Optimal. Leaf size=239 \[ \frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 (3 a B+A b) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}+\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d} \]
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Rubi [A] time = 0.886644, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 (3 a B+A b) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}+\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d} \]
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{5}{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{5}{2}}(c+d x)} \, dx\\ &=-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{1}{3} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{2} (-A b-3 a B)+\frac{3}{2} (a A-b B) \tan (c+d x)+A b \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 (A b+3 a B) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{4} a (a A-b B)-\frac{3}{4} a (A b+a B) \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{3 a}\\ &=-\frac{2 (A b+3 a B) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 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 (A b+3 a B) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 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 (A b+3 a B) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 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 (A b+3 a B) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{3 a d}-\frac{2 A \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 1.56611, size = 216, normalized size = 0.9 \[ \frac{\cot ^{\frac{3}{2}}(c+d x) \left (-2 \sqrt{a+b \tan (c+d x)} ((3 a B+A b) \tan (c+d x)+a A)-3 (-1)^{3/4} a \sqrt{-a-i b} (A+i B) \tan ^{\frac{3}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+3 \sqrt [4]{-1} a \sqrt{a-i b} (B+i A) \tan ^{\frac{3}{2}}(c+d x) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )}{3 a d} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.144, size = 21562, normalized size = 90.2 \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{5}{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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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