Optimal. Leaf size=241 \[ \frac {2 a^{5/2} c^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}} \]
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Rubi [A]
time = 0.14, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4025, 158, 152,
65, 212} \begin {gather*} \frac {2 a^{5/2} c^3 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \tan (e+f x) \left (d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)+2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )\right )}{105 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{7 f \sqrt {a \sec (e+f x)+a}}+\frac {2 a^2 (6 c+13 d) \tan (e+f x) (c+d \sec (e+f x))^2}{35 f \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 152
Rule 158
Rule 212
Rule 4025
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3 \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(a+a x) (c+d x)^3}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {(2 a \tan (e+f x)) \text {Subst}\left (\int \frac {(c+d x)^2 \left (-\frac {7 a^2 c}{2}-\frac {1}{2} a^2 (6 c+13 d) x\right )}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{7 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}-\frac {(4 \tan (e+f x)) \text {Subst}\left (\int \frac {(c+d x) \left (\frac {35 a^3 c^2}{4}+\frac {1}{4} a^3 \left (24 c^2+111 c d+52 d^2\right ) x\right )}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{35 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}}-\frac {\left (a^3 c^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}}+\frac {\left (2 a^2 c^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 a^{5/2} c^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (6 c+13 d) (c+d \sec (e+f x))^2 \tan (e+f x)}{35 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 (c+d \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt {a+a \sec (e+f x)}}+\frac {2 a^2 \left (2 \left (36 c^3+243 c^2 d+189 c d^2+52 d^3\right )+d \left (24 c^2+111 c d+52 d^2\right ) \sec (e+f x)\right ) \tan (e+f x)}{105 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 3.91, size = 219, normalized size = 0.91 \begin {gather*} \frac {a \sec \left (\frac {1}{2} (e+f x)\right ) \sec ^3(e+f x) \sqrt {a (1+\sec (e+f x))} \left (420 \sqrt {2} c^3 \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right ) \cos ^{\frac {7}{2}}(e+f x)+2 \left (210 c^2 d+378 c d^2+164 d^3+9 \left (35 c^3+175 c^2 d+154 c d^2+52 d^3\right ) \cos (e+f x)+2 d \left (105 c^2+189 c d+52 d^2\right ) \cos (2 (e+f x))+105 c^3 \cos (3 (e+f x))+525 c^2 d \cos (3 (e+f x))+378 c d^2 \cos (3 (e+f x))+104 d^3 \cos (3 (e+f x))\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{420 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(538\) vs.
\(2(216)=432\).
time = 3.30, size = 539, normalized size = 2.24
method | result | size |
default | \(\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (105 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \sqrt {2}\, c^{3}+315 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \sqrt {2}\, c^{3}+315 \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right ) \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \sqrt {2}\, c^{3}+105 \left (-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}\right )^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) c^{3} \sin \left (f x +e \right )-1680 \left (\cos ^{4}\left (f x +e \right )\right ) c^{3}-8400 \left (\cos ^{4}\left (f x +e \right )\right ) c^{2} d -6048 \left (\cos ^{4}\left (f x +e \right )\right ) c \,d^{2}-1664 \left (\cos ^{4}\left (f x +e \right )\right ) d^{3}+1680 \left (\cos ^{3}\left (f x +e \right )\right ) c^{3}+6720 \left (\cos ^{3}\left (f x +e \right )\right ) c^{2} d +3024 \left (\cos ^{3}\left (f x +e \right )\right ) c \,d^{2}+832 \left (\cos ^{3}\left (f x +e \right )\right ) d^{3}+1680 \left (\cos ^{2}\left (f x +e \right )\right ) c^{2} d +2016 \left (\cos ^{2}\left (f x +e \right )\right ) c \,d^{2}+208 \left (\cos ^{2}\left (f x +e \right )\right ) d^{3}+1008 \cos \left (f x +e \right ) c \,d^{2}+384 \cos \left (f x +e \right ) d^{3}+240 d^{3}\right ) a}{840 f \cos \left (f x +e \right )^{3} \sin \left (f x +e \right )}\) | \(539\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.41, size = 513, normalized size = 2.13 \begin {gather*} \left [\frac {105 \, {\left (a c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \cos \left (f x + e\right )^{3}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) + 2 \, {\left (15 \, a d^{3} + {\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (21 \, a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{105 \, {\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\right )}}, -\frac {2 \, {\left (105 \, {\left (a c^{3} \cos \left (f x + e\right )^{4} + a c^{3} \cos \left (f x + e\right )^{3}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - {\left (15 \, a d^{3} + {\left (105 \, a c^{3} + 525 \, a c^{2} d + 378 \, a c d^{2} + 104 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{2} d + 189 \, a c d^{2} + 52 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (21 \, a c d^{2} + 13 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )\right )}}{105 \, {\left (f \cos \left (f x + e\right )^{4} + f \cos \left (f x + e\right )^{3}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sec {\left (e + f x \right )}\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 525 vs.
\(2 (216) = 432\).
time = 1.87, size = 525, normalized size = 2.18 \begin {gather*} -\frac {\frac {105 \, \sqrt {-a} a^{2} c^{3} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{{\left | a \right |}} + \frac {2 \, {\left (105 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 630 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 210 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (315 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1680 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1260 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 280 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (315 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 1470 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 882 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 266 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - {\left (105 \, \sqrt {2} a^{5} c^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 420 \, \sqrt {2} a^{5} c^{2} d \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 252 \, \sqrt {2} a^{5} c d^{2} \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) + 76 \, \sqrt {2} a^{5} d^{3} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}}}{105 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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