Optimal. Leaf size=364 \[ \frac{b \left (62 a^2 A b^3+11 a^4 A b-40 a^3 b^2 B-4 a^5 B-20 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{b \left (27 a^2 A b-12 a^3 B-20 a b^2 B+35 A b^3\right )}{12 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{\left (8 a^2 A+20 a b B-35 A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{9/2} d}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 1.63428, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3609, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{b \left (62 a^2 A b^3+11 a^4 A b-40 a^3 b^2 B-4 a^5 B-20 a b^4 B+35 A b^5\right )}{4 a^4 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{b \left (27 a^2 A b-12 a^3 B-20 a b^2 B+35 A b^3\right )}{12 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{\left (8 a^2 A+20 a b B-35 A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{9/2} d}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3609
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx &=-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac{\int \frac{\cot ^2(c+d x) \left (\frac{1}{2} (7 A b-4 a B)+2 a A \tan (c+d x)+\frac{7}{2} A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{5/2}} \, dx}{2 a}\\ &=\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{\int \frac{\cot (c+d x) \left (\frac{1}{4} \left (-8 a^2 A+35 A b^2-20 a b B\right )-2 a^2 B \tan (c+d x)+\frac{5}{4} b (7 A b-4 a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{5/2}} \, dx}{2 a^2}\\ &=\frac{b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{\int \frac{\cot (c+d x) \left (-\frac{3}{8} \left (a^2+b^2\right ) \left (8 a^2 A-35 A b^2+20 a b B\right )+3 a^3 (A b-a B) \tan (c+d x)+\frac{3}{8} b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 a^3 \left (a^2+b^2\right )}\\ &=\frac{b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{2 \int \frac{\cot (c+d x) \left (-\frac{3}{16} \left (a^2+b^2\right )^2 \left (8 a^2 A-35 A b^2+20 a b B\right )+\frac{3}{2} a^4 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+\frac{3}{16} b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{3 a^4 \left (a^2+b^2\right )^2}\\ &=\frac{b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{2 \int \frac{\frac{3}{2} a^4 \left (2 a A b-a^2 B+b^2 B\right )+\frac{3}{2} a^4 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{3 a^4 \left (a^2+b^2\right )^2}-\frac{\left (8 a^2 A-35 A b^2+20 a b B\right ) \int \frac{\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{8 a^4}\\ &=\frac{b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{(i A-B) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}-\frac{(i A+B) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}-\frac{\left (8 a^2 A-35 A b^2+20 a b B\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{8 a^4 d}\\ &=\frac{b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{(A-i B) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}+\frac{(A+i B) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}-\frac{\left (8 a^2 A-35 A b^2+20 a b 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 )}{4 a^4 b d}\\ &=\frac{\left (8 a^2 A-35 A b^2+20 a b B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{9/2} d}+\frac{b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{(i (A+i B)) \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 )}{(a+i b)^2 b d}+\frac{(i A+B) \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 )}{(a-i b)^2 b d}\\ &=\frac{\left (8 a^2 A-35 A b^2+20 a b B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{9/2} d}-\frac{(A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{(a-i b)^{5/2} d}-\frac{(A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{(a+i b)^{5/2} d}+\frac{b \left (27 a^2 A b+35 A b^3-12 a^3 B-20 a b^2 B\right )}{12 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{(7 A b-4 a B) \cot (c+d x)}{4 a^2 d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}+\frac{b \left (11 a^4 A b+62 a^2 A b^3+35 A b^5-4 a^5 B-40 a^3 b^2 B-20 a b^4 B\right )}{4 a^4 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 6.29067, size = 593, normalized size = 1.63 \[ -\frac{A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac{-\frac{(7 A b-4 a B) \cot (c+d x)}{2 a d (a+b \tan (c+d x))^{3/2}}-\frac{\frac{2 \left (\frac{1}{4} b^2 \left (-8 a^2 A-20 a b B+35 A b^2\right )-a \left (-2 a^2 b B-\frac{5}{4} a b (7 A b-4 a B)\right )\right )}{3 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{2 \left (\frac{2 \left (-\frac{3}{8} b^2 \left (a^2+b^2\right ) \left (8 a^2 A+20 a b B-35 A b^2\right )-a \left (3 a^3 b (A b-a B)-\frac{3}{8} a b \left (27 a^2 A b-12 a^3 B-20 a b^2 B+35 A b^3\right )\right )\right )}{a d \left (a^2+b^2\right ) \sqrt{a+b \tan (c+d x)}}+\frac{2 \left (\frac{3 \left (a^2+b^2\right )^2 \left (8 a^2 A+20 a b B-35 A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{8 \sqrt{a} d}+\frac{i \sqrt{a-i b} \left (\frac{3}{2} a^4 \left (a^2 (-B)+2 a A b+b^2 B\right )-\frac{3}{2} i a^4 \left (a^2 A+2 a b B-A b^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (-a+i b)}-\frac{i \sqrt{a+i b} \left (\frac{3}{2} a^4 \left (a^2 (-B)+2 a A b+b^2 B\right )+\frac{3}{2} i a^4 \left (a^2 A+2 a b B-A b^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (-a-i b)}\right )}{a \left (a^2+b^2\right )}\right )}{3 a \left (a^2+b^2\right )}}{a}}{2 a} \]
Antiderivative was successfully verified.
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Maple [C] time = 10.691, size = 467680, normalized size = 1284.8 \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{{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{3}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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