Optimal. Leaf size=342 \[ -\frac{\left (-240 a^2 A b^2+128 a^4 A-320 a^3 b B+40 a b^3 B-5 A b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{64 a^{3/2} d}+\frac{\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}+\frac{\left (144 a^2 A b+64 a^3 B-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{(a-i b)^{5/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{(a+i b)^{5/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{a (8 a B+11 A b) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d} \]
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Rubi [A] time = 1.66868, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3605, 3645, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac{\left (-240 a^2 A b^2+128 a^4 A-320 a^3 b B+40 a b^3 B-5 A b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{64 a^{3/2} d}+\frac{\left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}+\frac{\left (144 a^2 A b+64 a^3 B-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{(a-i b)^{5/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{(a+i b)^{5/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{a (8 a B+11 A b) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 3605
Rule 3645
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \cot ^5(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) \sqrt{a+b \tan (c+d x)} \left (\frac{1}{2} a (11 A b+8 a B)-4 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac{1}{2} b (5 a A-8 b B) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac{1}{12} \int \frac{\cot ^3(c+d x) \left (-\frac{1}{4} a \left (48 a^2 A-59 A b^2-104 a b B\right )-12 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)-\frac{1}{4} b \left (85 a A b+40 a^2 B-48 b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=\frac{\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}-\frac{\int \frac{\cot ^2(c+d x) \left (\frac{3}{8} a \left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right )-24 a \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)-\frac{3}{8} a b \left (48 a^2 A-59 A b^2-104 a b B\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{24 a}\\ &=\frac{\left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac{\int \frac{\cot (c+d x) \left (\frac{3}{16} a \left (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 B\right )+24 a^2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \tan (c+d x)+\frac{3}{16} a b \left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{24 a^2}\\ &=\frac{\left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac{\int \frac{24 a^2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-24 a^2 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{24 a^2}+\frac{\left (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 B\right ) \int \frac{\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{128 a}\\ &=\frac{\left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac{1}{2} \left ((a-i b)^3 (i A+B)\right ) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx+\frac{\left (24 a^2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )-24 i a^2 \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{48 a^2}+\frac{\left (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 B\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{128 a d}\\ &=\frac{\left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}-\frac{\left ((a-i b)^3 (A-i 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)^3 (A+i 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 (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 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 )}{64 a b d}\\ &=-\frac{\left (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{64 a^{3/2} d}+\frac{\left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}+\frac{\left (i (a+i b)^3 (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)^3 (i A+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 (128 a^4 A-240 a^2 A b^2-5 A b^4-320 a^3 b B+40 a b^3 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{64 a^{3/2} d}+\frac{(a-i b)^{5/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{(a+i b)^{5/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}+\frac{\left (144 a^2 A b-5 A b^3+64 a^3 B-88 a b^2 B\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{64 a d}+\frac{\left (48 a^2 A-59 A b^2-104 a b B\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{96 d}-\frac{a (11 A b+8 a B) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{a A \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{4 d}\\ \end{align*}
Mathematica [A] time = 6.45776, size = 622, normalized size = 1.82 \[ -\frac{2 b B \cot ^4(c+d x) (a+b \tan (c+d x))^{3/2}}{5 d}-\frac{2}{5} \left (\frac{b (2 a B+5 A b) \cot ^4(c+d x) \sqrt{a+b \tan (c+d x)}}{7 d}-\frac{2}{7} \left (-\frac{\left (35 a^2 A-72 a b B-40 A b^2\right ) \cot ^4(c+d x) \sqrt{a+b \tan (c+d x)}}{16 d}-\frac{\frac{7 a \left (40 a^2 B+85 a A b-48 b^2 B\right ) \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)}}{24 d}-\frac{\frac{35 a^2 \left (48 a^2 A-104 a b B-59 A b^2\right ) \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{32 d}-\frac{-\frac{105 a^2 \left (144 a^2 A b+64 a^3 B-88 a b^2 B-5 A b^3\right ) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{32 d}-\frac{-\frac{105 a^{5/2} \left (-240 a^2 A b^2+128 a^4 A-320 a^3 b B+40 a b^3 B-5 A b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{32 d}+\frac{i \sqrt{a-i b} \left (210 a^4 \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right )+210 i a^4 \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (-a+i b)}-\frac{i \sqrt{a+i b} \left (210 a^4 \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right )-210 i a^4 \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (-a-i b)}}{a}}{2 a}}{3 a}}{4 a}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 4.294, size = 227162, normalized size = 664.2 \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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