Optimal. Leaf size=261 \[ \frac{2 \left (15 a^2+2 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}+\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
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Rubi [A] time = 0.902624, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4241, 3568, 3649, 3616, 3615, 93, 203, 206} \[ \frac{2 \left (15 a^2+2 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}+\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3568
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \cot ^{\frac{7}{2}}(c+d x) \sqrt{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)}}{\tan ^{\frac{7}{2}}(c+d x)} \, dx\\ &=-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{1}{5} \left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{b}{2}+\frac{5}{2} a \tan (c+d x)+2 b \tan ^2(c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{1}{4} \left (-15 a^2-2 b^2\right )-\frac{15}{4} a b \tan (c+d x)-\frac{1}{2} b^2 \tan ^2(c+d x)}{\tan ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}} \, dx}{15 a}\\ &=\frac{2 \left (15 a^2+2 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}-\frac{\left (8 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{15 a^2 b}{8}-\frac{15}{8} a^3 \tan (c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{15 a^2}\\ &=\frac{2 \left (15 a^2+2 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}+\frac{1}{2} \left ((i a-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 ((i a+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 (15 a^2+2 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}+\frac{\left ((i a-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 ((i a+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 (15 a^2+2 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}+\frac{\left ((i a-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 ((i a+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} \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} \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 (15 a^2+2 b^2\right ) \sqrt{\cot (c+d x)} \sqrt{a+b \tan (c+d x)}}{15 a^2 d}-\frac{2 b \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{15 a d}-\frac{2 \cot ^{\frac{5}{2}}(c+d x) \sqrt{a+b \tan (c+d x)}}{5 d}\\ \end{align*}
Mathematica [A] time = 1.34242, size = 220, normalized size = 0.84 \[ -\frac{\sqrt{\cot (c+d x)} \left (2 \sqrt{a+b \tan (c+d x)} \left (3 a^2 \cot ^2(c+d x)-15 a^2+a b \cot (c+d x)-2 b^2\right )+15 \sqrt [4]{-1} a^2 \sqrt{-a-i b} \sqrt{\tan (c+d x)} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+15 \sqrt [4]{-1} a^2 \sqrt{a-i b} \sqrt{\tan (c+d x)} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )}{15 a^2 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.987, size = 21333, normalized size = 81.7 \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 \sqrt{b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac{7}{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 \sqrt{b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{\frac{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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