Optimal. Leaf size=386 \[ -\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac {\left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 a d \sqrt {a+b \sin (c+d x)}}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.10, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2725, 3047, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}-\frac {\left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{8 a d \sqrt {a+b \sin (c+d x)}}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2725
Rule 2805
Rule 2807
Rule 3002
Rule 3047
Rule 3049
Rule 3059
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx &=\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}-\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (\frac {1}{4} \left (32 a^2+b^2\right )+\frac {3}{2} a b \sin (c+d x)-\frac {3}{4} \left (8 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2}\\ &=\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}-\frac {\int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (\frac {3}{8} b \left (36 a^2+b^2\right )-\frac {3}{4} a \left (8 a^2-b^2\right ) \sin (c+d x)-\frac {9}{8} b \left (16 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{6 a^2}\\ &=-\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}-\frac {\int \frac {\csc (c+d x) \left (\frac {9}{16} a b \left (36 a^2+b^2\right )-\frac {9}{8} a^2 \left (8 a^2-11 b^2\right ) \sin (c+d x)-\frac {9}{16} a b \left (32 a^2-b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{9 a^2}\\ &=-\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}+\frac {\int \frac {\csc (c+d x) \left (-\frac {9}{16} a b^2 \left (36 a^2+b^2\right )-\frac {9}{16} a^2 b \left (16 a^2+21 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{9 a^2 b}+\frac {\left (32 a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{16 a}\\ &=-\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}+\frac {1}{16} \left (-16 a^2-21 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (b \left (36 a^2+b^2\right )\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{16 a}+\frac {\left (\left (32 a^2-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}{16 a \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=-\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}+\frac {\left (32 a^2-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)}}{8 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (\left (-16 a^2-21 b^2\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}{16 \sqrt {a+b \sin (c+d x)}}-\frac {\left (b \left (36 a^2+b^2\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}{16 a \sqrt {a+b \sin (c+d x)}}\\ &=-\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}+\frac {\left (32 a^2-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)}}{8 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a^2+21 b^2\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}}}{8 d \sqrt {a+b \sin (c+d x)}}-\frac {b \left (36 a^2+b^2\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}}}{8 a d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.62, size = 600, normalized size = 1.55 \[ \frac {\sqrt {a+b \sin (c+d x)} \left (\frac {\csc (c+d x) \left (32 a^2 \cos (c+d x)-3 b^2 \cos (c+d x)\right )}{24 a}-\frac {1}{3} a \cot (c+d x) \csc ^2(c+d x)-\frac {2}{3} b \cos (c+d x)-\frac {7}{12} b \cot (c+d x) \csc (c+d x)\right )}{d}+\frac {-\frac {2 \left (32 a^3-44 a b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (-40 a^2 b-3 b^3\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (-c-d x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (b^3-32 a^2 b\right ) \cos (c+d x) \cos (2 (c+d x)) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b \sin (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right )}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2+b^2\right ) \sqrt {-\frac {a^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2-b^2}{b^2}}}}{32 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 146.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cot \left (d x + c\right )^{4} \sin \left (d x + c\right ) + a \cot \left (d x + c\right )^{4}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 2.16, size = 1511, normalized size = 3.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________