Optimal. Leaf size=278 \[ \frac{\left (24 a^2 A b+16 a^3 B-6 a b^2 B+A b^3\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 a^{3/2} d}+\frac{\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(6 a B+7 A b) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d} \]
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Rubi [A] time = 1.31489, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3605, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{\left (24 a^2 A b+16 a^3 B-6 a b^2 B+A b^3\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 a^{3/2} d}+\frac{\left (8 a^2 A-10 a b B-A b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(a-i b)^{3/2} (B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{(a+i b)^{3/2} (-B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(6 a B+7 A b) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3605
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{1}{3} \int \frac{\cot ^3(c+d x) \left (\frac{1}{2} a (7 A b+6 a B)-3 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac{1}{2} b (5 a A-6 b B) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{(7 A b+6 a B) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{\int \frac{\cot ^2(c+d x) \left (\frac{3}{4} a \left (8 a^2 A-A b^2-10 a b B\right )+6 a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac{3}{4} a b (7 A b+6 a B) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{6 a}\\ &=\frac{\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(7 A b+6 a B) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{\int \frac{\cot (c+d x) \left (-\frac{3}{8} a \left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right )+6 a^2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+\frac{3}{8} a b \left (8 a^2 A-A b^2-10 a b B\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{6 a^2}\\ &=\frac{\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(7 A b+6 a B) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{\int \frac{6 a^2 \left (a^2 A-A b^2-2 a b B\right )+6 a^2 \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{6 a^2}-\frac{\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \int \frac{\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{16 a}\\ &=\frac{\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(7 A b+6 a B) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{1}{2} \left ((a-i b)^2 (A-i B)\right ) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx+\frac{1}{2} \left ((a+i b)^2 (A+i B)\right ) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx-\frac{\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{16 a d}\\ &=\frac{\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(7 A b+6 a B) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{\left ((a+i b)^2 (i A-B)\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}+\frac{\left ((a-i b)^2 (i A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}-\frac{\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{8 a b d}\\ &=\frac{\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 a^{3/2} d}+\frac{\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(7 A b+6 a B) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{\left ((a-i b)^2 (A-i B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\frac{i a}{b}+\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{b d}-\frac{\left ((a+i b)^2 (A+i B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i a}{b}-\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{b d}\\ &=\frac{\left (24 a^2 A b+A b^3+16 a^3 B-6 a b^2 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 a^{3/2} d}-\frac{(a-i b)^{3/2} (i A+B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{(a+i b)^{3/2} (i A-B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}+\frac{\left (8 a^2 A-A b^2-10 a b B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 a d}-\frac{(7 A b+6 a B) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a A \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 5.47373, size = 241, normalized size = 0.87 \[ \frac{3 \left (24 a^2 A b+16 a^3 B-6 a b^2 B+A b^3\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )+\sqrt{a} \left (-\cot (c+d x) \sqrt{a+b \tan (c+d x)} \left (8 a^2 A \cot ^2(c+d x)-24 a^2 A+2 a (6 a B+7 A b) \cot (c+d x)+30 a b B+3 A b^2\right )-24 i a (a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )+24 i a (a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )\right )}{24 a^{3/2} d} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.825, size = 145176, normalized size = 522.2 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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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