Optimal. Leaf size=148 \[ -\frac {a^2 \cos (c+d x)}{d}+\frac {4 a b E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}-\frac {a c \sin (c+d x) \cos (c+d x)}{d}+a c x+b^2 x+\frac {4 b c F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d}-\frac {4 b c \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}+\frac {c^2 \cos ^3(c+d x)}{3 d}-\frac {c^2 \cos (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4395, 4401, 2639, 2638, 2635, 2641, 8, 2633} \[ -\frac {a^2 \cos (c+d x)}{d}+\frac {4 a b E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d}-\frac {a c \sin (c+d x) \cos (c+d x)}{d}+a c x+b^2 x+\frac {4 b c F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d}-\frac {4 b c \sqrt {\sin (c+d x)} \cos (c+d x)}{3 d}+\frac {c^2 \cos ^3(c+d x)}{3 d}-\frac {c^2 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2633
Rule 2635
Rule 2638
Rule 2639
Rule 2641
Rule 4395
Rule 4401
Rubi steps
\begin {align*} \int \sin (c+d x) \left (a+\frac {b}{\sqrt {\sin (c+d x)}}+c \sin (c+d x)\right )^2 \, dx &=\int \left (b+a \sqrt {\sin (c+d x)}+c \sin ^{\frac {3}{2}}(c+d x)\right )^2 \, dx\\ &=\int \left (b^2+2 a b \sqrt {\sin (c+d x)}+a^2 \sin (c+d x)+2 b c \sin ^{\frac {3}{2}}(c+d x)+2 a c \sin ^2(c+d x)+c^2 \sin ^3(c+d x)\right ) \, dx\\ &=b^2 x+a^2 \int \sin (c+d x) \, dx+(2 a b) \int \sqrt {\sin (c+d x)} \, dx+(2 a c) \int \sin ^2(c+d x) \, dx+(2 b c) \int \sin ^{\frac {3}{2}}(c+d x) \, dx+c^2 \int \sin ^3(c+d x) \, dx\\ &=b^2 x-\frac {a^2 \cos (c+d x)}{d}+\frac {4 a b E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d}-\frac {4 b c \cos (c+d x) \sqrt {\sin (c+d x)}}{3 d}-\frac {a c \cos (c+d x) \sin (c+d x)}{d}+(a c) \int 1 \, dx+\frac {1}{3} (2 b c) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx-\frac {c^2 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=b^2 x+a c x-\frac {a^2 \cos (c+d x)}{d}-\frac {c^2 \cos (c+d x)}{d}+\frac {c^2 \cos ^3(c+d x)}{3 d}+\frac {4 a b E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d}+\frac {4 b c F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d}-\frac {4 b c \cos (c+d x) \sqrt {\sin (c+d x)}}{3 d}-\frac {a c \cos (c+d x) \sin (c+d x)}{d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.28, size = 137, normalized size = 0.93 \[ \frac {-12 a^2 \cos (c+d x)-48 a b E\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )+12 a c^2+12 a c d x-6 a c \sin (2 (c+d x))+12 b^2 c+12 b^2 d x-16 b c F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )-16 b c \sqrt {\sin (c+d x)} \cos (c+d x)-9 c^2 \cos (c+d x)+c^2 \cos (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-2 \, a c \cos \left (d x + c\right )^{2} + b^{2} + 2 \, a c - {\left (c^{2} \cos \left (d x + c\right )^{2} - a^{2} - c^{2}\right )} \sin \left (d x + c\right ) + 2 \, {\left (b c \sin \left (d x + c\right ) + a b\right )} \sqrt {\sin \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.32, size = 266, normalized size = 1.80 \[ b^{2} x -\frac {a^{2} \cos \left (d x +c \right )}{d}-\frac {c^{2} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3 d}+\frac {2 a c \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 b \left (3 a \sqrt {1+\sin \left (d x +c \right )}\, \sqrt {-2 \sin \left (d x +c \right )+2}\, \sqrt {-\sin \left (d x +c \right )}\, \EllipticF \left (\sqrt {1+\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {1+\sin \left (d x +c \right )}\, \sqrt {-2 \sin \left (d x +c \right )+2}\, \sqrt {-\sin \left (d x +c \right )}\, \EllipticF \left (\sqrt {1+\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) c -6 a \sqrt {1+\sin \left (d x +c \right )}\, \sqrt {-2 \sin \left (d x +c \right )+2}\, \sqrt {-\sin \left (d x +c \right )}\, \EllipticE \left (\sqrt {1+\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) c \right )}{3 \cos \left (d x +c \right ) \sqrt {\sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [B] time = 6.68, size = 129, normalized size = 0.87 \[ b^2\,x-\frac {a^2\,\cos \left (c+d\,x\right )}{d}-\frac {a\,c\,\left (\sin \left (2\,c+2\,d\,x\right )-2\,d\,x\right )}{2\,d}+\frac {4\,a\,b\,\mathrm {E}\left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {c^2\,\cos \left (c+d\,x\right )\,\left ({\cos \left (c+d\,x\right )}^2-3\right )}{3\,d}-\frac {2\,b\,c\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\left ({\sin \left (c+d\,x\right )}^2\right )}^{5/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + \frac {b}{\sqrt {\sin {\left (c + d x \right )}}} + c \sin {\left (c + d x \right )}\right )^{2} \sin {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________