Optimal. Leaf size=354 \[ \frac{2 \left (35 a^2 A-7 a b B+4 A b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}+\frac{2 \left (35 a^2 A b+105 a^3 B+14 a b^2 B-8 A b^3\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}-\frac{2 (7 a B+A b) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d} \]
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Rubi [A] time = 1.48774, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4241, 3608, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 \left (35 a^2 A-7 a b B+4 A b^2\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}+\frac{2 \left (35 a^2 A b+105 a^3 B+14 a b^2 B-8 A b^3\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}-\frac{2 (7 a B+A b) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3608
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \cot ^{\frac{9}{2}}(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac{9}{2}}(c+d x)} \, dx\\ &=-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}-\frac{1}{7} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{2} (-A b-7 a B)+\frac{7}{2} (a A-b B) \tan (c+d x)+3 A b \tan ^2(c+d x)}{\tan ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 (A b+7 a B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{4} \left (-35 a^2 A-4 A b^2+7 a b B\right )-\frac{35}{4} a (A b+a B) \tan (c+d x)-b (A b+7 a B) \tan ^2(c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{35 a}\\ &=\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 (A b+7 a B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}-\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{8} \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right )-\frac{105}{8} a^2 (a A-b B) \tan (c+d x)-\frac{1}{4} b \left (35 a^2 A+4 A b^2-7 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{105 a^2}\\ &=\frac{2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 (A b+7 a B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}+\frac{\left (16 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{105}{16} a^3 (a A-b B)+\frac{105}{16} a^3 (A b+a B) \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{105 a^3}\\ &=\frac{2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 (A b+7 a B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}+\frac{1}{2} \left ((a-i b) (A-i 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) (A+i 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 (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 (A b+7 a B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}+\frac{\left ((a-i b) (A-i 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) (A+i 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 (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 (A b+7 a B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}+\frac{\left ((a-i b) (A-i 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) (A+i 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{\sqrt{i a-b} (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)}}{d}-\frac{\sqrt{i a+b} (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 (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{105 a^3 d}+\frac{2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{105 a^2 d}-\frac{2 (A b+7 a B) \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{35 a d}-\frac{2 A \cot ^{\frac{7}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}\\ \end{align*}
Mathematica [A] time = 4.00942, size = 291, normalized size = 0.82 \[ \frac{\cot ^{\frac{7}{2}}(c+d x) \left (2 \sqrt{a+b \tan (c+d x)} \left (a \left (35 a^2 A-7 a b B+4 A b^2\right ) \tan ^2(c+d x)+\left (35 a^2 A b+105 a^3 B+14 a b^2 B-8 A b^3\right ) \tan ^3(c+d x)-3 a^2 (7 a B+A b) \tan (c+d x)-15 a^3 A\right )+105 (-1)^{3/4} a^3 \sqrt{-a-i b} (A+i B) \tan ^{\frac{7}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )-105 \sqrt [4]{-1} a^3 \sqrt{a-i b} (B+i A) \tan ^{\frac{7}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )}{105 a^3 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.038, size = 43931, normalized size = 124.1 \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 )} \sqrt{b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac{9}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt{b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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