Optimal. Leaf size=156 \[ -\frac{(7 A-27 B) \tan (c+d x)}{15 a^3 d}+\frac{(A-3 B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(A-3 B) \tan (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(4 A-9 B) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.428503, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4019, 4008, 3787, 3770, 3767, 8} \[ -\frac{(7 A-27 B) \tan (c+d x)}{15 a^3 d}+\frac{(A-3 B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(A-3 B) \tan (c+d x)}{d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(4 A-9 B) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 4019
Rule 4008
Rule 3787
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx &=\frac{(A-B) \sec ^3(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{\int \frac{\sec ^3(c+d x) (3 a (A-B)-a (A-6 B) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=\frac{(A-B) \sec ^3(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(4 A-9 B) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{\int \frac{\sec ^2(c+d x) \left (2 a^2 (4 A-9 B)-a^2 (7 A-27 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=\frac{(A-B) \sec ^3(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(4 A-9 B) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(A-3 B) \tan (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\int \sec (c+d x) \left (-15 a^3 (A-3 B)+a^3 (7 A-27 B) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=\frac{(A-B) \sec ^3(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(4 A-9 B) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(A-3 B) \tan (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(7 A-27 B) \int \sec ^2(c+d x) \, dx}{15 a^3}+\frac{(A-3 B) \int \sec (c+d x) \, dx}{a^3}\\ &=\frac{(A-3 B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{(A-B) \sec ^3(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(4 A-9 B) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(A-3 B) \tan (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{(7 A-27 B) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 a^3 d}\\ &=\frac{(A-3 B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 A-27 B) \tan (c+d x)}{15 a^3 d}+\frac{(A-B) \sec ^3(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{(4 A-9 B) \sec ^2(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(A-3 B) \tan (c+d x)}{d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 3.97864, size = 480, normalized size = 3.08 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec (c) \cos \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (5 (32 A-51 B) \sin \left (\frac{d x}{2}\right )+(567 B-167 A) \sin \left (\frac{3 d x}{2}\right )+170 A \sin \left (c-\frac{d x}{2}\right )-170 A \sin \left (c+\frac{d x}{2}\right )+160 A \sin \left (2 c+\frac{d x}{2}\right )+75 A \sin \left (c+\frac{3 d x}{2}\right )-167 A \sin \left (2 c+\frac{3 d x}{2}\right )+75 A \sin \left (3 c+\frac{3 d x}{2}\right )-95 A \sin \left (c+\frac{5 d x}{2}\right )+15 A \sin \left (2 c+\frac{5 d x}{2}\right )-95 A \sin \left (3 c+\frac{5 d x}{2}\right )+15 A \sin \left (4 c+\frac{5 d x}{2}\right )-22 A \sin \left (2 c+\frac{7 d x}{2}\right )-22 A \sin \left (4 c+\frac{7 d x}{2}\right )-600 B \sin \left (c-\frac{d x}{2}\right )+375 B \sin \left (c+\frac{d x}{2}\right )-480 B \sin \left (2 c+\frac{d x}{2}\right )-60 B \sin \left (c+\frac{3 d x}{2}\right )+402 B \sin \left (2 c+\frac{3 d x}{2}\right )-225 B \sin \left (3 c+\frac{3 d x}{2}\right )+315 B \sin \left (c+\frac{5 d x}{2}\right )+30 B \sin \left (2 c+\frac{5 d x}{2}\right )+240 B \sin \left (3 c+\frac{5 d x}{2}\right )-45 B \sin \left (4 c+\frac{5 d x}{2}\right )+72 B \sin \left (2 c+\frac{7 d x}{2}\right )+15 B \sin \left (3 c+\frac{7 d x}{2}\right )+57 B \sin \left (4 c+\frac{7 d x}{2}\right )\right )-960 (A-3 B) \cos ^6\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{120 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 245, normalized size = 1.6 \begin{align*} -{\frac{A}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{B}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{B}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{7\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{17\,B}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{A}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-3\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) B}{d{a}^{3}}}-{\frac{B}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{A}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+3\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) B}{d{a}^{3}}}-{\frac{B}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02667, size = 386, normalized size = 2.47 \begin{align*} \frac{3 \, B{\left (\frac{40 \, \sin \left (d x + c\right )}{{\left (a^{3} - \frac{a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} + \frac{\frac{85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )} - A{\left (\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.498247, size = 668, normalized size = 4.28 \begin{align*} \frac{15 \,{\left ({\left (A - 3 \, B\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (A - 3 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (A - 3 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (A - 3 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left ({\left (A - 3 \, B\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (A - 3 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (A - 3 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (A - 3 \, B\right )} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (2 \,{\left (11 \, A - 36 \, B\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (17 \, A - 57 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (32 \, A - 117 \, B\right )} \cos \left (d x + c\right ) - 15 \, B\right )} \sin \left (d x + c\right )}{30 \,{\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}} \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*} \frac{\int \frac{A \sec ^{4}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{B \sec ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36327, size = 251, normalized size = 1.61 \begin{align*} \frac{\frac{60 \,{\left (A - 3 \, B\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{60 \,{\left (A - 3 \, B\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} - \frac{120 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} a^{3}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 20 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 30 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 255 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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