Optimal. Leaf size=195 \[ -\frac {105 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {105 \cos (c+d x)}{256 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a \sin (c+d x)+a}}-\frac {7 \sec (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.29, 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 = {2681, 2687, 2650, 2649, 206} \[ -\frac {105 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {105 \cos (c+d x)}{256 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^3(c+d x)}{4 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}+\frac {35 \sec (c+d x)}{64 a d \sqrt {a \sin (c+d x)+a}}-\frac {7 \sec (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2681
Rule 2687
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 \operatorname {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] time = 0.35, size = 334, normalized size = 1.71 \[ \frac {\frac {192 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {32 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-123 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2+246 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {136 \sin \left (\frac {1}{2} (c+d x)\right )}{\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}-\frac {32}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {64 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+(315+315 i) (-1)^{3/4} \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (c+d x)\right )-1\right )\right )-68}{768 d (a (\sin (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 270, normalized size = 1.38 \[ \frac {315 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 252 \, \cos \left (d x + c\right )^{2} - 12 \, {\left (35 \, \cos \left (d x + c\right )^{2} + 16\right )} \sin \left (d x + c\right ) - 64\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3072 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 9.62, size = 914, normalized size = 4.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 289, normalized size = 1.48 \[ \frac {\left (-840 a^{\frac {9}{2}}-315 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+\left (-384 a^{\frac {9}{2}}+1260 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sin \left (d x +c \right )+630 a^{\frac {9}{2}} \left (\cos ^{4}\left (d x +c \right )\right )+\left (-504 a^{\frac {9}{2}}-945 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-128 a^{\frac {9}{2}}+1260 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}}{1536 a^{\frac {11}{2}} \left (\sin \left (d x +c \right )-1\right ) \left (1+\sin \left (d x +c \right )\right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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