Optimal. Leaf size=201 \[ \frac{\sqrt{-b+i a} (A+i B) \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{b} d}-\frac{\sqrt{b+i a} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d} \]
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Rubi [A] time = 1.54709, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.257, Rules used = {3610, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ \frac{\sqrt{-b+i a} (A+i B) \tan ^{-1}\left (\frac{\sqrt{-b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{b} d}-\frac{\sqrt{b+i a} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{b+i a} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3610
Rule 3655
Rule 6725
Rule 63
Rule 217
Rule 206
Rule 93
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx &=\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\int \frac{-\frac{a B}{2}+(a A-b B) \tan (c+d x)+\frac{1}{2} (2 A b+a B) \tan ^2(c+d x)}{\sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}} \, dx\\ &=\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{a B}{2}+(a A-b B) x+\frac{1}{2} (2 A b+a B) x^2}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{\operatorname{Subst}\left (\int \left (\frac{2 A b+a B}{2 \sqrt{x} \sqrt{a+b x}}-\frac{A b+a B-(a A-b B) x}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}-\frac{\operatorname{Subst}\left (\int \frac{A b+a B-(a A-b B) x}{\sqrt{x} \sqrt{a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}+\frac{(2 A b+a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}-\frac{\operatorname{Subst}\left (\int \left (\frac{a A-b B+i (A b+a B)}{2 (i-x) \sqrt{x} \sqrt{a+b x}}+\frac{-a A+b B+i (A b+a B)}{2 \sqrt{x} (i+x) \sqrt{a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac{(2 A b+a B) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{((a-i b) (A-i B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} (i+x) \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac{((a+i b) (A+i B)) \operatorname{Subst}\left (\int \frac{1}{(i-x) \sqrt{x} \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac{(2 A b+a B) \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 )}{d}\\ &=\frac{(2 A b+a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{b} d}+\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}+\frac{((a-i b) (A-i B)) \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{((a+i b) (A+i B)) \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} (A+i B) \tan ^{-1}\left (\frac{\sqrt{i a-b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{(2 A b+a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{\sqrt{b} d}-\frac{\sqrt{i a+b} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{i a+b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )}{d}+\frac{B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 3.00635, size = 241, normalized size = 1.2 \[ \frac{-\sqrt [4]{-1} \sqrt{-a-i b} (A+i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{-a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )-\sqrt [4]{-1} \sqrt{a-i b} (A-i B) \tanh ^{-1}\left (\frac{\sqrt [4]{-1} \sqrt{a-i b} \sqrt{\tan (c+d x)}}{\sqrt{a+b \tan (c+d x)}}\right )+\frac{(a B+2 A b) \sqrt{a+b \tan (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{\tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{\frac{b \tan (c+d x)}{a}+1}}+B \sqrt{\tan (c+d x)} \sqrt{a+b \tan (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.712, size = 2178530, normalized size = 10838.5 \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{\left (B \tan \left (d x + c\right ) + A\right )} \sqrt{b \tan \left (d x + c\right ) + a} \sqrt{\tan \left (d x + c\right )}\,{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 \left (A + B \tan{\left (c + d x \right )}\right ) \sqrt{a + b \tan{\left (c + d x \right )}} \sqrt{\tan{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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