Optimal. Leaf size=107 \[ -\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {7 d \sin ^3(a+b x)}{30 b (d \tan (a+b x))^{3/2}}+\frac {7 \sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{20 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 107, 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 = {2598, 2601, 2572, 2639} \[ -\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}-\frac {7 d \sin ^3(a+b x)}{30 b (d \tan (a+b x))^{3/2}}+\frac {7 \sin (a+b x) E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{20 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2572
Rule 2598
Rule 2601
Rule 2639
Rubi steps
\begin {align*} \int \frac {\sin ^5(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx &=-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}+\frac {7}{10} \int \frac {\sin ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=-\frac {7 d \sin ^3(a+b x)}{30 b (d \tan (a+b x))^{3/2}}-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}+\frac {7}{20} \int \frac {\sin (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=-\frac {7 d \sin ^3(a+b x)}{30 b (d \tan (a+b x))^{3/2}}-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}+\frac {\left (7 \sqrt {\sin (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)} \, dx}{20 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {7 d \sin ^3(a+b x)}{30 b (d \tan (a+b x))^{3/2}}-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}+\frac {(7 \sin (a+b x)) \int \sqrt {\sin (2 a+2 b x)} \, dx}{20 \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {7 d \sin ^3(a+b x)}{30 b (d \tan (a+b x))^{3/2}}-\frac {d \sin ^5(a+b x)}{5 b (d \tan (a+b x))^{3/2}}+\frac {7 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sin (a+b x)}{20 b \sqrt {\sin (2 a+2 b x)} \sqrt {d \tan (a+b x)}}\\ \end {align*}
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Mathematica [C] time = 0.87, size = 86, normalized size = 0.80 \[ \frac {\sin (a+b x) \left (28 \tan (a+b x) \sqrt {\sec ^2(a+b x)} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(a+b x)\right )-20 \sin (2 (a+b x))+3 \sin (4 (a+b x))\right )}{120 b \sqrt {d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt {d \tan \left (b x + a\right )} \sin \left (b x + a\right )}{d \tan \left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {\sin \left (b x + a\right )^{5}}{\sqrt {d \tan \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 550, normalized size = 5.14 \[ -\frac {\left (-1+\cos \left (b x +a \right )\right )^{2} \left (12 \sqrt {2}\, \left (\cos ^{6}\left (b x +a \right )\right )-38 \left (\cos ^{4}\left (b x +a \right )\right ) \sqrt {2}-21 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \cos \left (b x +a \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \cos \left (b x +a \right ) \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+42 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticE \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+47 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-21 \cos \left (b x +a \right ) \sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \sqrt {2}}{120 b \cos \left (b x +a \right ) \sin \left (b x +a \right )^{4} \sqrt {\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}}} \]
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 \[ \int \frac {\sin \left (b x + a\right )^{5}}{\sqrt {d \tan \left (b x + a\right )}}\,{d x} \]
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 \[ \int \frac {{\sin \left (a+b\,x\right )}^5}{\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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