Optimal. Leaf size=422 \[ \frac{2 \left (21 a^2 A-24 a b B-A b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}+\frac{2 \left (126 a^2 A b+105 a^3 B-9 a b^2 B+4 A b^3\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}-\frac{2 \left (-63 a^2 A b^2+315 a^4 A-420 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}-\frac{2 (9 a B+10 A b) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}+\frac{(-b+i a)^{3/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{(b+i a)^{3/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d} \]
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Rubi [A] time = 2.02452, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4241, 3605, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 \left (21 a^2 A-24 a b B-A b^2\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}+\frac{2 \left (126 a^2 A b+105 a^3 B-9 a b^2 B+4 A b^3\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}-\frac{2 \left (-63 a^2 A b^2+315 a^4 A-420 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}-\frac{2 (9 a B+10 A b) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}+\frac{(-b+i a)^{3/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{(b+i a)^{3/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3605
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \cot ^{\frac{11}{2}}(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x))}{\tan ^{\frac{11}{2}}(c+d x)} \, dx\\ &=-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{1}{9} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{2} a (10 A b+9 a B)-\frac{9}{2} \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac{1}{2} b (8 a A-9 b B) \tan ^2(c+d x)}{\tan ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{3}{4} a \left (21 a^2 A-A b^2-24 a b B\right )+\frac{63}{4} a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac{3}{2} a b (10 A b+9 a B) \tan ^2(c+d x)}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{63 a}\\ &=\frac{2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{8} a \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right )+\frac{315}{8} a^2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac{3}{2} a b \left (21 a^2 A-A b^2-24 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{315 a^2}\\ &=\frac{2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{\left (16 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{16} a \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right )-\frac{945}{16} a^3 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)-\frac{3}{8} a b \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{945 a^3}\\ &=-\frac{2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}+\frac{2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}+\frac{\left (32 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{945}{32} a^4 \left (2 a A b+a^2 B-b^2 B\right )-\frac{945}{32} a^4 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{945 a^4}\\ &=-\frac{2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}+\frac{2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{1}{2} \left ((a+i b)^2 (i A-B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1-i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx+\frac{1}{2} \left ((a-i b)^2 (i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1+i \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}+\frac{2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{\left ((a+i b)^2 (i A-B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left ((a-i b)^2 (i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac{2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}+\frac{2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}-\frac{\left ((a+i b)^2 (i A-B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-(-i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{\left ((a-i b)^2 (i A+B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-(i a+b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}\\ &=-\frac{(a+i b)^2 (i A-B) \tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{\sqrt{i a-b} d}-\frac{(i a+b)^{3/2} (i A+B) \tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{d}-\frac{2 \left (315 a^4 A-63 a^2 A b^2+8 A b^4-420 a^3 b B-18 a b^3 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{315 a^3 d}+\frac{2 \left (126 a^2 A b+4 A b^3+105 a^3 B-9 a b^2 B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{315 a^2 d}+\frac{2 \left (21 a^2 A-A b^2-24 a b B\right ) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a d}-\frac{2 (10 A b+9 a B) \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{63 d}-\frac{2 a A \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{9 d}\\ \end{align*}
Mathematica [A] time = 6.59843, size = 495, normalized size = 1.17 \[ \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (-\frac{b B \sqrt{a+b \tan (c+d x)}}{4 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{1}{4} \left (-\frac{(8 a A-9 b B) \sqrt{a+b \tan (c+d x)}}{9 d \tan ^{\frac{9}{2}}(c+d x)}+\frac{2 \left (-\frac{4 a (9 a B+10 A b) \sqrt{a+b \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2 \left (-\frac{6 a \left (21 a^2 A-24 a b B-A b^2\right ) \sqrt{a+b \tan (c+d x)}}{5 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 \left (\frac{a \left (126 a^2 A b+105 a^3 B-9 a b^2 B+4 A b^3\right ) \sqrt{a+b \tan (c+d x)}}{d \tan ^{\frac{3}{2}}(c+d x)}-\frac{2 \left (\frac{3 a \left (-63 a^2 A b^2+315 a^4 A-420 a^3 b B-18 a b^3 B+8 A b^4\right ) \sqrt{a+b \tan (c+d x)}}{2 d \sqrt{\tan (c+d x)}}+\frac{945 a^4 \left (\sqrt [4]{-1} (-a-i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )-\sqrt [4]{-1} (a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )}{4 d}\right )}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 2.586, size = 74462, normalized size = 176.5 \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] 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 )}^{\frac{3}{2}} \cot \left (d x + c\right )^{\frac{11}{2}}\,{d x} \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(-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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