Optimal. Leaf size=154 \[ \frac{B \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}-\frac{B \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d}+\frac{2 B \sqrt{\tan (c+d x)}}{d}+\frac{B \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{B \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d} \]
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Rubi [A] time = 0.104313, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {21, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{B \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}-\frac{B \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{\sqrt{2} d}+\frac{2 B \sqrt{\tan (c+d x)}}{d}+\frac{B \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d}-\frac{B \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )}{2 \sqrt{2} d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3473
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\tan ^{\frac{3}{2}}(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=B \int \tan ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 B \sqrt{\tan (c+d x)}}{d}-B \int \frac{1}{\sqrt{\tan (c+d x)}} \, dx\\ &=\frac{2 B \sqrt{\tan (c+d x)}}{d}-\frac{B \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{2 B \sqrt{\tan (c+d x)}}{d}-\frac{(2 B) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{2 B \sqrt{\tan (c+d x)}}{d}-\frac{B \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}-\frac{B \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\tan (c+d x)}\right )}{d}\\ &=\frac{2 B \sqrt{\tan (c+d x)}}{d}-\frac{B \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d}-\frac{B \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 d}+\frac{B \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} d}+\frac{B \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\tan (c+d x)}\right )}{2 \sqrt{2} d}\\ &=\frac{B \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}-\frac{B \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{2 B \sqrt{\tan (c+d x)}}{d}-\frac{B \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{B \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}\\ &=\frac{B \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}-\frac{B \tan ^{-1}\left (1+\sqrt{2} \sqrt{\tan (c+d x)}\right )}{\sqrt{2} d}+\frac{B \log \left (1-\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}-\frac{B \log \left (1+\sqrt{2} \sqrt{\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt{2} d}+\frac{2 B \sqrt{\tan (c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.11429, size = 138, normalized size = 0.9 \[ \frac{B \left (2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (c+d x)}\right )-2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (c+d x)}+1\right )+8 \sqrt{\tan (c+d x)}+\sqrt{2} \log \left (\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right )-\sqrt{2} \log \left (\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 118, normalized size = 0.8 \begin{align*} 2\,{\frac{B\sqrt{\tan \left ( dx+c \right ) }}{d}}-{\frac{B\sqrt{2}}{2\,d}\arctan \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }-{\frac{B\sqrt{2}}{2\,d}\arctan \left ( -1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) } \right ) }-{\frac{B\sqrt{2}}{4\,d}\ln \left ({ \left ( 1+\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) \left ( 1-\sqrt{2}\sqrt{\tan \left ( dx+c \right ) }+\tan \left ( dx+c \right ) \right ) ^{-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.68618, size = 166, normalized size = 1.08 \begin{align*} -\frac{2 \, \sqrt{2} B \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt{2} B \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\tan \left (d x + c\right )}\right )}\right ) + \sqrt{2} B \log \left (\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt{2} B \log \left (-\sqrt{2} \sqrt{\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 8 \, B \sqrt{\tan \left (d x + c\right )}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.29899, size = 1311, normalized size = 8.51 \begin{align*} \frac{4 \, \sqrt{2} d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} B d^{3} \left (\frac{B^{4}}{d^{4}}\right )^{\frac{3}{4}} \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - \sqrt{2} d^{3} \left (\frac{B^{4}}{d^{4}}\right )^{\frac{3}{4}} \sqrt{\frac{\sqrt{2} B d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + d^{2} \sqrt{\frac{B^{4}}{d^{4}}} \cos \left (d x + c\right ) + B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} + B^{4}}{B^{4}}\right ) + 4 \, \sqrt{2} d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} B d^{3} \left (\frac{B^{4}}{d^{4}}\right )^{\frac{3}{4}} \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - \sqrt{2} d^{3} \left (\frac{B^{4}}{d^{4}}\right )^{\frac{3}{4}} \sqrt{-\frac{\sqrt{2} B d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - d^{2} \sqrt{\frac{B^{4}}{d^{4}}} \cos \left (d x + c\right ) - B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}} - B^{4}}{B^{4}}\right ) - \sqrt{2} d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \log \left (\frac{\sqrt{2} B d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) + d^{2} \sqrt{\frac{B^{4}}{d^{4}}} \cos \left (d x + c\right ) + B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + \sqrt{2} d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{\sqrt{2} B d \left (\frac{B^{4}}{d^{4}}\right )^{\frac{1}{4}} \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) - d^{2} \sqrt{\frac{B^{4}}{d^{4}}} \cos \left (d x + c\right ) - B^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right ) + 8 \, B \sqrt{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right )}}}{4 \, d} \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*} B \int \tan ^{\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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