2 (b c-a d)^{5/2}}-\frac{(i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (a-i b)^2 (c-i d)^{3/2}}-\frac{(B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (a+i b)^2 (c+i d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \frac{A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{\int \frac{\frac{1}{2} \left (3 A b^2 d-2 a A (b c-a d)-(b B-a C) (2 b c+a d)\right )+(A b-a B-b C) (b c-a d) \tan (e+f x)+\frac{3}{2} \left (A b^2-a (b B-a C)\right ) d \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx}{\left (a^2+b^2\right ) (b c-a d)}\\ &=-\frac{d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{2 \int \frac{\frac{1}{4} \left (-2 a^3 d^2 (A c-c C+B d)+a^2 b (4 A-C) d \left (c^2+d^2\right )-b^3 (2 B c-3 A d) \left (c^2+d^2\right )+a b^2 \left (2 c^3 C+B c^2 d+4 c C d^2-B d^3-2 A \left (c^3+2 c d^2\right )\right )\right )+\frac{1}{2} (b c-a d)^2 (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} b d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )}\\ &=-\frac{d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{2 \int \frac{-\frac{1}{2} (b c-a d)^2 \left (a^2 (A c-c C+B d)-b^2 (A c-c C+B d)+2 a b (B c-(A-C) d)\right )+\frac{1}{2} (b c-a d)^2 \left (2 a b (A c-c C+B d)-a^2 (B c-(A-C) d)+b^2 (B c-(A-C) d)\right ) \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)^2 \left (c^2+d^2\right )}-\frac{\left (2 \left (-\frac{1}{2} a b (b c-a d)^2 (A b c-a B c-b c C+a A d+b B d-a C d)+\frac{1}{4} a^2 b d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )+\frac{1}{4} b^2 \left (-2 a^3 d^2 (A c-c C+B d)+a^2 b (4 A-C) d \left (c^2+d^2\right )-b^3 (2 B c-3 A d) \left (c^2+d^2\right )+a b^2 \left (2 c^3 C+B c^2 d+4 c C d^2-B d^3-2 A \left (c^3+2 c d^2\right )\right )\right )\right )\right ) \int \frac{1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2 (b c-a d)^2 \left (c^2+d^2\right )}\\ &=-\frac{d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}+\frac{(A-i B-C) \int \frac{1+i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2 (c-i d)}+\frac{(A+i B-C) \int \frac{1-i \tan (e+f x)}{\sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2 (c+i d)}-\frac{\left (2 \left (-\frac{1}{2} a b (b c-a d)^2 (A b c-a B c-b c C+a A d+b B d-a C d)+\frac{1}{4} a^2 b d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )+\frac{1}{4} b^2 \left (-2 a^3 d^2 (A c-c C+B d)+a^2 b (4 A-C) d \left (c^2+d^2\right )-b^3 (2 B c-3 A d) \left (c^2+d^2\right )+a b^2 \left (2 c^3 C+B c^2 d+4 c C d^2-B d^3-2 A \left (c^3+2 c d^2\right )\right )\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{\left (a^2+b^2\right )^2 (b c-a d)^2 \left (c^2+d^2\right ) f}\\ &=-\frac{d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}+\frac{(i A+B-i C) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 (c-i d) f}-\frac{(i (A+i B-C)) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 (c+i d) f}-\frac{\left (4 \left (-\frac{1}{2} a b (b c-a d)^2 (A b c-a B c-b c C+a A d+b B d-a C d)+\frac{1}{4} a^2 b d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )+\frac{1}{4} b^2 \left (-2 a^3 d^2 (A c-c C+B d)+a^2 b (4 A-C) d \left (c^2+d^2\right )-b^3 (2 B c-3 A d) \left (c^2+d^2\right )+a b^2 \left (2 c^3 C+B c^2 d+4 c C d^2-B d^3-2 A \left (c^3+2 c d^2\right )\right )\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d \tan (e+f x)}\right )}{\left (a^2+b^2\right )^2 d (b c-a d)^2 \left (c^2+d^2\right ) f}\\ &=-\frac{\sqrt{b} \left (5 a^3 b B d-3 a^4 C d+b^4 (2 B c-3 A d)+a b^3 (4 A c-4 c C+B d)-a^2 b^2 (2 B c+7 A d-C d)\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{5/2} f}-\frac{d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}-\frac{(A-i B-C) \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 )}{(a-i b)^2 (c-i d) d f}-\frac{(A+i B-C) \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 )}{(a+i b)^2 (c+i d) d f}\\ &=-\frac{(i A+B-i C) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{(a-i b)^2 (c-i d)^{3/2} f}-\frac{(B-i (A-C)) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{(a+i b)^2 (c+i d)^{3/2} f}-\frac{\sqrt{b} \left (5 a^3 b B d-3 a^4 C d+b^4 (2 B c-3 A d)+a b^3 (4 A c-4 c C+B d)-a^2 b^2 (2 B c+7 A d-C d)\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\left (a^2+b^2\right )^2 (b c-a d)^{5/2} f}-\frac{d \left (2 a^2 A d^2+2 b^2 c (c C-B d)-a b B \left (c^2+d^2\right )+A b^2 \left (c^2+3 d^2\right )+a^2 \left (3 c^2 C-2 B c d+C d^2\right )\right )}{\left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right ) f \sqrt{c+d \tan (e+f x)}}-\frac{A b^2-a (b B-a C)}{\left (a^2+b^2\right ) (b c-a d) f (a+b \tan (e+f x)) \sqrt{c+d \tan (e+f x)}}\\ \end{align*}
Mathematica [B] time = 6.25202, size = 2078, normalized size = 4.65 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.263, size = 40619, normalized size = 90.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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