Optimal. Leaf size=324 \[ \frac {2 d^3 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^3 \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-d)^2 (c+2 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \]
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Rubi [A]
time = 0.16, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {4025, 186, 65,
212, 44} \begin {gather*} \frac {2 c^3 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right )}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {(c-d)^3 \tan (e+f x)}{2 a f (\sec (e+f x)+1) \sqrt {a \sec (e+f x)+a}}-\frac {(c-d)^3 \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} \sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {\sqrt {2} (c-d)^2 (c+2 d) \tan (e+f x) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {2 d^3 \tan (e+f x)}{a f \sqrt {a \sec (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 186
Rule 212
Rule 4025
Rubi steps
\begin {align*} \int \frac {(c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^{3/2}} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {(c+d x)^3}{x \sqrt {a-a x} (a+a x)^2} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=-\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \left (\frac {d^3}{a^2 \sqrt {a-a x}}+\frac {c^3}{a^2 x \sqrt {a-a x}}-\frac {(c-d)^3}{a^2 (1+x)^2 \sqrt {a-a x}}-\frac {(c-d)^2 (c+2 d)}{a^2 (1+x) \sqrt {a-a x}}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 d^3 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}-\frac {\left (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 {\left ((c-d)^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x)^2 \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 {\left ((c-d)^2 (c+2 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(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 d^3 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^3 \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {\left (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 )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left ((c-d)^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (2 (c-d)^2 (c+2 d) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 d^3 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^3 \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-d)^2 (c+2 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left ((c-d)^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{2 a f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {2 d^3 \tan (e+f x)}{a f \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^3 \tan (e+f x)}{2 a f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 c^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(c-d)^3 \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-d)^2 (c+2 d) \tanh ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 29.47, size = 21121, normalized size = 65.19 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(956\) vs.
\(2(280)=560\).
time = 1.72, size = 957, normalized size = 2.95
method | result | size |
default | \(\text {Expression too large to display}\) | \(957\) |
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 = 34.81, size = 744, normalized size = 2.30 \begin {gather*} \left [-\frac {\sqrt {2} {\left (5 \, c^{3} - 3 \, c^{2} d - 9 \, c d^{2} + 7 \, d^{3} + {\left (5 \, c^{3} - 3 \, c^{2} d - 9 \, c d^{2} + 7 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, c^{3} - 3 \, c^{2} d - 9 \, c d^{2} + 7 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {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 ) - 3 \, a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, {\left (c^{3} \cos \left (f x + e\right )^{2} + 2 \, c^{3} \cos \left (f x + e\right ) + c^{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 ) - 4 \, {\left (4 \, d^{3} - {\left (c^{3} - 3 \, c^{2} d + 3 \, c d^{2} - 5 \, 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 )}{8 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )}}, \frac {\sqrt {2} {\left (5 \, c^{3} - 3 \, c^{2} d - 9 \, c d^{2} + 7 \, d^{3} + {\left (5 \, c^{3} - 3 \, c^{2} d - 9 \, c d^{2} + 7 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, c^{3} - 3 \, c^{2} d - 9 \, c d^{2} + 7 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \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 ) - 8 \, {\left (c^{3} \cos \left (f x + e\right )^{2} + 2 \, c^{3} \cos \left (f x + e\right ) + c^{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 ) + 2 \, {\left (4 \, d^{3} - {\left (c^{3} - 3 \, c^{2} d + 3 \, c d^{2} - 5 \, 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 )}{4 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\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 \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{3}}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \frac {{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^3}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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