Optimal. Leaf size=426 \[ \frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}+\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {a \left (20 a^2+759 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b d}+\frac {5 a^2 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {a \left (20 a^4+739 a^2 b^2+816 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (20 a^4+1689 a^2 b^2-168 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d} \]
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Rubi [A] time = 1.42, antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2894, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}+\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {a \left (20 a^2+759 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b d}+\frac {a \left (739 a^2 b^2+20 a^4+816 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (1689 a^2 b^2+20 a^4-168 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{315 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {5 a^2 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2805
Rule 2807
Rule 2894
Rule 3002
Rule 3049
Rule 3059
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \cot ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}-\frac {2 \int \csc (c+d x) (a+b \sin (c+d x))^{5/2} \left (-\frac {45 b^2}{4}+\frac {11}{2} a b \sin (c+d x)+\frac {1}{4} \left (4 a^2+63 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{9 a b}\\ &=\frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}-\frac {4 \int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (-\frac {315 a b^2}{8}+\frac {87}{4} a^2 b \sin (c+d x)+\frac {1}{8} a \left (20 a^2+469 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{63 a b}\\ &=\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}-\frac {8 \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-\frac {1575}{16} a^2 b^2+\frac {3}{8} a b \left (155 a^2-28 b^2\right ) \sin (c+d x)+\frac {3}{16} a^2 \left (20 a^2+759 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{315 a b}\\ &=\frac {a \left (20 a^2+759 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b d}+\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}-\frac {16 \int \frac {\csc (c+d x) \left (-\frac {4725}{32} a^3 b^2+\frac {3}{16} a^2 b \left (475 a^2-492 b^2\right ) \sin (c+d x)+\frac {3}{32} a \left (20 a^4+1689 a^2 b^2-168 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{945 a b}\\ &=\frac {a \left (20 a^2+759 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b d}+\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}+\frac {16 \int \frac {\csc (c+d x) \left (\frac {4725 a^3 b^3}{32}+\frac {3}{32} a^2 \left (20 a^4+739 a^2 b^2+816 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{945 a b^2}-\frac {\left (20 a^4+1689 a^2 b^2-168 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{630 b^2}\\ &=\frac {a \left (20 a^2+759 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b d}+\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}+\frac {1}{2} \left (5 a^2 b\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx+\frac {\left (a \left (20 a^4+739 a^2 b^2+816 b^4\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{630 b^2}-\frac {\left (\left (20 a^4+1689 a^2 b^2-168 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{630 b^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {a \left (20 a^2+759 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b d}+\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}-\frac {\left (20 a^4+1689 a^2 b^2-168 b^4\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)}}{315 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (5 a^2 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{2 \sqrt {a+b \sin (c+d x)}}+\frac {\left (a \left (20 a^4+739 a^2 b^2+816 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}{630 b^2 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {a \left (20 a^2+759 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b d}+\frac {\left (20 a^2+469 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{315 b d}+\frac {\left (4 a^2+63 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{63 a b d}-\frac {2 \cos (c+d x) (a+b \sin (c+d x))^{7/2}}{9 b d}-\frac {\cot (c+d x) (a+b \sin (c+d x))^{7/2}}{a d}-\frac {\left (20 a^4+1689 a^2 b^2-168 b^4\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)}}{315 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {a \left (20 a^4+739 a^2 b^2+816 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}}}{315 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {5 a^2 b \Pi \left (2;\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}}}{d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 4.79, size = 496, normalized size = 1.16 \[ \frac {\frac {8 a b \left (475 a^2-492 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (20 a^4-1461 a^2 b^2-168 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (-20 a^4-1689 a^2 b^2+168 b^4\right ) \sec (c+d x) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {\frac {b (\sin (c+d x)+1)}{b-a}} \left (b \left (b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )}{a b^2 \sqrt {-\frac {1}{a+b}}}-\sqrt {a+b \sin (c+d x)} \left (\left (40 a^3-2202 a b^2\right ) \cos (c+d x)+2 b \left (\sin (2 (c+d x)) \left (150 a^2-35 b^2 \cos (2 (c+d x))-119 b^2\right )+630 a^2 \cot (c+d x)-95 a b \cos (3 (c+d x))\right )\right )}{1260 b d} \]
Antiderivative was successfully verified.
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fricas [F] time = 100.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, a b \cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (b^{2} \cos \left (d x + c\right )^{4} - {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \cot \left (d x + c\right )^{2}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 1.95, size = 865, normalized size = 2.03 \[ -\frac {70 b^{6} \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+\left (-340 a^{2} b^{4}+14 b^{6}\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (10 a^{4} b^{2}+57 a^{2} b^{4}-84 b^{6}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, \left (20 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{6}+1669 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b^{2}-1857 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{4}+168 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{6}-1575 \EllipticPi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{4}+1575 \EllipticPi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{5}-20 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5} b -930 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b^{2}-739 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{3}+2673 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{4}-816 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{5}-168 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b^{6}\right ) \sin \left (d x +c \right )+260 a \,b^{5} \left (\cos ^{6}\left (d x +c \right )\right )+\left (-160 a^{3} b^{3}+232 a \,b^{5}\right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (475 a^{3} b^{3}-492 a \,b^{5}\right ) \left (\cos ^{2}\left (d x +c \right )\right )}{315 \sin \left (d x +c \right ) b^{3} \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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^2\,{\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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