3.1.59 \(\int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx\) [59]

Optimal. Leaf size=356 \[ -\frac {2 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 b^{5/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {6 b^{5/2} \sqrt {6 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{5 d^{7/2}}+\frac {2 b^{5/2} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}} \]

[Out]

-4/5*b*cos(b*x+a)*sin(b*x+a)^2/d^2/(d*x+c)^(3/2)-2/5*sin(b*x+a)^3/d/(d*x+c)^(5/2)-2/5*b^(5/2)*cos(a-b*c/d)*Fre
snelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*2^(1/2)*Pi^(1/2)/d^(7/2)+2/5*b^(5/2)*FresnelS(b^(1/2)*2^
(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(a-b*c/d)*2^(1/2)*Pi^(1/2)/d^(7/2)+6/5*b^(5/2)*cos(3*a-3*b*c/d)*Fresn
elC(b^(1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*6^(1/2)*Pi^(1/2)/d^(7/2)-6/5*b^(5/2)*FresnelS(b^(1/2)*6^(1
/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)*6^(1/2)*Pi^(1/2)/d^(7/2)-16/5*b^2*sin(b*x+a)/d^3/(d*x+c)^
(1/2)+24/5*b^2*sin(b*x+a)^3/d^3/(d*x+c)^(1/2)

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Rubi [A]
time = 0.54, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3395, 3378, 3387, 3386, 3432, 3385, 3433, 3394} \begin {gather*} -\frac {2 \sqrt {2 \pi } b^{5/2} \cos \left (a-\frac {b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 \sqrt {6 \pi } b^{5/2} \cos \left (3 a-\frac {3 b c}{d}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {b} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {6 \sqrt {6 \pi } b^{5/2} \sin \left (3 a-\frac {3 b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {2 \sqrt {2 \pi } b^{5/2} \sin \left (a-\frac {b c}{d}\right ) S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \sin ^2(a+b x) \cos (a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^3/(c + d*x)^(7/2),x]

[Out]

(-2*b^(5/2)*Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) + (6
*b^(5/2)*Sqrt[6*Pi]*Cos[3*a - (3*b*c)/d]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(5*d^(7/2)) - (
6*b^(5/2)*Sqrt[6*Pi]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[3*a - (3*b*c)/d])/(5*d^(7/2)) +
(2*b^(5/2)*Sqrt[2*Pi]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(5*d^(7/2)) - (16
*b^2*Sin[a + b*x])/(5*d^3*Sqrt[c + d*x]) - (4*b*Cos[a + b*x]*Sin[a + b*x]^2)/(5*d^2*(c + d*x)^(3/2)) - (2*Sin[
a + b*x]^3)/(5*d*(c + d*x)^(5/2)) + (24*b^2*Sin[a + b*x]^3)/(5*d^3*Sqrt[c + d*x])

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 3395

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*Si
n[e + f*x])^n/(d*(m + 1))), x] + (Dist[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[f^2*(n^2/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1)*(m + 2))), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rubi steps

\begin {align*} \int \frac {\sin ^3(a+b x)}{(c+d x)^{7/2}} \, dx &=-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {\sin (a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2}-\frac {\left (12 b^2\right ) \int \frac {\sin ^3(a+b x)}{(c+d x)^{3/2}} \, dx}{5 d^2}\\ &=-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {\left (16 b^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (72 b^3\right ) \int \left (\frac {\cos (a+b x)}{4 \sqrt {c+d x}}-\frac {\cos (3 a+3 b x)}{4 \sqrt {c+d x}}\right ) \, dx}{5 d^3}\\ &=-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {\left (18 b^3\right ) \int \frac {\cos (a+b x)}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (18 b^3\right ) \int \frac {\cos (3 a+3 b x)}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (16 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (16 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}\\ &=-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {\left (18 b^3 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}+\frac {\left (32 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}-\frac {\left (18 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (18 b^3 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}-\frac {\left (32 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}+\frac {\left (18 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}} \, dx}{5 d^3}\\ &=\frac {16 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {16 b^{5/2} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}+\frac {\left (36 b^3 \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}-\frac {\left (36 b^3 \cos \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}-\frac {\left (36 b^3 \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}+\frac {\left (36 b^3 \sin \left (a-\frac {b c}{d}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {b x^2}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{5 d^4}\\ &=-\frac {2 b^{5/2} \sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}+\frac {6 b^{5/2} \sqrt {6 \pi } \cos \left (3 a-\frac {3 b c}{d}\right ) C\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{5 d^{7/2}}-\frac {6 b^{5/2} \sqrt {6 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{5 d^{7/2}}+\frac {2 b^{5/2} \sqrt {2 \pi } S\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{5 d^{7/2}}-\frac {16 b^2 \sin (a+b x)}{5 d^3 \sqrt {c+d x}}-\frac {4 b \cos (a+b x) \sin ^2(a+b x)}{5 d^2 (c+d x)^{3/2}}-\frac {2 \sin ^3(a+b x)}{5 d (c+d x)^{5/2}}+\frac {24 b^2 \sin ^3(a+b x)}{5 d^3 \sqrt {c+d x}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1429\) vs. \(2(356)=712\).
time = 6.22, size = 1429, normalized size = 4.01 \begin {gather*} \frac {3}{4} \left (\cos (a) \left (\frac {2 \left (\frac {b}{d}\right )^{5/2} \sin \left (\frac {b c}{d}\right ) \left (\frac {\cos \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}-\frac {2}{3} \left (2 \left (\frac {\cos \left (\frac {b (c+d x)}{d}\right )}{\sqrt {\frac {b}{d}} \sqrt {c+d x}}+\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )\right )+\frac {\sin \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}\right )\right )}{5 d}-\frac {2 \left (\frac {b}{d}\right )^{5/2} \cos \left (\frac {b c}{d}\right ) \left (\frac {\sin \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}+\frac {2}{3} \left (\frac {\cos \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}-2 \left (-\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )+\frac {\sin \left (\frac {b (c+d x)}{d}\right )}{\sqrt {\frac {b}{d}} \sqrt {c+d x}}\right )\right )\right )}{5 d}\right )+\sin (a) \left (-\frac {2 \left (\frac {b}{d}\right )^{5/2} \cos \left (\frac {b c}{d}\right ) \left (\frac {\cos \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}-\frac {2}{3} \left (2 \left (\frac {\cos \left (\frac {b (c+d x)}{d}\right )}{\sqrt {\frac {b}{d}} \sqrt {c+d x}}+\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )\right )+\frac {\sin \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}\right )\right )}{5 d}-\frac {2 \left (\frac {b}{d}\right )^{5/2} \sin \left (\frac {b c}{d}\right ) \left (\frac {\sin \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}+\frac {2}{3} \left (\frac {\cos \left (\frac {b (c+d x)}{d}\right )}{\left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}-2 \left (-\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}\right )+\frac {\sin \left (\frac {b (c+d x)}{d}\right )}{\sqrt {\frac {b}{d}} \sqrt {c+d x}}\right )\right )\right )}{5 d}\right )\right )+\frac {1}{4} \left (-\cos (3 a) \left (\frac {18 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} \sin \left (\frac {3 b c}{d}\right ) \left (\frac {\cos \left (\frac {3 b (c+d x)}{d}\right )}{9 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}-\frac {2}{3} \left (2 \left (\frac {\cos \left (\frac {3 b (c+d x)}{d}\right )}{\sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x}}+\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )\right )+\frac {\sin \left (\frac {3 b (c+d x)}{d}\right )}{3 \sqrt {3} \left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}\right )\right )}{5 d}-\frac {18 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} \cos \left (\frac {3 b c}{d}\right ) \left (\frac {\sin \left (\frac {3 b (c+d x)}{d}\right )}{9 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}+\frac {2}{3} \left (\frac {\cos \left (\frac {3 b (c+d x)}{d}\right )}{3 \sqrt {3} \left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}-2 \left (-\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+\frac {\sin \left (\frac {3 b (c+d x)}{d}\right )}{\sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x}}\right )\right )\right )}{5 d}\right )-\sin (3 a) \left (-\frac {18 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} \cos \left (\frac {3 b c}{d}\right ) \left (\frac {\cos \left (\frac {3 b (c+d x)}{d}\right )}{9 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}-\frac {2}{3} \left (2 \left (\frac {\cos \left (\frac {3 b (c+d x)}{d}\right )}{\sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x}}+\sqrt {2 \pi } S\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )\right )+\frac {\sin \left (\frac {3 b (c+d x)}{d}\right )}{3 \sqrt {3} \left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}\right )\right )}{5 d}-\frac {18 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} \sin \left (\frac {3 b c}{d}\right ) \left (\frac {\sin \left (\frac {3 b (c+d x)}{d}\right )}{9 \sqrt {3} \left (\frac {b}{d}\right )^{5/2} (c+d x)^{5/2}}+\frac {2}{3} \left (\frac {\cos \left (\frac {3 b (c+d x)}{d}\right )}{3 \sqrt {3} \left (\frac {b}{d}\right )^{3/2} (c+d x)^{3/2}}-2 \left (-\sqrt {2 \pi } C\left (\sqrt {\frac {b}{d}} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}\right )+\frac {\sin \left (\frac {3 b (c+d x)}{d}\right )}{\sqrt {3} \sqrt {\frac {b}{d}} \sqrt {c+d x}}\right )\right )\right )}{5 d}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^3/(c + d*x)^(7/2),x]

[Out]

(3*(Cos[a]*((2*(b/d)^(5/2)*Sin[(b*c)/d]*(Cos[(b*(c + d*x))/d]/((b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[(b*(c
+ d*x))/d]/(Sqrt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c +
d*x))/d]/((b/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (2*(b/d)^(5/2)*Cos[(b*c)/d]*(Sin[(b*(c + d*x))/d]/((b/d)
^(5/2)*(c + d*x)^(5/2)) + (2*(Cos[(b*(c + d*x))/d]/((b/d)^(3/2)*(c + d*x)^(3/2)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sq
rt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c + d*x))/d]/(Sqrt[b/d]*Sqrt[c + d*x]))))/3))/(5*d)) + Sin[a]*((-
2*(b/d)^(5/2)*Cos[(b*c)/d]*(Cos[(b*(c + d*x))/d]/((b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[(b*(c + d*x))/d]/(
Sqrt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c + d*x))/d]/((b
/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (2*(b/d)^(5/2)*Sin[(b*c)/d]*(Sin[(b*(c + d*x))/d]/((b/d)^(5/2)*(c + d
*x)^(5/2)) + (2*(Cos[(b*(c + d*x))/d]/((b/d)^(3/2)*(c + d*x)^(3/2)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[
2/Pi]*Sqrt[c + d*x]]) + Sin[(b*(c + d*x))/d]/(Sqrt[b/d]*Sqrt[c + d*x]))))/3))/(5*d))))/4 + (-(Cos[3*a]*((18*Sq
rt[3]*(b/d)^(5/2)*Sin[(3*b*c)/d]*(Cos[(3*b*(c + d*x))/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[
(3*b*(c + d*x))/d]/(Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]
) + Sin[(3*b*(c + d*x))/d]/(3*Sqrt[3]*(b/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (18*Sqrt[3]*(b/d)^(5/2)*Cos[(
3*b*c)/d]*(Sin[(3*b*(c + d*x))/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) + (2*(Cos[(3*b*(c + d*x))/d]/(3*Sqrt
[3]*(b/d)^(3/2)*(c + d*x)^(3/2)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) + Sin[(3*b*(c
+ d*x))/d]/(Sqrt[3]*Sqrt[b/d]*Sqrt[c + d*x]))))/3))/(5*d))) - Sin[3*a]*((-18*Sqrt[3]*(b/d)^(5/2)*Cos[(3*b*c)/
d]*(Cos[(3*b*(c + d*x))/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) - (2*(2*(Cos[(3*b*(c + d*x))/d]/(Sqrt[3]*Sq
rt[b/d]*Sqrt[c + d*x]) + Sqrt[2*Pi]*FresnelS[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) + Sin[(3*b*(c + d*x))/d]/(3*
Sqrt[3]*(b/d)^(3/2)*(c + d*x)^(3/2))))/3))/(5*d) - (18*Sqrt[3]*(b/d)^(5/2)*Sin[(3*b*c)/d]*(Sin[(3*b*(c + d*x))
/d]/(9*Sqrt[3]*(b/d)^(5/2)*(c + d*x)^(5/2)) + (2*(Cos[(3*b*(c + d*x))/d]/(3*Sqrt[3]*(b/d)^(3/2)*(c + d*x)^(3/2
)) - 2*(-(Sqrt[2*Pi]*FresnelC[Sqrt[b/d]*Sqrt[6/Pi]*Sqrt[c + d*x]]) + Sin[(3*b*(c + d*x))/d]/(Sqrt[3]*Sqrt[b/d]
*Sqrt[c + d*x]))))/3))/(5*d)))/4

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Maple [A]
time = 0.03, size = 450, normalized size = 1.26

method	result	size
derivativedivides	\(\frac {-\frac {3 \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}+\frac {3 b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}+\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}\right )}{5 d}}{d}\)	\(450\)
default	\(\frac {-\frac {3 \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}+\frac {3 b \left (-\frac {\cos \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {b \left (d x +c \right )}{d}+\frac {d a -c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {d a -c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {d a -c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}+\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{10 \left (d x +c \right )^{\frac {5}{2}}}-\frac {3 b \left (-\frac {\cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 b \left (-\frac {\sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 d a -3 c b}{d}\right )}{\sqrt {d x +c}}+\frac {b \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 d a -3 c b}{d}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {3 d a -3 c b}{d}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{d}\right )}{5 d}}{d}\)	\(450\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^3/(d*x+c)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/d*(-3/20/(d*x+c)^(5/2)*sin(1/d*b*(d*x+c)+(a*d-b*c)/d)+3/10*b/d*(-1/3/(d*x+c)^(3/2)*cos(1/d*b*(d*x+c)+(a*d-b*
c)/d)-2/3*b/d*(-1/(d*x+c)^(1/2)*sin(1/d*b*(d*x+c)+(a*d-b*c)/d)+b/d*2^(1/2)*Pi^(1/2)/(b/d)^(1/2)*(cos((a*d-b*c)
/d)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin((a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^
(1/2)*b*(d*x+c)^(1/2)/d))))+1/20/(d*x+c)^(5/2)*sin(3/d*b*(d*x+c)+3*(a*d-b*c)/d)-3/10*b/d*(-1/3/(d*x+c)^(3/2)*c
os(3/d*b*(d*x+c)+3*(a*d-b*c)/d)-2*b/d*(-1/(d*x+c)^(1/2)*sin(3/d*b*(d*x+c)+3*(a*d-b*c)/d)+b/d*2^(1/2)*Pi^(1/2)*
3^(1/2)/(b/d)^(1/2)*(cos(3*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)-sin(3
*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)))))

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Maxima [C] Result contains complex when optimal does not.
time = 0.68, size = 254, normalized size = 0.71 \begin {gather*} -\frac {3 \, {\left (3 \, \sqrt {3} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {3 i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {3 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}} - {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) - \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {5}{2}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {5}{2}}\right )}}{16 \, {\left (d x + c\right )}^{\frac {5}{2}} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="maxima")

[Out]

-3/16*(3*sqrt(3)*(((I - 1)*sqrt(2)*gamma(-5/2, 3*I*(d*x + c)*b/d) - (I + 1)*sqrt(2)*gamma(-5/2, -3*I*(d*x + c)
*b/d))*cos(-3*(b*c - a*d)/d) + ((I + 1)*sqrt(2)*gamma(-5/2, 3*I*(d*x + c)*b/d) - (I - 1)*sqrt(2)*gamma(-5/2, -
3*I*(d*x + c)*b/d))*sin(-3*(b*c - a*d)/d))*((d*x + c)*b/d)^(5/2) - (((I - 1)*sqrt(2)*gamma(-5/2, I*(d*x + c)*b
/d) - (I + 1)*sqrt(2)*gamma(-5/2, -I*(d*x + c)*b/d))*cos(-(b*c - a*d)/d) + ((I + 1)*sqrt(2)*gamma(-5/2, I*(d*x
+ c)*b/d) - (I - 1)*sqrt(2)*gamma(-5/2, -I*(d*x + c)*b/d))*sin(-(b*c - a*d)/d))*((d*x + c)*b/d)^(5/2))/((d*x
+ c)^(5/2)*d)

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Fricas [A]
time = 0.42, size = 549, normalized size = 1.54 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) + \sqrt {2} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 3 \, \sqrt {6} {\left (\pi b^{2} d^{3} x^{3} + 3 \, \pi b^{2} c d^{2} x^{2} + 3 \, \pi b^{2} c^{2} d x + \pi b^{2} c^{3}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + {\left (2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) + {\left (4 \, b^{2} d^{2} x^{2} + 8 \, b^{2} c d x + 4 \, b^{2} c^{2} - {\left (12 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 12 \, b^{2} c^{2} - d^{2}\right )} \cos \left (b x + a\right )^{2} - d^{2}\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}\right )}}{5 \, {\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="fricas")

[Out]

2/5*(3*sqrt(6)*(pi*b^2*d^3*x^3 + 3*pi*b^2*c*d^2*x^2 + 3*pi*b^2*c^2*d*x + pi*b^2*c^3)*sqrt(b/(pi*d))*cos(-3*(b*
c - a*d)/d)*fresnel_cos(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) - sqrt(2)*(pi*b^2*d^3*x^3 + 3*pi*b^2*c*d^2*x^2 +
3*pi*b^2*c^2*d*x + pi*b^2*c^3)*sqrt(b/(pi*d))*cos(-(b*c - a*d)/d)*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(p
i*d))) + sqrt(2)*(pi*b^2*d^3*x^3 + 3*pi*b^2*c*d^2*x^2 + 3*pi*b^2*c^2*d*x + pi*b^2*c^3)*sqrt(b/(pi*d))*fresnel_
sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-(b*c - a*d)/d) - 3*sqrt(6)*(pi*b^2*d^3*x^3 + 3*pi*b^2*c*d^2*x^2
+ 3*pi*b^2*c^2*d*x + pi*b^2*c^3)*sqrt(b/(pi*d))*fresnel_sin(sqrt(6)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c
- a*d)/d) + (2*(b*d^2*x + b*c*d)*cos(b*x + a)^3 - 2*(b*d^2*x + b*c*d)*cos(b*x + a) + (4*b^2*d^2*x^2 + 8*b^2*c
*d*x + 4*b^2*c^2 - (12*b^2*d^2*x^2 + 24*b^2*c*d*x + 12*b^2*c^2 - d^2)*cos(b*x + a)^2 - d^2)*sin(b*x + a))*sqrt
(d*x + c))/(d^6*x^3 + 3*c*d^5*x^2 + 3*c^2*d^4*x + c^3*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)**3/(d*x+c)**(7/2),x)

[Out]

Integral(sin(a + b*x)**3/(c + d*x)**(7/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^3/(d*x+c)^(7/2),x, algorithm="giac")

[Out]

integrate(sin(b*x + a)^3/(d*x + c)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^3/(c + d*x)^(7/2),x)

[Out]

int(sin(a + b*x)^3/(c + d*x)^(7/2), x)

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