Optimal. Leaf size=195 \[ -\frac {105 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {105 \cos (c+d x)}{256 d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{32 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.20, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2760, 2766,
2729, 2728, 212} \begin {gather*} -\frac {105 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {105 \cos (c+d x)}{256 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a \sin (c+d x)+a}}-\frac {7 \sec (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rule 2760
Rule 2766
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {3 \int \frac {\sec ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{4 a}\\ &=-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a+a \sin (c+d x)}}+\frac {7}{8} \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {7 \sec (c+d x)}{32 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a+a \sin (c+d x)}}+\frac {35 \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{64 a}\\ &=-\frac {7 \sec (c+d x)}{32 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a+a \sin (c+d x)}}+\frac {105}{128} \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {105 \cos (c+d x)}{256 d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{32 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a+a \sin (c+d x)}}+\frac {105 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{512 a}\\ &=-\frac {105 \cos (c+d x)}{256 d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{32 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a+a \sin (c+d x)}}-\frac {105 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{256 a d}\\ &=-\frac {105 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {105 \cos (c+d x)}{256 d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{32 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^3(c+d x)}{6 d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 334, normalized size = 1.71 \begin {gather*} \frac {-68+\frac {64 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {32}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {136 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+246 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-123 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2+(315+315 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3+\frac {32 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {192 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}}{768 d (a (1+\sin (c+d x)))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.62, size = 289, normalized size = 1.48
method | result | size |
default | \(\frac {\left (-840 a^{\frac {9}{2}}-315 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+\left (-384 a^{\frac {9}{2}}+1260 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (d x +c \right )+630 a^{\frac {9}{2}} \left (\cos ^{4}\left (d x +c \right )\right )+\left (-504 a^{\frac {9}{2}}-945 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-128 a^{\frac {9}{2}}+1260 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}}{1536 a^{\frac {11}{2}} \left (\sin \left (d x +c \right )-1\right ) \left (1+\sin \left (d x +c \right )\right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(289\) |
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 = 0.40, size = 270, normalized size = 1.38 \begin {gather*} \frac {315 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 252 \, \cos \left (d x + c\right )^{2} - 12 \, {\left (35 \, \cos \left (d x + c\right )^{2} + 16\right )} \sin \left (d x + c\right ) - 64\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3072 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\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 {\sec ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d 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 [A]
time = 6.82, size = 220, normalized size = 1.13 \begin {gather*} \frac {\sqrt {a} {\left (\frac {315 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {315 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, {\left (315 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 840 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 693 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 144 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, \sqrt {2}\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{3072 \, d} \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 = -0.01 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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