Optimal. Leaf size=110 \[ -\frac {4 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{5 b d^4 \sqrt {\sin (2 a+2 b x)}}-\frac {4 \cos (a+b x)}{5 b d^3 \sqrt {d \tan (a+b x)}}-\frac {2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2608, 2615, 2572, 2639} \[ -\frac {4 \cos (a+b x)}{5 b d^3 \sqrt {d \tan (a+b x)}}-\frac {4 \cos (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (a+b x)}}{5 b d^4 \sqrt {\sin (2 a+2 b x)}}-\frac {2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2572
Rule 2608
Rule 2615
Rule 2639
Rubi steps
\begin {align*} \int \frac {\sec ^3(a+b x)}{(d \tan (a+b x))^{7/2}} \, dx &=-\frac {2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}}+\frac {2 \int \frac {\sec (a+b x)}{(d \tan (a+b x))^{3/2}} \, dx}{5 d^2}\\ &=-\frac {2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}}-\frac {4 \cos (a+b x)}{5 b d^3 \sqrt {d \tan (a+b x)}}-\frac {4 \int \cos (a+b x) \sqrt {d \tan (a+b x)} \, dx}{5 d^4}\\ &=-\frac {2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}}-\frac {4 \cos (a+b x)}{5 b d^3 \sqrt {d \tan (a+b x)}}-\frac {\left (4 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{5 d^4 \sqrt {\sin (a+b x)}}\\ &=-\frac {2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}}-\frac {4 \cos (a+b x)}{5 b d^3 \sqrt {d \tan (a+b x)}}-\frac {\left (4 \cos (a+b x) \sqrt {d \tan (a+b x)}\right ) \int \sqrt {\sin (2 a+2 b x)} \, dx}{5 d^4 \sqrt {\sin (2 a+2 b x)}}\\ &=-\frac {2 \sec (a+b x)}{5 b d (d \tan (a+b x))^{5/2}}-\frac {4 \cos (a+b x)}{5 b d^3 \sqrt {d \tan (a+b x)}}-\frac {4 \cos (a+b x) E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {d \tan (a+b x)}}{5 b d^4 \sqrt {\sin (2 a+2 b x)}}\\ \end {align*}
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Mathematica [C] time = 1.54, size = 103, normalized size = 0.94 \[ -\frac {2 \sin (a+b x) \sqrt {d \tan (a+b x)} \left (4 \sec ^2(a+b x) \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(a+b x)\right )+3 \left (\csc ^4(a+b x)+\csc ^2(a+b x)-2\right ) \sqrt {\sec ^2(a+b x)}\right )}{15 b d^4 \sqrt {\sec ^2(a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \tan \left (b x + a\right )} \sec \left (b x + a\right )^{3}}{d^{4} \tan \left (b x + a\right )^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.62, size = 986, normalized size = 8.96 \[ \text {result too large to display} \]
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 \[ \int \frac {\sec \left (b x + a\right )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (a+b\,x\right )}^3\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\sec ^{3}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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