Optimal. Leaf size=457 \[ \frac{\left (-5 a^2 B+40 a A b-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a^2 A b-5 a^3 B-112 a b^2 B-64 A b^3\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a^3 A b-240 a^2 b^2 B-5 a^4 B-320 a A b^3+128 b^4 B\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac{(-b+i a)^{5/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}-\frac{(b+i a)^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}} \]
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Rubi [A] time = 3.01098, antiderivative size = 457, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {4241, 3607, 3647, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ \frac{\left (-5 a^2 B+40 a A b-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a^2 A b-5 a^3 B-112 a b^2 B-64 A b^3\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a^3 A b-240 a^2 b^2 B-5 a^4 B-320 a A b^3+128 b^4 B\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{64 b^{3/2} d}-\frac{(-b+i a)^{5/2} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}-\frac{(b+i a)^{5/2} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3607
Rule 3647
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \tan ^{\frac{3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\\ &=\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+b \tan (c+d x))^{5/2} \left (-\frac{a B}{2}-4 b B \tan (c+d x)+\frac{1}{2} (8 A b-a B) \tan ^2(c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{4 b}\\ &=\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{(a+b \tan (c+d x))^{3/2} \left (-\frac{1}{4} a (8 A b+5 a B)-12 b (A b+a B) \tan (c+d x)+\frac{1}{4} \left (40 a A b-5 a^2 B-48 b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{12 b}\\ &=\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+b \tan (c+d x)} \left (-\frac{3}{8} a \left (24 a A b+5 a^2 B-16 b^2 B\right )-24 b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac{3}{8} \left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \tan ^2(c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{24 b}\\ &=\frac{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{3}{16} a \left (88 a^2 A b-64 A b^3+5 a^3 B-144 a b^2 B\right )-24 b \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} \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{24 b}\\ &=\frac{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{-\frac{3}{16} a \left (88 a^2 A b-64 A b^3+5 a^3 B-144 a b^2 B\right )-24 b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x+\frac{3}{16} \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) x^2}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=\frac{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{3 \left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right )}{16 \sqrt{x} \sqrt{a+b x}}-\frac{24 \left (b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x\right )}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{24 b d}\\ &=\frac{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b d}+\frac{\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{128 b d}\\ &=\frac{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \left (\frac{-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b d}+\frac{\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{64 b d}\\ &=\frac{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left ((i a+b)^3 (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{\left (\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{64 b d}-\frac{\left (\left (-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=\frac{\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{64 b^{3/2} d}+\frac{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\left ((i a+b)^3 (A-i B) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{i-(-a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}-\frac{\left (\left (-b \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+i b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{i-(a+i b) x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{b d}\\ &=-\frac{(i a-b)^{5/2} (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{\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{64 b^{3/2} d}-\frac{(i a+b)^{5/2} (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{\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt{a+b \tan (c+d x)}}{64 b d \sqrt{\cot (c+d x)}}+\frac{\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt{\cot (c+d x)}}+\frac{(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt{\cot (c+d x)}}\\ \end{align*}
Mathematica [A] time = 5.32911, size = 431, normalized size = 0.94 \[ \frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (-2 \left (5 a^2 B-40 a A b+48 b^2 B\right ) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{3/2}-3 \left (-40 a^2 A b+5 a^3 B+112 a b^2 B+64 A b^3\right ) \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}-\frac{3 \sqrt{a} \left (-40 a^3 A b+240 a^2 b^2 B+5 a^4 B+320 a A b^3-128 b^4 B\right ) \sqrt{\frac{b \tan (c+d x)}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{b} \sqrt{a+b \tan (c+d x)}}+8 (8 A b-a B) \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{5/2}-192 (-1)^{3/4} b (-a-i b)^{5/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+192 \sqrt [4]{-1} b (a-i b)^{5/2} (B+i A) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+48 B \sqrt{\tan (c+d x)} (a+b \tan (c+d x))^{7/2}\right )}{192 b d} \]
Antiderivative was successfully verified.
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Maple [C] time = 3.753, size = 39339, normalized size = 86.1 \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 \frac{{\left (B \tan \left (d x + c\right ) + A\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cot \left (d x + c\right )^{\frac{3}{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(-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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