Optimal. Leaf size=91 \[ \frac{a \left (2 a^2-3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \sec (c+d x) \left (4 \left (4 a^2-b^2\right )+5 a b \tan (c+d x)\right )}{6 d}+\frac{b \sec (c+d x) (a+b \tan (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.0851151, antiderivative size = 109, normalized size of antiderivative = 1.2, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3512, 743, 780, 215} \[ \frac{b \sec (c+d x) \left (4 \left (4 a^2-b^2\right )+5 a b \tan (c+d x)\right )}{6 d}+\frac{a \left (2 a^2-3 b^2\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{2 d \sqrt{\sec ^2(c+d x)}}+\frac{b \sec (c+d x) (a+b \tan (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 743
Rule 780
Rule 215
Rubi steps
\begin{align*} \int \sec (c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac{\sec (c+d x) \operatorname{Subst}\left (\int \frac{(a+x)^3}{\sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{b \sec (c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac{(b \sec (c+d x)) \operatorname{Subst}\left (\int \frac{(a+x) \left (-2+\frac{3 a^2}{b^2}+\frac{5 a x}{b^2}\right )}{\sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{3 d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{b \sec (c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac{b \sec (c+d x) \left (4 \left (4 a^2-b^2\right )+5 a b \tan (c+d x)\right )}{6 d}-\frac{\left (a \left (3-\frac{2 a^2}{b^2}\right ) b \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{2 d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{a \left (2 a^2-3 b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{2 d \sqrt{\sec ^2(c+d x)}}+\frac{b \sec (c+d x) (a+b \tan (c+d x))^2}{3 d}+\frac{b \sec (c+d x) \left (4 \left (4 a^2-b^2\right )+5 a b \tan (c+d x)\right )}{6 d}\\ \end{align*}
Mathematica [B] time = 1.55495, size = 293, normalized size = 3.22 \[ \frac{-6 a \left (2 a^2-3 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+2 b \sin ^2\left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (\left (18 a^2-5 b^2\right ) \cos (2 (c+d x))+18 a^2+2 b^2 \cos (c+d x)-b^2\right )+36 a^2 b+12 a^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{9 a b^2}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}-\frac{9 a b^2}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-18 a b^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{b^3}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{b^3}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}-10 b^3}{12 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 187, normalized size = 2.1 \begin{align*}{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\,d\cos \left ( dx+c \right ) }}-{\frac{{b}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3\,d}}-{\frac{2\,{b}^{3}\cos \left ( dx+c \right ) }{3\,d}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a{b}^{2}\sin \left ( dx+c \right ) }{2\,d}}-{\frac{3\,a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{b{a}^{2}}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12219, size = 150, normalized size = 1.65 \begin{align*} -\frac{9 \, a b^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} + \log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - \frac{36 \, a^{2} b}{\cos \left (d x + c\right )} + \frac{4 \,{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} b^{3}}{\cos \left (d x + c\right )^{3}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94229, size = 304, normalized size = 3.34 \begin{align*} \frac{3 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 18 \, a b^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 4 \, b^{3} + 12 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}}{12 \, d \cos \left (d x + c\right )^{3}} \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 )^{3} \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.83717, size = 231, normalized size = 2.54 \begin{align*} \frac{3 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{2 \,{\left (9 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 18 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 36 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 9 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 18 \, a^{2} b + 4 \, b^{3}\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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