Optimal. Leaf size=261 \[ -\frac {8 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{35 b^4 d}+\frac {8 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.42, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2863, 2865, 2752, 2663, 2661, 2655, 2653} \[ -\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-24 a b \sin (c+d x)-5 b^2\right )}{35 b^4 d}+\frac {8 \left (-37 a^2 b^2+32 a^4+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {8 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2863
Rule 2865
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {12 \int \frac {\cos ^2(c+d x) \left (-\frac {b}{2}-4 a \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{7 b^2}\\ &=\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac {16 \int \frac {\frac {1}{4} b \left (8 a^2-5 b^2\right )+\frac {1}{4} a \left (32 a^2-29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{35 b^4}\\ &=\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac {\left (4 a \left (32 a^2-29 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{35 b^5}+\frac {\left (4 \left (32 a^4-37 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{35 b^5}\\ &=\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}-\frac {\left (4 a \left (32 a^2-29 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{35 b^5 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{35 b^5 \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {8 a \left (32 a^2-29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{35 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (32 a^4-37 a^2 b^2+5 b^4\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{35 b^5 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{7 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^2-5 b^2-24 a b \sin (c+d x)\right )}{35 b^4 d}\\ \end {align*}
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Mathematica [A] time = 4.16, size = 222, normalized size = 0.85 \[ \frac {-16 \left (32 a^4-37 a^2 b^2+5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+b \cos (c+d x) \left (-256 a^3+\left (45 b^3-64 a^2 b\right ) \sin (c+d x)-16 a b^2 \cos (2 (c+d x))+216 a b^2+5 b^3 \sin (3 (c+d x))\right )+16 a \left (32 a^3+32 a^2 b-29 a b^2-29 b^3\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{70 b^5 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.65, size = 943, normalized size = 3.61 \[ -\frac {2 \left (-5 b^{5} \left (\sin ^{5}\left (d x +c \right )\right )+128 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b -96 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-148 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}+96 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+20 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}-128 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}+244 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}-116 \sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (d x +c \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (d x +c \right )\right ) b}{a -b}}\, \EllipticE \left (\sqrt {\frac {a +b \sin \left (d x +c \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+8 a \,b^{4} \left (\sin ^{4}\left (d x +c \right )\right )-16 a^{2} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )+20 b^{5} \left (\sin ^{3}\left (d x +c \right )\right )-64 a^{3} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )+42 a \,b^{4} \left (\sin ^{2}\left (d x +c \right )\right )+16 a^{2} b^{3} \sin \left (d x +c \right )-15 b^{5} \sin \left (d x +c \right )+64 a^{3} b^{2}-50 a \,b^{4}\right )}{35 b^{6} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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