Optimal. Leaf size=310 \[ -\frac{(2 A-(5+7 i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}+\frac{(2 A-(5+7 i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}-\frac{(2 A+(5-7 i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{(2 A+(5-7 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{(-B+i A) \sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (a \cot (c+d x)+i a)^2} \]
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Rubi [A] time = 0.756826, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3581, 3596, 3534, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{(2 A-(5+7 i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}+\frac{(2 A-(5+7 i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}-\frac{(2 A+(5-7 i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{(2 A+(5-7 i) B) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{16 \sqrt{2} a^3 d}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{(-B+i A) \sqrt{\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (a \cot (c+d x)+i a)^2} \]
Antiderivative was successfully verified.
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Rule 3581
Rule 3596
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx &=\int \frac{B+A \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^3} \, dx\\ &=\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{\int \frac{\frac{1}{2} a (A-11 i B)-\frac{5}{2} a (i A-B) \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{\int \frac{-3 a^2 (i A+6 B)-3 a^2 (A+4 i B) \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))} \, dx}{24 a^4}\\ &=\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{-3 a^3 (2 A-7 i B)-15 a^3 B \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{48 a^6}\\ &=\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{3 a^3 (2 A-7 i B)+15 a^3 B x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{24 a^6 d}\\ &=\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{(2 A-(5+7 i) B) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 a^3 d}+\frac{(2 A+(5-7 i) B) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 a^3 d}\\ &=\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{(2 A-(5+7 i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 \sqrt{2} a^3 d}-\frac{(2 A-(5+7 i) B) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 \sqrt{2} a^3 d}+\frac{(2 A+(5-7 i) B) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 a^3 d}+\frac{(2 A+(5-7 i) B) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{32 a^3 d}\\ &=\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{(2 A-(5+7 i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{(2 A-(5+7 i) B) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{(2 A+(5-7 i) B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}-\frac{(2 A+(5-7 i) B) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}\\ &=-\frac{(2 A+(5-7 i) B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{(2 A+(5-7 i) B) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{16 \sqrt{2} a^3 d}+\frac{(i A-B) \sqrt{\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac{(A+4 i B) \sqrt{\cot (c+d x)}}{12 a d (i a+a \cot (c+d x))^2}+\frac{5 B \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}-\frac{(2 A-(5+7 i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+\frac{(2 A-(5+7 i) B) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}\\ \end{align*}
Mathematica [A] time = 4.22937, size = 415, normalized size = 1.34 \[ \frac{\cot ^{\frac{3}{2}}(c+d x) \csc ^2(c+d x) \sec ^3(c+d x) (A \cos (c+d x)+B \sin (c+d x)) \left (-(A+19 i B) \cos (4 (c+d x))+(3+3 i) ((1+i) A+(6-i) B) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (3 (c+d x))-i \cos (3 (c+d x)))+6 i A \sin (2 (c+d x))-3 i A \sin (4 (c+d x))+6 i A \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+6 A \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+A-12 B \sin (2 (c+d x))+21 B \sin (4 (c+d x))+(21-15 i) B \sqrt{\sin (2 (c+d x))} \sin (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )-(15+21 i) B \sqrt{\sin (2 (c+d x))} \cos (3 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+19 i B\right )}{96 a^3 d (\cot (c+d x)+i)^3 (A+B \tan (c+d x))} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.506, size = 5731, normalized size = 18.5 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.61743, size = 1813, normalized size = 5.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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{B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \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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