Optimal. Leaf size=324 \[ \frac{\left (a^2 (-B)+4 a A b-8 b^2 B\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 b^{3/2} d}+\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 b d \sqrt{\cot (c+d x)}} \]
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Rubi [A] time = 1.99145, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ \frac{\left (a^2 (-B)+4 a A b-8 b^2 B\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{4 b^{3/2} d}+\frac{\sqrt{-b+i a} (-B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{\sqrt{b+i a} (B+i A) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 b d \sqrt{\cot (c+d x)}} \]
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{\sqrt{a+b \tan (c+d x)} (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) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\\ &=\frac{B (a+b \tan (c+d x))^{3/2}}{2 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{a B}{2}-2 b B \tan (c+d x)+\frac{1}{2} (4 A b-a B) \tan ^2(c+d x)\right )}{\sqrt{\tan (c+d x)}} \, dx}{2 b}\\ &=\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 b d \sqrt{\cot (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{-\frac{1}{4} a (4 A b+a B)-2 b (A b+a B) \tan (c+d x)+\frac{1}{4} \left (4 a A b-a^2 B-8 b^2 B\right ) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx}{2 b}\\ &=\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 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{1}{4} a (4 A b+a B)-2 b (A b+a B) x+\frac{1}{4} \left (4 a A b-a^2 B-8 b^2 B\right ) x^2}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 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{4 a A b-a^2 B-8 b^2 B}{4 \sqrt{x} \sqrt{a+b x}}-\frac{2 (b (a A-b B)+b (A b+a B) x)}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 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 (a A-b B)+b (A b+a B) x}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b d}+\frac{\left (\left (4 a A b-a^2 B-8 b^2 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 )}{8 b d}\\ &=\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 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 (A b+a B)+i b (a A-b B)}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{b (A b+a B)+i b (a A-b B)}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b d}+\frac{\left (\left (4 a A b-a^2 B-8 b^2 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 )}{4 b d}\\ &=\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 b d \sqrt{\cot (c+d x)}}-\frac{\left ((i a+b) (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 ((i a-b) (A+i B) \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 d}+\frac{\left (\left (4 a A b-a^2 B-8 b^2 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 )}{4 b d}\\ &=\frac{\left (4 a A b-a^2 B-8 b^2 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)}}{4 b^{3/2} d}+\frac{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 b d \sqrt{\cot (c+d x)}}-\frac{\left ((i a+b) (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 ((i a-b) (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{\sqrt{i a-b} (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 (4 a A b-a^2 B-8 b^2 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)}}{4 b^{3/2} d}+\frac{\sqrt{i a+b} (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{(4 A b-a B) \sqrt{a+b \tan (c+d x)}}{4 b d \sqrt{\cot (c+d x)}}+\frac{B (a+b \tan (c+d x))^{3/2}}{2 b d \sqrt{\cot (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.73364, size = 356, normalized size = 1.1 \[ \frac{\sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \left (-\left (a^2 B-4 a A b+8 b^2 B\right ) (a+b \tan (c+d x)) \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )+\sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1} \left (\sqrt{\tan (c+d x)} (a+b \tan (c+d x)) (a B+4 A b+2 b B \tan (c+d x))-4 (-1)^{3/4} b \sqrt{-a-i b} (A+i B) \sqrt{a+b \tan (c+d x)} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+4 \sqrt [4]{-1} b \sqrt{a-i b} (B+i A) \sqrt{a+b \tan (c+d x)} \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )\right )\right )}{4 \sqrt{a} b^{3/2} d \sqrt{a+b \tan (c+d x)} \sqrt{\frac{b \tan (c+d x)}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.514, size = 28218, normalized size = 87.1 \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 )} \sqrt{b \tan \left (d x + c\right ) + a}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \tan{\left (c + d x \right )}\right ) \sqrt{a + b \tan{\left (c + d x \right )}}}{\cot ^{\frac{3}{2}}{\left (c + d x \right )}}\, dx \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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