3.2.0 ∫ (e+fx) sin3(c+dx) a+a sin(c+dx) dx [193]

Integrand size = 26, antiderivative size = 158 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 e x}{2 a}+\frac {3 f x^2}{4 a}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \sin (c+d x)}{a d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2} \]

Rubi [A] (veriﬁed)

Time = 0.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4611, 3391, 3377, 2717, 3399, 4269, 3556} \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {f \sin ^2(c+d x)}{4 a d^2}-\frac {f \sin (c+d x)}{a d^2}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )\right )}{a d^2}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {3 e x}{2 a}+\frac {3 f x^2}{4 a} \]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \sin ^2(c+d x) \, dx}{a}-\int \frac {(e+f x) \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx \\ & = -\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2}+\frac {\int (e+f x) \, dx}{2 a}-\frac {\int (e+f x) \sin (c+d x) \, dx}{a}+\int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx \\ & = \frac {e x}{2 a}+\frac {f x^2}{4 a}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2}+\frac {\int (e+f x) \, dx}{a}-\frac {f \int \cos (c+d x) \, dx}{a d}-\int \frac {e+f x}{a+a \sin (c+d x)} \, dx \\ & = \frac {3 e x}{2 a}+\frac {3 f x^2}{4 a}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {f \sin (c+d x)}{a d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2}-\frac {\int (e+f x) \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a} \\ & = \frac {3 e x}{2 a}+\frac {3 f x^2}{4 a}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {f \sin (c+d x)}{a d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2}-\frac {f \int \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d} \\ & = \frac {3 e x}{2 a}+\frac {3 f x^2}{4 a}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \sin (c+d x)}{a d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 6.79 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.89 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))+2 \left (-8 d e+6 c d e+4 c f-3 c^2 f+6 d^2 e x-4 d f x+3 d^2 f x^2-8 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 f \sin (c+d x)-d (e+f x) \sin (2 (c+d x))\right )\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))+2 \left (6 c d e+4 c f-3 c^2 f+6 d^2 e x+4 d f x+3 d^2 f x^2-8 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 f \sin (c+d x)-d (e+f x) \sin (2 (c+d x))\right )\right )\right )}{8 a d^2 (1+\sin (c+d x))} \]

Maple [C] (veriﬁed)

Time = 0.59 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.18


method	result	size

risch	\(\frac {3 f \,x^{2}}{4 a}+\frac {3 e x}{2 a}+\frac {\left (d x f +d e +i f \right ) {\mathrm e}^{i \left (d x +c \right )}}{2 a \,d^{2}}+\frac {\left (d x f +d e -i f \right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 a \,d^{2}}+\frac {2 i f x}{a d}+\frac {2 i f c}{a \,d^{2}}+\frac {2 f x +2 e}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {2 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{2}}-\frac {f \cos \left (2 d x +2 c \right )}{8 a \,d^{2}}-\frac {\left (f x +e \right ) \sin \left (2 d x +2 c \right )}{4 d a}\)	\(187\)
parallelrisch	\(\frac {16 f \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (12 d^{2} x^{2}+24 d x +26\right ) f +24 d^{2} e x +8 d e \right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (12 d^{2} x^{2}-24 d x +26\right ) f +24 d^{2} e x -40 d e \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (6 d x f +6 d e +7 f \right ) \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\left (2 d x f +2 d e -f \right ) \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\left (6 d x f +6 d e -7 f \right ) \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-2 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right ) \left (d x f +d e +\frac {1}{2} f \right )}{16 a \,d^{2} \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\)	\(266\)
default	\(-\frac {-\frac {4 e \left (\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{d}-\frac {2 f \,x^{2}+2 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 f \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 f \,x^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 f x}{d}-\frac {4 f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 f x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 f x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {8 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d^{2}}-\frac {4 f \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{2}}+\frac {4 e \left (\frac {-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{d}+\frac {4 f \left (\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {3 \left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}+\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )-c \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (d x +c \right ) c +\left (d x +c \right ) c \right )}{d^{2}}}{4 a}\)	\(412\)
norman	\(\frac {\frac {d e +2 f}{a \,d^{2}}-\frac {2 e \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {5 f \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-3 d e +2 f \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-6 d e +5 f \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-5 d e +3 f \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-d e +3 f \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {3 f \,x^{2}}{4 a}+\frac {3 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a}+\frac {9 f \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {9 f \,x^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {9 f \,x^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {9 f \,x^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {3 f \,x^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {3 f \,x^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {\left (3 d e +4 f \right ) x}{2 d a}+\frac {\left (3 d e -4 f \right ) x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (3 d e -2 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {3 \left (3 d e -2 f \right ) x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {3 \left (3 d e +2 f \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (3 d e +2 f \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (9 d e -4 f \right ) x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (9 d e +4 f \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {f \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}-\frac {2 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a \,d^{2}}\)	\(589\)

Fricas [A] (veriﬁcation not implemented)

Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.58 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {6 \, d^{2} f x^{2} + 2 \, {\left (2 \, d f x + 2 \, d e - f\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (4 \, d f x + 4 \, d e + 3 \, f\right )} \cos \left (d x + c\right )^{2} + 8 \, d e + 4 \, {\left (3 \, d^{2} e + 2 \, d f\right )} x + {\left (6 \, d^{2} f x^{2} + 12 \, d e + 12 \, {\left (d^{2} e + d f\right )} x + f\right )} \cos \left (d x + c\right ) - 8 \, {\left (f \cos \left (d x + c\right ) + f \sin \left (d x + c\right ) + f\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, d^{2} f x^{2} - 2 \, {\left (2 \, d f x + 2 \, d e + f\right )} \cos \left (d x + c\right )^{2} - 8 \, d e + 4 \, {\left (3 \, d^{2} e - 2 \, d f\right )} x + 4 \, {\left (d f x + d e - 2 \, f\right )} \cos \left (d x + c\right ) - 7 \, f\right )} \sin \left (d x + c\right ) - 7 \, f}{8 \, {\left (a d^{2} \cos \left (d x + c\right ) + a d^{2} \sin \left (d x + c\right ) + a d^{2}\right )}} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4653 vs. \(2 (134) = 268\).

Time = 2.15 (sec) , antiderivative size = 4653, normalized size of antiderivative = 29.45 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7564 vs. \(2 (138) = 276\).

Time = 0.98 (sec) , antiderivative size = 7564, normalized size of antiderivative = 47.87 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]

Mupad [B] (veriﬁcation not implemented)

Time = 1.90 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.56 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx={\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {d\,e+f\,1{}\mathrm {i}}{2\,a\,d^2}+\frac {f\,x}{2\,a\,d}\right )-{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {-d\,e+f\,1{}\mathrm {i}}{2\,a\,d^2}-\frac {f\,x}{2\,a\,d}\right )+{\mathrm {e}}^{-c\,2{}\mathrm {i}-d\,x\,2{}\mathrm {i}}\,\left (\frac {\left (-2\,d\,e+f\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a\,d^2}-\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\left (\frac {\left (2\,d\,e+f\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a\,d^2}+\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+\frac {3\,f\,x^2}{4\,a}-\frac {2\,f\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}{a\,d^2}+\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}+\frac {x\,\left (3\,d\,e+f\,4{}\mathrm {i}\right )}{2\,a\,d} \]

\(\int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [193]

Optimal result