Optimal. Leaf size=156 \[ \frac{8 d^3 \sin (a+b x) \sqrt{d \cos (a+b x)}}{231 b}+\frac{8 d^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{231 b \sqrt{d \cos (a+b x)}}-\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{9/2}}{15 b d}-\frac{4 \sin (a+b x) (d \cos (a+b x))^{9/2}}{55 b d}+\frac{8 d \sin (a+b x) (d \cos (a+b x))^{5/2}}{385 b} \]
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Rubi [A] time = 0.147477, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2568, 2635, 2642, 2641} \[ \frac{8 d^3 \sin (a+b x) \sqrt{d \cos (a+b x)}}{231 b}+\frac{8 d^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{231 b \sqrt{d \cos (a+b x)}}-\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{9/2}}{15 b d}-\frac{4 \sin (a+b x) (d \cos (a+b x))^{9/2}}{55 b d}+\frac{8 d \sin (a+b x) (d \cos (a+b x))^{5/2}}{385 b} \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (d \cos (a+b x))^{7/2} \sin ^4(a+b x) \, dx &=-\frac{2 (d \cos (a+b x))^{9/2} \sin ^3(a+b x)}{15 b d}+\frac{2}{5} \int (d \cos (a+b x))^{7/2} \sin ^2(a+b x) \, dx\\ &=-\frac{4 (d \cos (a+b x))^{9/2} \sin (a+b x)}{55 b d}-\frac{2 (d \cos (a+b x))^{9/2} \sin ^3(a+b x)}{15 b d}+\frac{4}{55} \int (d \cos (a+b x))^{7/2} \, dx\\ &=\frac{8 d (d \cos (a+b x))^{5/2} \sin (a+b x)}{385 b}-\frac{4 (d \cos (a+b x))^{9/2} \sin (a+b x)}{55 b d}-\frac{2 (d \cos (a+b x))^{9/2} \sin ^3(a+b x)}{15 b d}+\frac{1}{77} \left (4 d^2\right ) \int (d \cos (a+b x))^{3/2} \, dx\\ &=\frac{8 d^3 \sqrt{d \cos (a+b x)} \sin (a+b x)}{231 b}+\frac{8 d (d \cos (a+b x))^{5/2} \sin (a+b x)}{385 b}-\frac{4 (d \cos (a+b x))^{9/2} \sin (a+b x)}{55 b d}-\frac{2 (d \cos (a+b x))^{9/2} \sin ^3(a+b x)}{15 b d}+\frac{1}{231} \left (4 d^4\right ) \int \frac{1}{\sqrt{d \cos (a+b x)}} \, dx\\ &=\frac{8 d^3 \sqrt{d \cos (a+b x)} \sin (a+b x)}{231 b}+\frac{8 d (d \cos (a+b x))^{5/2} \sin (a+b x)}{385 b}-\frac{4 (d \cos (a+b x))^{9/2} \sin (a+b x)}{55 b d}-\frac{2 (d \cos (a+b x))^{9/2} \sin ^3(a+b x)}{15 b d}+\frac{\left (4 d^4 \sqrt{\cos (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{231 \sqrt{d \cos (a+b x)}}\\ &=\frac{8 d^4 \sqrt{\cos (a+b x)} F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{231 b \sqrt{d \cos (a+b x)}}+\frac{8 d^3 \sqrt{d \cos (a+b x)} \sin (a+b x)}{231 b}+\frac{8 d (d \cos (a+b x))^{5/2} \sin (a+b x)}{385 b}-\frac{4 (d \cos (a+b x))^{9/2} \sin (a+b x)}{55 b d}-\frac{2 (d \cos (a+b x))^{9/2} \sin ^3(a+b x)}{15 b d}\\ \end{align*}
Mathematica [C] time = 0.0970417, size = 57, normalized size = 0.37 \[ \frac{\cos ^2(a+b x)^{3/4} \tan ^5(a+b x) (d \cos (a+b x))^{7/2} \, _2F_1\left (-\frac{5}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 262, normalized size = 1.7 \begin{align*} -{\frac{8\,{d}^{4}}{1155\,b}\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 ( 4928\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{17}-22176\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{15}+41216\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{13}-40768\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{11}+22868\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{9}-6994\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}+926\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}+5\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+5\,\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -5\,\cos \left ( 1/2\,bx+a/2 \right ) \right ){\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 \left (d \cos \left (b x + a\right )\right )^{\frac{7}{2}} \sin \left (b x + a\right )^{4}\,{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 ({\left (d^{3} \cos \left (b x + a\right )^{7} - 2 \, d^{3} \cos \left (b x + a\right )^{5} + d^{3} \cos \left (b x + a\right )^{3}\right )} \sqrt{d \cos \left (b x + a\right )}, 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 \left (d \cos \left (b x + a\right )\right )^{\frac{7}{2}} \sin \left (b x + a\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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