Optimal. Leaf size=292 \[ \frac{\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) (a+b \tan (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{b \tan (c+d x)}{a}+1\right )}{2 a^3 d (n+1)}-\frac{(B+i A) (a+b \tan (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (b+i a)}-\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}-\frac{\cot (c+d x) (2 a B-A b (1-n)) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{2 a d} \]
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Rubi [A] time = 0.805643, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {3609, 3649, 3653, 3539, 3537, 68, 3634, 65} \[ \frac{\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right )}{2 a^3 d (n+1)}-\frac{\cot (c+d x) (2 a B-A b (1-n)) (a+b \tan (c+d x))^{n+1}}{2 a^2 d}-\frac{(B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (b+i a)}-\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (a+i b)}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{2 a d} \]
Antiderivative was successfully verified.
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Rule 3609
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 68
Rule 3634
Rule 65
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx &=-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}-\frac{\int \cot ^2(c+d x) (a+b \tan (c+d x))^n \left (-2 a B+A (b-b n)+2 a A \tan (c+d x)+A b (1-n) \tan ^2(c+d x)\right ) \, dx}{2 a}\\ &=-\frac{(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac{\int \cot (c+d x) (a+b \tan (c+d x))^n \left (-2 a^2 A+2 a b B n-A b^2 (1-n) n-2 a^2 B \tan (c+d x)+b n (2 a B-A (b-b n)) \tan ^2(c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac{(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac{\int \left (-2 a^2 B+2 a^2 A \tan (c+d x)\right ) (a+b \tan (c+d x))^n \, dx}{2 a^2}+\frac{\left (-2 a^2 A+2 a b B n-A b^2 (1-n) n\right ) \int \cot (c+d x) (a+b \tan (c+d x))^n \left (1+\tan ^2(c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac{(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac{1}{2} (-i A-B) \int (1+i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx+\frac{1}{2} (i A-B) \int (1-i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx-\frac{\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^n}{x} \, dx,x,\tan (c+d x)\right )}{2 a^2 d}\\ &=-\frac{(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}+\frac{\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) \, _2F_1\left (1,1+n;2+n;1+\frac{b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{2 a^3 d (1+n)}+\frac{(A-i B) \operatorname{Subst}\left (\int \frac{(a-i b x)^n}{-1+x} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac{(A+i B) \operatorname{Subst}\left (\int \frac{(a+i b x)^n}{-1+x} \, dx,x,-i \tan (c+d x)\right )}{2 d}\\ &=-\frac{(2 a B-A b (1-n)) \cot (c+d x) (a+b \tan (c+d x))^{1+n}}{2 a^2 d}-\frac{A \cot ^2(c+d x) (a+b \tan (c+d x))^{1+n}}{2 a d}-\frac{(A-i B) \, _2F_1\left (1,1+n;2+n;\frac{a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a-i b) d (1+n)}-\frac{(A+i B) \, _2F_1\left (1,1+n;2+n;\frac{a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a+i b) d (1+n)}+\frac{\left (2 a^2 A-2 a b B n+A b^2 (1-n) n\right ) \, _2F_1\left (1,1+n;2+n;1+\frac{b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^{1+n}}{2 a^3 d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.474324, size = 230, normalized size = 0.79 \[ -\frac{(a+b \tan (c+d x))^{n+1} \left (a^3 (a+i b) (A-i B) \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a-i b}\right )+(a-i b) \left (a^3 (A+i B) \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a+i b}\right )-2 (a+i b) \left (a^2 A \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{b \tan (c+d x)}{a}+1\right )+b \left (a B \text{Hypergeometric2F1}\left (2,n+1,n+2,\frac{b \tan (c+d x)}{a}+1\right )-A b \text{Hypergeometric2F1}\left (3,n+1,n+2,\frac{b \tan (c+d x)}{a}+1\right )\right )\right )\right )\right )}{2 a^3 d (n+1) (a-i b) (a+i b)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.653, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( dx+c \right ) \right ) ^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n} \left ( A+B\tan \left ( dx+c \right ) \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 (B \tan \left (d x + c\right ) + A\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3}\,{d x} \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 ({\left (B \cot \left (d x + c\right )^{3} \tan \left (d x + c\right ) + A \cot \left (d x + c\right )^{3}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{n}, 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{\left (B \tan \left (d x + c\right ) + A\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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