Optimal. Leaf size=551 \[ -\frac{b \left (-176 a^2 b^2+720 a^4+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}+\frac{b \left (1696 a^2 b^2+816 a^4+5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1536 a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left (-176 a^2 b^2+720 a^4+15 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1536 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (720 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{512 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{\left (-56 a^2 b^2+16 a^4+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d} \]
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Rubi [A] time = 1.97142, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.355, Rules used = {2893, 3047, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{b \left (-176 a^2 b^2+720 a^4+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}+\frac{b \left (1696 a^2 b^2+816 a^4+5 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1536 a^2 d \sqrt{a+b \sin (c+d x)}}-\frac{b \left (-176 a^2 b^2+720 a^4+15 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{1536 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (720 a^4 b^2+60 a^2 b^4+64 a^6-5 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{512 a^3 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}-\frac{\left (-56 a^2 b^2+16 a^4+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d} \]
Antiderivative was successfully verified.
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Rule 2893
Rule 3047
Rule 3055
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac{\int \csc ^5(c+d x) (a+b \sin (c+d x))^{5/2} \left (\frac{5}{4} \left (28 a^2-3 b^2\right )+\frac{5}{2} a b \sin (c+d x)-\frac{5}{4} \left (24 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx}{30 a^2}\\ &=\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac{\int \csc ^4(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac{15}{8} b \left (52 a^2-5 b^2\right )-\frac{15}{4} a \left (4 a^2-b^2\right ) \sin (c+d x)-\frac{5}{8} b \left (164 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{120 a^2}\\ &=\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac{\int \csc ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{45}{16} \left (16 a^4-56 a^2 b^2+5 b^4\right )-\frac{15}{8} a b \left (84 a^2-b^2\right ) \sin (c+d x)-\frac{15}{16} b^2 \left (276 a^2-5 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{360 a^2}\\ &=-\frac{\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac{\int \frac{\csc ^2(c+d x) \left (-\frac{15}{32} b \left (720 a^4-176 a^2 b^2+15 b^4\right )-\frac{15}{16} a \left (48 a^4+720 a^2 b^2+b^4\right ) \sin (c+d x)-\frac{15}{32} b \left (48 a^4+936 a^2 b^2-5 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{720 a^2}\\ &=-\frac{b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}-\frac{\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac{\int \frac{\csc (c+d x) \left (-\frac{45}{64} \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right )-\frac{15}{32} a b \left (48 a^4+936 a^2 b^2-5 b^4\right ) \sin (c+d x)+\frac{15}{64} b^2 \left (720 a^4-176 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{720 a^3}\\ &=-\frac{b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}-\frac{\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}+\frac{\int \frac{\csc (c+d x) \left (\frac{45}{64} b \left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right )+\frac{15}{64} a b^2 \left (816 a^4+1696 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{720 a^3 b}-\frac{\left (b \left (720 a^4-176 a^2 b^2+15 b^4\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{3072 a^3}\\ &=-\frac{b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}-\frac{\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}+\frac{\left (b \left (816 a^4+1696 a^2 b^2+5 b^4\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{3072 a^2}+\frac{\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{1024 a^3}-\frac{\left (b \left (720 a^4-176 a^2 b^2+15 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}{3072 a^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=-\frac{b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}-\frac{\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac{b \left (720 a^4-176 a^2 b^2+15 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)}}{1536 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (b \left (816 a^4+1696 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}{3072 a^2 \sqrt{a+b \sin (c+d x)}}+\frac{\left (\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \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}{1024 a^3 \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{b \left (720 a^4-176 a^2 b^2+15 b^4\right ) \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{1536 a^3 d}-\frac{\left (16 a^4-56 a^2 b^2+5 b^4\right ) \cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{256 a^2 d}+\frac{b \left (52 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac{\left (28 a^2-3 b^2\right ) \cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac{b \cot (c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac{\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^{7/2}}{6 a d}-\frac{b \left (720 a^4-176 a^2 b^2+15 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)}}{1536 a^3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{b \left (816 a^4+1696 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}}}{1536 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{\left (64 a^6+720 a^4 b^2+60 a^2 b^4-5 b^6\right ) \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}}}{512 a^3 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.69572, size = 771, normalized size = 1.4 \[ \frac{\sqrt{a+b \sin (c+d x)} \left (\frac{1}{96} \csc ^4(c+d x) \left (28 a^2 \cos (c+d x)-27 b^2 \cos (c+d x)\right )+\frac{\csc ^3(c+d x) \left (164 a^2 b \cos (c+d x)-b^3 \cos (c+d x)\right )}{192 a}+\frac{\csc ^2(c+d x) \left (600 a^2 b^2 \cos (c+d x)-48 a^4 \cos (c+d x)+5 b^4 \cos (c+d x)\right )}{768 a^2}+\frac{\csc (c+d x) \left (176 a^2 b^3 \cos (c+d x)-720 a^4 b \cos (c+d x)-15 b^5 \cos (c+d x)\right )}{1536 a^3}-\frac{1}{6} a^2 \cot (c+d x) \csc ^5(c+d x)-\frac{5}{12} a b \cot (c+d x) \csc ^4(c+d x)\right )}{d}+\frac{-\frac{2 \left (3744 a^3 b^3+192 a^5 b-20 a b^5\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 \left (3600 a^4 b^2+536 a^2 b^4+384 a^6-45 b^6\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (-c-d x+\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{2 i \left (-176 a^2 b^4+720 a^4 b^2+15 b^6\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \sin (c+d x)}{a+b}} \sqrt{-\frac{b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt{-\frac{a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{6144 a^3 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 2.677, size = 2458, normalized size = 4.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [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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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(-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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