Optimal. Leaf size=237 \[ \frac{5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 \sqrt{a} d}+\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d} \]
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Rubi [A] time = 1.0665, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {3565, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 \sqrt{a} d}+\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d} \]
Antiderivative was successfully verified.
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Rule 3565
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))^{5/2} \, dx &=-\frac{a^2 \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{13 a^2 b}{2}-3 a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac{1}{2} b \left (5 a^2-6 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a^2 \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^2 \left (8 a^2-11 b^2\right )+6 a b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac{39}{4} a^2 b^2 \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{6 a}\\ &=\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{\int \frac{\cot (c+d x) \left (-\frac{15}{8} a^2 b \left (8 a^2-b^2\right )+6 a^3 \left (a^2-3 b^2\right ) \tan (c+d x)+\frac{3}{8} a^2 b \left (8 a^2-11 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{6 a^2}\\ &=\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{\int \frac{6 a^3 \left (a^2-3 b^2\right )+6 a^2 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{6 a^2}-\frac{1}{16} \left (5 b \left (8 a^2-b^2\right )\right ) \int \frac{\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{1}{2} (a-i b)^3 \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx+\frac{1}{2} (a+i b)^3 \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx-\frac{\left (5 b \left (8 a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}+\frac{(i a-b)^3 \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac{(i a+b)^3 \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 (5 \left (8 a^2-b^2\right )\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 d}\\ &=\frac{5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 \sqrt{a} d}+\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}-\frac{(a-i b)^3 \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{(a+i b)^3 \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{5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 \sqrt{a} d}-\frac{i (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{i (a+i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}+\frac{\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{8 d}-\frac{13 a b \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{12 d}-\frac{a^2 \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 3.52187, size = 185, normalized size = 0.78 \[ -\frac{\frac{15 b \left (b^2-8 a^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a}}+\cot (c+d x) \sqrt{a+b \tan (c+d x)} \left (8 a^2 \cot ^2(c+d x)-24 a^2+26 a b \cot (c+d x)+33 b^2\right )+24 i (a-i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )-24 i (a+i b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.677, size = 88284, normalized size = 372.5 \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: 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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