Optimal. Leaf size=159 \[ \frac{3 a \left (2 a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}+\frac{a \left (2 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{3 a \left (2 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.14451, antiderivative size = 177, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3512, 743, 780, 195, 215} \[ \frac{b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}+\frac{a \left (2 a^2-b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{3 a \left (2 a^2-b^2\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{3 a \left (2 a^2-b^2\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{16 d \sqrt{\sec ^2(c+d x)}}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 3512
Rule 743
Rule 780
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a+b \tan (c+d x))^3 \, dx &=\frac{\sec (c+d x) \operatorname{Subst}\left (\int (a+x)^3 \left (1+\frac{x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac{(b \sec (c+d x)) \operatorname{Subst}\left (\int (a+x) \left (-2+\frac{7 a^2}{b^2}+\frac{9 a x}{b^2}\right ) \left (1+\frac{x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{7 d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac{b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac{\left (\left (\frac{9 a}{b^2}-\frac{6 a \left (-2+\frac{7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \left (1+\frac{x^2}{b^2}\right )^{3/2} \, dx,x,b \tan (c+d x)\right )}{42 d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac{b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac{\left (\left (\frac{9 a}{b^2}-\frac{6 a \left (-2+\frac{7 a^2}{b^2}\right )}{b^2}\right ) b^3 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \sqrt{1+\frac{x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{56 d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{3 a \left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac{b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}-\frac{\left (\left (\frac{9 a}{b^2}-\frac{6 a \left (-2+\frac{7 a^2}{b^2}\right )}{b^2}\right ) b^3 \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 )}{112 d \sqrt{\sec ^2(c+d x)}}\\ &=\frac{3 a \left (2 a^2-b^2\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{16 d \sqrt{\sec ^2(c+d x)}}+\frac{3 a \left (2 a^2-b^2\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a \left (2 a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{b \sec ^5(c+d x) (a+b \tan (c+d x))^2}{7 d}+\frac{b \sec ^5(c+d x) \left (4 \left (8 a^2-b^2\right )+15 a b \tan (c+d x)\right )}{70 d}\\ \end{align*}
Mathematica [B] time = 2.14269, size = 637, normalized size = 4.01 \[ \frac{\sec ^7(c+d x) \left (3584 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-3675 a \left (2 a^2-b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+10752 a^2 b+4340 a^3 \sin (2 (c+d x))+2800 a^3 \sin (4 (c+d x))+420 a^3 \sin (6 (c+d x))-4410 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-1470 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-210 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4410 a^3 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1470 a^3 \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+210 a^3 \cos (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+6790 a b^2 \sin (2 (c+d x))-1400 a b^2 \sin (4 (c+d x))-210 a b^2 \sin (6 (c+d x))+2205 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+735 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-2205 a b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-735 a b^2 \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-105 a b^2 \cos (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1536 b^3\right )}{35840 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.074, size = 328, normalized size = 2.1 \begin{align*}{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{7\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{3\,{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d\cos \left ( dx+c \right ) }}-{\frac{{b}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{35\,d}}-{\frac{2\,{b}^{3}\cos \left ( dx+c \right ) }{35\,d}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a{b}^{2}\sin \left ( dx+c \right ) }{16\,d}}-{\frac{3\,a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{3\,b{a}^{2}}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17441, size = 281, normalized size = 1.77 \begin{align*} \frac{35 \, a b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 70 \, a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac{672 \, a^{2} b}{\cos \left (d x + c\right )^{5}} - \frac{32 \,{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} b^{3}}{\cos \left (d x + c\right )^{7}}}{1120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08916, size = 410, normalized size = 2.58 \begin{align*} \frac{105 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 160 \, b^{3} + 224 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 70 \,{\left (3 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 8 \, a b^{2} \cos \left (d x + c\right ) + 2 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1120 \, d \cos \left (d x + c\right )^{7}} \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 ^{5}{\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 = 1.89232, size = 628, normalized size = 3.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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