Optimal. Leaf size=102 \[ \frac{4 \sin (a+b x) \sqrt{d \cos (a+b x)}}{3 b d^3}-\frac{8 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{2 \sin ^3(a+b x)}{3 b d (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.10562, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2566, 2568, 2642, 2641} \[ \frac{4 \sin (a+b x) \sqrt{d \cos (a+b x)}}{3 b d^3}-\frac{8 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{2 \sin ^3(a+b x)}{3 b d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2568
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{\sin ^4(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx &=\frac{2 \sin ^3(a+b x)}{3 b d (d \cos (a+b x))^{3/2}}-\frac{2 \int \frac{\sin ^2(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx}{d^2}\\ &=\frac{4 \sqrt{d \cos (a+b x)} \sin (a+b x)}{3 b d^3}+\frac{2 \sin ^3(a+b x)}{3 b d (d \cos (a+b x))^{3/2}}-\frac{4 \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx}{3 d^2}\\ &=\frac{4 \sqrt{d \cos (a+b x)} \sin (a+b x)}{3 b d^3}+\frac{2 \sin ^3(a+b x)}{3 b d (d \cos (a+b x))^{3/2}}-\frac{\left (4 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{3 d^2 \sqrt{d \cos (a+b x)}}\\ &=-\frac{8 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{4 \sqrt{d \cos (a+b x)} \sin (a+b x)}{3 b d^3}+\frac{2 \sin ^3(a+b x)}{3 b d (d \cos (a+b x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0688563, size = 60, normalized size = 0.59 \[ \frac{\sin ^5(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left (\frac{7}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right )}{5 b d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 286, normalized size = 2.8 \begin{align*} -{\frac{8}{3\,{d}^{2}b} \left ( -2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}\cos \left ( 1/2\,bx+a/2 \right ) +2\,\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+2\,\cos \left ( 1/2\,bx+a/2 \right ) \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}-\sqrt{ \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) ,\sqrt{2} \right ) - \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}\cos \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) \sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-1}{\frac{1}{\sqrt{-d \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2} \right ) }}} \left ( \sin \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )^{4}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sqrt{d \cos \left (b x + a\right )}}{d^{3} \cos \left (b x + a\right )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )^{4}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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