Optimal. Leaf size=184 \[ \frac {a^2 b \left (41 a^2+26 b^2\right ) \tan (d+e x) \sec (d+e x)}{24 e}+\frac {\left (4 a^2+7 b^2\right ) \tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )^2}{12 a e}+\frac {b \tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )^3}{4 a^2 e}+\frac {a \left (4 a^4+50 a^2 b^2+19 b^4\right ) \tan (d+e x)}{6 e}+\frac {b \left (19 a^4+56 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (d+e x))}{8 e}+a b^4 x \]
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Rubi [A] time = 0.43, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.180, Rules used = {4172, 3918, 4056, 4048, 3770, 3767, 8} \[ \frac {a \left (50 a^2 b^2+4 a^4+19 b^4\right ) \tan (d+e x)}{6 e}+\frac {b \left (56 a^2 b^2+19 a^4+8 b^4\right ) \tanh ^{-1}(\sin (d+e x))}{8 e}+\frac {a^2 b \left (41 a^2+26 b^2\right ) \tan (d+e x) \sec (d+e x)}{24 e}+\frac {\left (4 a^2+7 b^2\right ) \tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )^2}{12 a e}+\frac {b \tan (d+e x) \left (a^2 \sec (d+e x)+a b\right )^3}{4 a^2 e}+a b^4 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3918
Rule 4048
Rule 4056
Rule 4172
Rubi steps
\begin {align*} \int (a+b \sec (d+e x)) \left (b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right )^2 \, dx &=\frac {\int \left (2 a b+2 a^2 \sec (d+e x)\right )^4 (a+b \sec (d+e x)) \, dx}{16 a^4}\\ &=\frac {b \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{4 a^2 e}+\frac {\int \left (2 a b+2 a^2 \sec (d+e x)\right )^2 \left (16 a^3 b^2+4 a^2 b \left (11 a^2+4 b^2\right ) \sec (d+e x)+4 a^3 \left (4 a^2+7 b^2\right ) \sec ^2(d+e x)\right ) \, dx}{64 a^4}\\ &=\frac {\left (4 a^2+7 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{12 a e}+\frac {b \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{4 a^2 e}+\frac {\int \left (2 a b+2 a^2 \sec (d+e x)\right ) \left (96 a^4 b^3+8 a^3 \left (8 a^4+59 a^2 b^2+12 b^4\right ) \sec (d+e x)+8 a^4 b \left (41 a^2+26 b^2\right ) \sec ^2(d+e x)\right ) \, dx}{192 a^4}\\ &=\frac {a^2 b \left (41 a^2+26 b^2\right ) \sec (d+e x) \tan (d+e x)}{24 e}+\frac {\left (4 a^2+7 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{12 a e}+\frac {b \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{4 a^2 e}+\frac {\int \left (384 a^5 b^4+48 a^4 b \left (19 a^4+56 a^2 b^2+8 b^4\right ) \sec (d+e x)+64 a^5 \left (4 a^4+50 a^2 b^2+19 b^4\right ) \sec ^2(d+e x)\right ) \, dx}{384 a^4}\\ &=a b^4 x+\frac {a^2 b \left (41 a^2+26 b^2\right ) \sec (d+e x) \tan (d+e x)}{24 e}+\frac {\left (4 a^2+7 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{12 a e}+\frac {b \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{4 a^2 e}+\frac {1}{8} \left (b \left (19 a^4+56 a^2 b^2+8 b^4\right )\right ) \int \sec (d+e x) \, dx+\frac {1}{6} \left (a \left (4 a^4+50 a^2 b^2+19 b^4\right )\right ) \int \sec ^2(d+e x) \, dx\\ &=a b^4 x+\frac {b \left (19 a^4+56 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (d+e x))}{8 e}+\frac {a^2 b \left (41 a^2+26 b^2\right ) \sec (d+e x) \tan (d+e x)}{24 e}+\frac {\left (4 a^2+7 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{12 a e}+\frac {b \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{4 a^2 e}-\frac {\left (a \left (4 a^4+50 a^2 b^2+19 b^4\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (d+e x))}{6 e}\\ &=a b^4 x+\frac {b \left (19 a^4+56 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (d+e x))}{8 e}+\frac {a \left (4 a^4+50 a^2 b^2+19 b^4\right ) \tan (d+e x)}{6 e}+\frac {a^2 b \left (41 a^2+26 b^2\right ) \sec (d+e x) \tan (d+e x)}{24 e}+\frac {\left (4 a^2+7 b^2\right ) \left (a b+a^2 \sec (d+e x)\right )^2 \tan (d+e x)}{12 a e}+\frac {b \left (a b+a^2 \sec (d+e x)\right )^3 \tan (d+e x)}{4 a^2 e}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 130, normalized size = 0.71 \[ \frac {3 b \left (19 a^4+56 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (d+e x))+8 a^3 \left (a^2+4 b^2\right ) \tan ^3(d+e x)+3 a \tan (d+e x) \left (2 a^3 b \sec ^3(d+e x)+a b \left (19 a^2+24 b^2\right ) \sec (d+e x)+8 \left (a^4+10 a^2 b^2+4 b^4\right )\right )+24 a b^4 e x}{24 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.43, size = 198, normalized size = 1.08 \[ \frac {48 \, a b^{4} e x \cos \left (e x + d\right )^{4} + 3 \, {\left (19 \, a^{4} b + 56 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (e x + d\right )^{4} \log \left (\sin \left (e x + d\right ) + 1\right ) - 3 \, {\left (19 \, a^{4} b + 56 \, a^{2} b^{3} + 8 \, b^{5}\right )} \cos \left (e x + d\right )^{4} \log \left (-\sin \left (e x + d\right ) + 1\right ) + 2 \, {\left (6 \, a^{4} b + 16 \, {\left (a^{5} + 13 \, a^{3} b^{2} + 6 \, a b^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (19 \, a^{4} b + 24 \, a^{2} b^{3}\right )} \cos \left (e x + d\right )^{2} + 8 \, {\left (a^{5} + 4 \, a^{3} b^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{48 \, e \cos \left (e x + d\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 470, normalized size = 2.55 \[ \frac {1}{24} \, {\left (24 \, {\left (x e + d\right )} a b^{4} + 3 \, {\left (19 \, a^{4} b + 56 \, a^{2} b^{3} + 8 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1 \right |}\right ) - 3 \, {\left (19 \, a^{4} b + 56 \, a^{2} b^{3} + 8 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{7} - 63 \, a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{7} + 240 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{7} - 72 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{7} + 96 \, a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{7} - 40 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 39 \, a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 592 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 72 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 288 \, a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 40 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 39 \, a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 592 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 72 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 288 \, a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 24 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 63 \, a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 240 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 72 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 96 \, a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 1\right )}^{4}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 246, normalized size = 1.34 \[ a \,b^{4} x +\frac {a \,b^{4} d}{e}+\frac {7 a^{2} b^{3} \ln \left (\sec \left (e x +d \right )+\tan \left (e x +d \right )\right )}{e}+\frac {26 a^{3} b^{2} \tan \left (e x +d \right )}{3 e}+\frac {19 a^{4} b \sec \left (e x +d \right ) \tan \left (e x +d \right )}{8 e}+\frac {19 a^{4} b \ln \left (\sec \left (e x +d \right )+\tan \left (e x +d \right )\right )}{8 e}+\frac {2 a^{5} \tan \left (e x +d \right )}{3 e}+\frac {a^{5} \tan \left (e x +d \right ) \left (\sec ^{2}\left (e x +d \right )\right )}{3 e}+\frac {b^{5} \ln \left (\sec \left (e x +d \right )+\tan \left (e x +d \right )\right )}{e}+\frac {4 a \,b^{4} \tan \left (e x +d \right )}{e}+\frac {3 a^{2} b^{3} \sec \left (e x +d \right ) \tan \left (e x +d \right )}{e}+\frac {4 a^{3} b^{2} \tan \left (e x +d \right ) \left (\sec ^{2}\left (e x +d \right )\right )}{3 e}+\frac {a^{4} b \tan \left (e x +d \right ) \left (\sec ^{3}\left (e x +d \right )\right )}{4 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 299, normalized size = 1.62 \[ \frac {16 \, {\left (\tan \left (e x + d\right )^{3} + 3 \, \tan \left (e x + d\right )\right )} a^{5} + 64 \, {\left (\tan \left (e x + d\right )^{3} + 3 \, \tan \left (e x + d\right )\right )} a^{3} b^{2} + 48 \, {\left (e x + d\right )} a b^{4} - 3 \, a^{4} b {\left (\frac {2 \, {\left (3 \, \sin \left (e x + d\right )^{3} - 5 \, \sin \left (e x + d\right )\right )}}{\sin \left (e x + d\right )^{4} - 2 \, \sin \left (e x + d\right )^{2} + 1} - 3 \, \log \left (\sin \left (e x + d\right ) + 1\right ) + 3 \, \log \left (\sin \left (e x + d\right ) - 1\right )\right )} - 48 \, a^{4} b {\left (\frac {2 \, \sin \left (e x + d\right )}{\sin \left (e x + d\right )^{2} - 1} - \log \left (\sin \left (e x + d\right ) + 1\right ) + \log \left (\sin \left (e x + d\right ) - 1\right )\right )} - 72 \, a^{2} b^{3} {\left (\frac {2 \, \sin \left (e x + d\right )}{\sin \left (e x + d\right )^{2} - 1} - \log \left (\sin \left (e x + d\right ) + 1\right ) + \log \left (\sin \left (e x + d\right ) - 1\right )\right )} + 192 \, a^{2} b^{3} \log \left (\sec \left (e x + d\right ) + \tan \left (e x + d\right )\right ) + 48 \, b^{5} \log \left (\sec \left (e x + d\right ) + \tan \left (e x + d\right )\right ) + 288 \, a^{3} b^{2} \tan \left (e x + d\right ) + 192 \, a b^{4} \tan \left (e x + d\right )}{48 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.30, size = 323, normalized size = 1.76 \[ \frac {2\,a^5\,\sin \left (d+e\,x\right )}{3\,e\,\cos \left (d+e\,x\right )}+\frac {a^5\,\sin \left (d+e\,x\right )}{3\,e\,{\cos \left (d+e\,x\right )}^3}+\frac {2\,a\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )}{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}\right )}{e}+\frac {4\,a\,b^4\,\sin \left (d+e\,x\right )}{e\,\cos \left (d+e\,x\right )}+\frac {19\,a^4\,b\,\sin \left (d+e\,x\right )}{8\,e\,{\cos \left (d+e\,x\right )}^2}+\frac {a^4\,b\,\sin \left (d+e\,x\right )}{4\,e\,{\cos \left (d+e\,x\right )}^4}+\frac {26\,a^3\,b^2\,\sin \left (d+e\,x\right )}{3\,e\,\cos \left (d+e\,x\right )}+\frac {3\,a^2\,b^3\,\sin \left (d+e\,x\right )}{e\,{\cos \left (d+e\,x\right )}^2}+\frac {4\,a^3\,b^2\,\sin \left (d+e\,x\right )}{3\,e\,{\cos \left (d+e\,x\right )}^3}-\frac {b^5\,\mathrm {atan}\left (\frac {\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}\right )\,2{}\mathrm {i}}{e}-\frac {a^2\,b^3\,\mathrm {atan}\left (\frac {\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}\right )\,14{}\mathrm {i}}{e}-\frac {a^4\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {d}{2}+\frac {e\,x}{2}\right )\,1{}\mathrm {i}}{\cos \left (\frac {d}{2}+\frac {e\,x}{2}\right )}\right )\,19{}\mathrm {i}}{4\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec {\left (d + e x \right )}\right ) \left (a \sec {\left (d + e x \right )} + b\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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