Optimal. Leaf size=119 \[ \frac{\sin ^2(c+d x) \left (\left (-6 a^2 b^2+a^4+b^4\right ) \cot (c+d x)+4 a b \left (a^2-b^2\right )\right )}{2 d}+\frac{1}{2} x \left (6 a^2 b^2+a^4-3 b^4\right )-\frac{4 a b^3 \log (\sin (c+d x))}{d}+\frac{4 a b^3 \log (\tan (c+d x))}{d}+\frac{b^4 \tan (c+d x)}{d} \]
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Rubi [A] time = 0.183296, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3088, 1805, 1802, 635, 203, 260} \[ \frac{\sin ^2(c+d x) \left (\left (-6 a^2 b^2+a^4+b^4\right ) \cot (c+d x)+4 a b \left (a^2-b^2\right )\right )}{2 d}+\frac{1}{2} x \left (6 a^2 b^2+a^4-3 b^4\right )-\frac{4 a b^3 \log (\sin (c+d x))}{d}+\frac{4 a b^3 \log (\tan (c+d x))}{d}+\frac{b^4 \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 1805
Rule 1802
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^4}{x^2 \left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{\left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{-2 b^4-8 a b^3 x-\left (a^4+6 a^2 b^2-b^4\right ) x^2}{x^2 \left (1+x^2\right )} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac{\left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac{\operatorname{Subst}\left (\int \left (-\frac{2 b^4}{x^2}-\frac{8 a b^3}{x}+\frac{-a^4-6 a^2 b^2+3 b^4+8 a b^3 x}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac{4 a b^3 \log (\tan (c+d x))}{d}+\frac{\left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac{b^4 \tan (c+d x)}{d}+\frac{\operatorname{Subst}\left (\int \frac{-a^4-6 a^2 b^2+3 b^4+8 a b^3 x}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac{4 a b^3 \log (\tan (c+d x))}{d}+\frac{\left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac{b^4 \tan (c+d x)}{d}+\frac{\left (4 a b^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (a^4+6 a^2 b^2-3 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac{1}{2} \left (a^4+6 a^2 b^2-3 b^4\right ) x-\frac{4 a b^3 \log (\sin (c+d x))}{d}+\frac{4 a b^3 \log (\tan (c+d x))}{d}+\frac{\left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{2 d}+\frac{b^4 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 6.26261, size = 477, normalized size = 4.01 \[ \frac{b^3 \left (\frac{\cos ^2(c+d x) (a+b \tan (c+d x))^5 \left (a b \tan (c+d x)+b^2\right )}{2 b^4 \left (a^2+b^2\right )}-\frac{\left (3 b^2-5 a^2\right ) \left (b \left (6 a^2-b^2\right ) \tan (c+d x)+\frac{1}{2} \left (\frac{-6 a^2 b^2+a^4+b^4}{\sqrt{-b^2}}+4 a (a-b) (a+b)\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+\frac{1}{2} \left (4 a (a-b) (a+b)-\frac{-6 a^2 b^2+a^4+b^4}{\sqrt{-b^2}}\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )+2 a b^2 \tan ^2(c+d x)+\frac{1}{3} b^3 \tan ^3(c+d x)\right )+4 a \left (\frac{1}{2} b^2 \left (10 a^2-b^2\right ) \tan ^2(c+d x)+5 a b \left (2 a^2-b^2\right ) \tan (c+d x)+\frac{1}{2} \left (-10 a^2 b^2+\frac{-10 a^3 b^2+a^5+5 a b^4}{\sqrt{-b^2}}+5 a^4+b^4\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+\frac{1}{2} \left (-10 a^2 b^2-\frac{-10 a^3 b^2+a^5+5 a b^4}{\sqrt{-b^2}}+5 a^4+b^4\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )+\frac{5}{3} a b^3 \tan ^3(c+d x)+\frac{1}{4} b^4 \tan ^4(c+d x)\right )}{2 b^2 \left (a^2+b^2\right )}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.133, size = 210, normalized size = 1.8 \begin{align*}{\frac{{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{4}x}{2}}+{\frac{{a}^{4}c}{2\,d}}-2\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{3}b}{d}}-3\,{\frac{{a}^{2}{b}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+3\,{a}^{2}{b}^{2}x+3\,{\frac{{a}^{2}{b}^{2}c}{d}}-2\,{\frac{a{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{a{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{d\cos \left ( dx+c \right ) }}+{\frac{{b}^{4}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{3\,{b}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}-{\frac{3\,{b}^{4}x}{2}}-{\frac{3\,{b}^{4}c}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78476, size = 182, normalized size = 1.53 \begin{align*} \frac{8 \, a^{3} b \sin \left (d x + c\right )^{2} +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 6 \,{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b^{2} - 8 \,{\left (\sin \left (d x + c\right )^{2} + \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} a b^{3} - 2 \,{\left (3 \, d x + 3 \, c - \frac{\tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2} + 1} - 2 \, \tan \left (d x + c\right )\right )} b^{4}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.52448, size = 312, normalized size = 2.62 \begin{align*} -\frac{8 \, a b^{3} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 4 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3} -{\left (2 \, a^{3} b - 2 \, a b^{3} +{\left (a^{4} + 6 \, a^{2} b^{2} - 3 \, b^{4}\right )} d x\right )} \cos \left (d x + c\right ) -{\left (2 \, b^{4} +{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.21104, size = 173, normalized size = 1.45 \begin{align*} \frac{4 \, a b^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, b^{4} \tan \left (d x + c\right ) +{\left (a^{4} + 6 \, a^{2} b^{2} - 3 \, b^{4}\right )}{\left (d x + c\right )} - \frac{4 \, a b^{3} \tan \left (d x + c\right )^{2} - a^{4} \tan \left (d x + c\right ) + 6 \, a^{2} b^{2} \tan \left (d x + c\right ) - b^{4} \tan \left (d x + c\right ) + 4 \, a^{3} b}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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