Optimal. Leaf size=403 \[ -\frac{(a+b \tan (e+f x))^{m+1} \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b \left (A d^2 \left (c^2 (2-m)-d^2 m\right )-B c d \left (c^2 (1-m)-d^2 (m+1)\right )+c^2 (-C) \left (c^2 m+d^2 (m+2)\right )\right )\right ) \text{Hypergeometric2F1}\left (1,m+1,m+2,-\frac{d (a+b \tan (e+f x))}{b c-a d}\right )}{f (m+1) \left (c^2+d^2\right )^2 (b c-a d)^2}+\frac{(A-i B-C) (a+b \tan (e+f x))^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,\frac{a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) (c-i d)^2}+\frac{(i A-B-i C) (a+b \tan (e+f x))^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,\frac{a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b) (c+i d)^2}+\frac{\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))} \]
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Rubi [A] time = 1.21502, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3649, 3653, 3539, 3537, 68, 3634} \[ -\frac{(a+b \tan (e+f x))^{m+1} \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-A d^4 m-B \left (c^3 d (1-m)-c d^3 (m+1)\right )-c^2 C d^2 (m+2)+c^4 (-C) m\right )\right ) \, _2F_1\left (1,m+1;m+2;-\frac{d (a+b \tan (e+f x))}{b c-a d}\right )}{f (m+1) \left (c^2+d^2\right )^2 (b c-a d)^2}+\frac{\left (A d^2-B c d+c^2 C\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))}+\frac{(A-i B-C) (a+b \tan (e+f x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a) (c-i d)^2}+\frac{(i A-B-i C) (a+b \tan (e+f x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (a+i b) (c+i d)^2} \]
Antiderivative was successfully verified.
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Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 68
Rule 3634
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^m \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^2} \, dx &=\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{\int \frac{(a+b \tan (e+f x))^m \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )+(b c-a d) (B c-(A-C) d) \tan (e+f x)-b \left (c^2 C-B c d+A d^2\right ) m \tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{(b c-a d) \left (c^2+d^2\right )}\\ &=\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{\int (a+b \tan (e+f x))^m \left (-(b c-a d) \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-(b c-a d) \left (2 c (A-C) d-B \left (c^2-d^2\right )\right ) \tan (e+f x)\right ) \, dx}{(b c-a d) \left (c^2+d^2\right )^2}+\frac{\left (-c d (b c-a d) (B c-(A-C) d)-b c^2 \left (c^2 C-B c d+A d^2\right ) m+d^2 \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \int \frac{(a+b \tan (e+f x))^m \left (1+\tan ^2(e+f x)\right )}{c+d \tan (e+f x)} \, dx}{(b c-a d) \left (c^2+d^2\right )^2}\\ &=\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{(A-i B-C) \int (1+i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c-i d)^2}+\frac{(A+i B-C) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx}{2 (c+i d)^2}+\frac{\left (-c d (b c-a d) (B c-(A-C) d)-b c^2 \left (c^2 C-B c d+A d^2\right ) m+d^2 \left ((c C-B d) (a d-b c (1+m))-A \left (a c d-b \left (c^2-d^2 m\right )\right )\right )\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^m}{c+d x} \, dx,x,\tan (e+f x)\right )}{(b c-a d) \left (c^2+d^2\right )^2 f}\\ &=-\frac{\left (a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-c^4 C m-A d^4 m-c^2 C d^2 (2+m)-B \left (c^3 d (1-m)-c d^3 (1+m)\right )\right )\right ) \, _2F_1\left (1,1+m;2+m;-\frac{d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (1+m)}+\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{(i A+B-i C) \operatorname{Subst}\left (\int \frac{(a-i b x)^m}{-1+x} \, dx,x,i \tan (e+f x)\right )}{2 (c-i d)^2 f}-\frac{(i (A+i B-C)) \operatorname{Subst}\left (\int \frac{(a+i b x)^m}{-1+x} \, dx,x,-i \tan (e+f x)\right )}{2 (c+i d)^2 f}\\ &=-\frac{(i A+B-i C) \, _2F_1\left (1,1+m;2+m;\frac{a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) (c-i d)^2 f (1+m)}-\frac{(A+i B-C) \, _2F_1\left (1,1+m;2+m;\frac{a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a-b) (c+i d)^2 f (1+m)}-\frac{\left (a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )-b \left (A c^2 d^2 (2-m)-c^4 C m-A d^4 m-c^2 C d^2 (2+m)-B \left (c^3 d (1-m)-c d^3 (1+m)\right )\right )\right ) \, _2F_1\left (1,1+m;2+m;-\frac{d (a+b \tan (e+f x))}{b c-a d}\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d)^2 \left (c^2+d^2\right )^2 f (1+m)}+\frac{\left (c^2 C-B c d+A d^2\right ) (a+b \tan (e+f x))^{1+m}}{(b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))}\\ \end{align*}
Mathematica [A] time = 6.18405, size = 563, normalized size = 1.4 \[ -\frac{\left (A d^2-c (B d-c C)\right ) (a+b \tan (e+f x))^{m+1}}{f \left (c^2+d^2\right ) (a d-b c) (c+d \tan (e+f x))}-\frac{-\frac{(a+b \tan (e+f x))^{m+1} \left (d^2 \left ((c C-B d) (a d-b c (m+1))-A \left (a c d-b \left (c^2-d^2 m\right )\right )\right )-c d (b c-a d) (B c-d (A-C))-b c^2 m \left (A d^2-B c d+c^2 C\right )\right ) \text{Hypergeometric2F1}\left (1,m+1,m+2,\frac{d (a+b \tan (e+f x))}{a d-b c}\right )}{f (m+1) \left (c^2+d^2\right ) (a d-b c)}+\frac{\frac{i \left (-(b c-a d) \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )-i (b c-a d) \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,\frac{-i a-i b \tan (e+f x)}{b-i a}\right )}{2 f (m+1) (a+i b)}-\frac{i \left (-(b c-a d) \left (-A \left (c^2-d^2\right )-2 B c d+c^2 C-C d^2\right )+i (b c-a d) \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )\right ) (a+b \tan (e+f x))^{m+1} \text{Hypergeometric2F1}\left (1,m+1,m+2,-\frac{i a+i b \tan (e+f x)}{-b-i a}\right )}{2 f (m+1) (a-i b)}}{c^2+d^2}}{\left (c^2+d^2\right ) (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.641, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\tan \left ( fx+e \right ) \right ) ^{m} \left ( A+B\tan \left ( fx+e \right ) +C \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) }{ \left ( c+d\tan \left ( fx+e \right ) \right ) ^{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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}, x\right ) \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{{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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