Optimal. Leaf size=534 \[ \frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac{a \left (a^2 A b+3 a^3 B+11 a b^2 B-7 A b^3\right ) \sqrt{\cot (c+d x)}}{4 b^2 d \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)}+\frac{\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3}-\frac{\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3}-\frac{\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}-\frac{\sqrt{a} \left (18 a^2 A b^3+a^4 A b+6 a^3 b^2 B+3 a^5 B+35 a b^4 B-15 A b^5\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{4 b^{5/2} d \left (a^2+b^2\right )^3} \]
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Rubi [A] time = 1.37878, antiderivative size = 534, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.424, Rules used = {3581, 3609, 3649, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ \frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b d \left (a^2+b^2\right ) (a \cot (c+d x)+b)^2}-\frac{a \left (a^2 A b+3 a^3 B+11 a b^2 B-7 A b^3\right ) \sqrt{\cot (c+d x)}}{4 b^2 d \left (a^2+b^2\right )^2 (a \cot (c+d x)+b)}+\frac{\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3}-\frac{\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{2 \sqrt{2} d \left (a^2+b^2\right )^3}-\frac{\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} d \left (a^2+b^2\right )^3}-\frac{\sqrt{a} \left (18 a^2 A b^3+a^4 A b+6 a^3 b^2 B+3 a^5 B+35 a b^4 B-15 A b^5\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{4 b^{5/2} d \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3581
Rule 3609
Rule 3649
Rule 3653
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 3634
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{\cot ^{\frac{5}{2}}(c+d x) (a+b \tan (c+d x))^3} \, dx &=\int \frac{B+A \cot (c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))^3} \, dx\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{\int \frac{\frac{1}{2} \left (-a A b-3 a^2 B-4 b^2 B\right )-2 b (A b-a B) \cot (c+d x)+\frac{3}{2} a (A b-a B) \cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt{\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac{\int \frac{\frac{1}{4} \left (a^3 A b+9 a A b^3+3 a^4 B+3 a^2 b^2 B+8 b^4 B\right )-2 b^2 \left (a^2 A-A b^2+2 a b B\right ) \cot (c+d x)+\frac{1}{4} a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt{\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac{\int \frac{-2 b^2 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )-2 b^2 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{2 b^2 \left (a^2+b^2\right )^3}+\frac{\left (a \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right )\right ) \int \frac{1+\cot ^2(c+d x)}{\sqrt{\cot (c+d x)} (b+a \cot (c+d x))} \, dx}{8 b^2 \left (a^2+b^2\right )^3}\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt{\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac{\operatorname{Subst}\left (\int \frac{2 b^2 \left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right )+2 b^2 \left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{b^2 \left (a^2+b^2\right )^3 d}+\frac{\left (a \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x} (b-a x)} \, dx,x,-\cot (c+d x)\right )}{8 b^2 \left (a^2+b^2\right )^3 d}\\ &=\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt{\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}-\frac{\left (a \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right )^3 d}-\frac{\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}+\frac{\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\left (a^2+b^2\right )^3 d}\\ &=-\frac{\sqrt{a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt{\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac{\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^3 d}+\frac{\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \sqrt{2} \left (a^2+b^2\right )^3 d}+\frac{\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}+\frac{\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{2 \left (a^2+b^2\right )^3 d}\\ &=-\frac{\sqrt{a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt{\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac{\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^3 d}-\frac{\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^3 d}+\frac{\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d}-\frac{\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d}\\ &=-\frac{\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d}+\frac{\left (a^3 (A-B)-3 a b^2 (A-B)+3 a^2 b (A+B)-b^3 (A+B)\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} \left (a^2+b^2\right )^3 d}-\frac{\sqrt{a} \left (a^4 A b+18 a^2 A b^3-15 A b^5+3 a^5 B+6 a^3 b^2 B+35 a b^4 B\right ) \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{\cot (c+d x)}}{\sqrt{b}}\right )}{4 b^{5/2} \left (a^2+b^2\right )^3 d}+\frac{a (A b-a B) \sqrt{\cot (c+d x)}}{2 b \left (a^2+b^2\right ) d (b+a \cot (c+d x))^2}-\frac{a \left (a^2 A b-7 A b^3+3 a^3 B+11 a b^2 B\right ) \sqrt{\cot (c+d x)}}{4 b^2 \left (a^2+b^2\right )^2 d (b+a \cot (c+d x))}+\frac{\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^3 d}-\frac{\left (3 a^2 b (A-B)-b^3 (A-B)-a^3 (A+B)+3 a b^2 (A+B)\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt{2} \left (a^2+b^2\right )^3 d}\\ \end{align*}
Mathematica [A] time = 6.44473, size = 592, normalized size = 1.11 \[ \frac{2 \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (\frac{\sqrt{a} \left (a^2 A b-2 a^3 B-4 a b^2 B+3 A b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{2 b^{5/2} \left (a^2+b^2\right )^2}+\frac{\sqrt{a} \left (a^2 A b^3+3 a^3 b^2 B+a^5 B+6 a b^4 B-3 A b^5\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{b^{5/2} \left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)-b^3 (A+B)\right ) \left (\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{4 \left (a^2+b^2\right )^3}-\frac{3 (A b-a B) \left (\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{a \sqrt{\tan (c+d x)}}{a+b \tan (c+d x)}\right )}{8 b^2 \left (a^2+b^2\right )}-\frac{a^2 (A b-a B) \sqrt{\tan (c+d x)}}{4 b^2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a \left (a^2 A b-2 a^3 B-4 a b^2 B+3 A b^3\right ) \sqrt{\tan (c+d x)}}{2 b^2 \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{\left (3 a^2 b (A-B)+a^3 (-(A+B))+3 a b^2 (A+B)-b^3 (A-B)\right ) \left (\sqrt{2} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\sqrt{2} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{8 \left (a^2+b^2\right )^3}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.564, size = 102109, normalized size = 191.2 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \tan \left (d x + c\right ) + A}{{\left (b \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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